// Numbas version: exam_results_page_options {"showstudentname": true, "question_groups": [{"name": "Group", "questions": [{"name": "Decimals: how to read", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"place": {"definition": "random([placedig*0.001, \"tenths\", \"hundredths\", \"thousandths\" ]\n,[placedig*0.01, \"tenths\", \"thousandths\", \"hundredths\"],\n [placedig*0.1, \"thousandths\", \"hundredths\", \"tenths\"])", "group": "Ungrouped variables", "templateType": "anything", "name": "place", "description": ""}, "pron": {"definition": "random(\n[dpformat(0.10, 2), \"zero point one zero\", \"zero point ten\"], \n[dpformat(0.11, 2), \"zero point one one\", \"zero point eleven\"], \n[dpformat(0.12, 2), \"zero point one two\", \"zero point twelve\"],\n[dpformat(0.13, 2), \"zero point one three\", \"zero point thirteen\"],\n[dpformat(0.14, 2), \"zero point one four\", \"zero point fourteen\"],\n[dpformat(0.15, 2), \"zero point one five\", \"zero point fifteen\"],\n[dpformat(0.16, 2), \"zero point one six\", \"zero point sixteen\"],\n[dpformat(0.17, 2), \"zero point one seven\", \"zero point seventeen\"],\n[dpformat(0.18, 2), \"zero point one eight\", \"zero point eighteen\"],\n[dpformat(0.19, 2), \"zero point one nine\", \"zero point ninteen\"], \n\n[dpformat(0.20, 2), \"zero point two zero\", \"zero point twenty\"], \n[dpformat(0.21, 2), \"zero point two one\", \"zero point twenty one\"], \n[dpformat(0.22, 2), \"zero point two two\", \"zero point twenty two\"],\n[dpformat(0.23, 2), \"zero point two three\", \"zero point twenty three\"],\n[dpformat(0.24, 2), \"zero point two four\", \"zero point twenty four\"], \n[dpformat(0.25, 2), \"zero point two five\", \"zero point twenty five\"],\n[dpformat(0.26, 2), \"zero point two six\", \"zero point twenty six\"],\n[dpformat(0.27, 2), \"zero point two seven\", \"zero point twenty seven\"],\n[dpformat(0.28, 2), \"zero point two eight\", \"zero point twenty eight\"],\n[dpformat(0.29, 2), \"zero point two nine\", \"zero point twenty nine\"], \n \n[dpformat(0.30, 2), \"zero point three zero\", \"zero point thirty\"], \n[dpformat(0.31, 2), \"zero point three one\", \"zero point thirty one\"], \n[dpformat(0.32, 2), \"zero point three two\", \"zero point thirty two\"],\n[dpformat(0.33, 2), \"zero point three three\", \"zero point thirty three\"],\n[dpformat(0.34, 2), \"zero point three four\", \"zero point thirty four\"], \n[dpformat(0.35, 2), \"zero point three five\", \"zero point thirty five\"],\n[dpformat(0.36, 2), \"zero point three six\", \"zero point thirty six\"],\n[dpformat(0.37, 2), \"zero point three seven\", \"zero point thirty seven\"],\n[dpformat(0.38, 2), \"zero point three eight\", \"zero point thirty eight\"],\n[dpformat(0.39, 2), \"zero point three nine\", \"zero point thirty nine\"], \n \n[dpformat(0.40, 2), \"zero point four zero\", \"zero point forty\"], \n[dpformat(0.41, 2), \"zero point four one\", \"zero point forty one\"], \n[dpformat(0.42, 2), \"zero point four two\", \"zero point forty two\"],\n[dpformat(0.43, 2), \"zero point four three\", \"zero point forty three\"],\n[dpformat(0.44, 2), \"zero point four four\", \"zero point forty four\"], \n[dpformat(0.45, 2), \"zero point four five\", \"zero point forty five\"],\n[dpformat(0.46, 2), \"zero point four six\", \"zero point forty six\"],\n[dpformat(0.47, 2), \"zero point four seven\", \"zero point forty seven\"],\n[dpformat(0.48, 2), \"zero point four eight\", \"zero point forty eight\"],\n[dpformat(0.49, 2), \"zero point four nine\", \"zero point forty nine\"], \n \n[dpformat(0.50, 2), \"zero point five zero\", \"zero point fifty\"], \n[dpformat(0.51, 2), \"zero point five one\", \"zero point fifty one\"], \n[dpformat(0.52, 2), \"zero point five two\", \"zero point fifty two\"],\n[dpformat(0.53, 2), \"zero point five three\", \"zero point fifty three\"],\n[dpformat(0.54, 2), \"zero point five four\", \"zero point fifty four\"], \n[dpformat(0.55, 2), \"zero point five five\", \"zero point fifty five\"],\n[dpformat(0.56, 2), \"zero point five six\", \"zero point fifty six\"],\n[dpformat(0.57, 2), \"zero point five seven\", \"zero point fifty seven\"],\n[dpformat(0.58, 2), \"zero point five eight\", \"zero point fifty eight\"],\n[dpformat(0.59, 2), \"zero point five nine\", \"zero point fifty nine\"], \n \n[0.60, \"zero point six zero\", \"zero point sixty\"], \n\n\n[0.70, \"zero point seven zero\", \"zero point seventy\"], \n \n\n[0.80, \"zero point eight zero\", \"zero point eighty\"], \n\n \n[0.90, \"zero point nine zero\", \"zero point ninety\"], \n\n \n[dpformat(0.100, 3), \"zero point one zero zero\", \"zero point one hundred\"], \n[dpformat(0.200, 3), \"zero point two zero zero\", \"zero point two hundred\"], \n[dpformat(0.300, 3), \"zero point three zero zero\", \"zero point three hundred\"],\n[dpformat(0.400, 3), \"zero point four zero zero\", \"zero point four hundred\"],\n[dpformat(0.500, 3), \"zero point five zero zero\", \"zero point five hundred\"], \n[dpformat(0.600, 3), \"zero point six zero zero\", \"zero point six hundred\"],\n[dpformat(0.700, 3), \"zero point seven zero zero\", \"zero point seven hundred\"],\n[dpformat(0.800, 3), \"zero point eight zero zero\", \"zero point eight hundred\"],\n[dpformat(0.900, 3), \"zero point nine zero zero\", \"zero point nine hundred\"]\n)", "group": "Ungrouped variables", "templateType": "anything", "name": "pron", "description": ""}, "placedig": {"definition": "random(1..9)", "group": "Ungrouped variables", "templateType": "anything", "name": "placedig", "description": ""}}, "statement": "

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "type": "question", "preamble": {"js": "", "css": ""}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "ungrouped_variables": ["pron", "placedig", "place"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "parts": [{"steps": [{"unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "prompt": "

Say each digit individually after the decimal point.

\n

\n
\n

\n

It makes no sense to call 0.500, \"zero point five hundred\" since that sounds a lot bigger than \"zero point five\", or \"zero point fifty\", but these are all equal to the same number! Pronouncing decimals like this is misleading and doesn't help with your intuition.

\n

\n
\n

\n

That is, $\\var{pron}$ is read as {pron}.

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{pron}

", "

{pron}

"], "maxMarks": "1", "distractors": ["", "Please see the steps"], "marks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "type": "1_n_2", "customMarkingAlgorithm": "", "showCellAnswerState": true, "minMarks": 0, "unitTests": [], "scripts": {}, "prompt": "

The decimal $\\var{pron}$ should be read as

", "showCorrectAnswer": true, "stepsPenalty": "1", "shuffleChoices": true}, {"steps": [{"unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "prompt": "

The decimal 0.1 is also known as \"one tenth\" (notice you need ten of them to make a whole).

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The decimal 0.01 is also known as \"one hundredth\" (notice you need a hundred of them to make a whole).

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The decimal 0.001 is also known as \"one thousandth\" (notice you need a thousand of them to make a whole).

\n

\n
\n

\n

That is, the digit $\\var{placedig}$ in the decimal $\\var{place}$ is in the {place} column and so represents $\\var{placedig}$ {place}.

", "variableReplacements": [], "type": "information", "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": ""}], "matrix": [0, 0, "1"], "displayType": "radiogroup", "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "choices": ["

$\\var{placedig}$ {place}

", "

$\\var{placedig}$ {place}

", "

$\\var{placedig}$ {place}

"], "maxMarks": "1", "distractors": ["", "", ""], "marks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "type": "1_n_2", "customMarkingAlgorithm": "", "showCellAnswerState": true, "minMarks": 0, "unitTests": [], "scripts": {}, "prompt": "

The digit $\\var{placedig}$ in the decimal $\\var{place}$ represents

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first non-zero digit

"}, "trailshort": {"definition": "random(0.1..0.9#0.1)", "group": "Ungrouped variables", "templateType": "anything", "name": "trailshort", "description": ""}}, "statement": "

Use the drop-down menu to create the correct sentence.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "type": "question", "preamble": {"js": "", "css": ""}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "ungrouped_variables": ["trail", "trailshort", "fnzdigsmall", "fnzdigbig", "fnz"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Some students believe a decimal is larger if it is longer, some believe a decimal is larger if its first non-zero digit is larger.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "prompt": "

The trailing zeros do not change the value of a decimal. In the same way that $42$ is no different to $000042$ (regardless of how many zeros are placed at the front), $\\var{trailshort}$ is no different to $\\var{trailshort}0000$ (regardless of how many zeros are placed at the back). This is why it is important to read things such as $0.200$ as \"zero point two zero zero\" and not as \"zero point two hundred\".

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In general, the length or number of digits in a decimal does not tell us anything about how big the decimal is. The only things that affect the actual value of a decimal are the non-zero digits and their placement relative to the decimal point (that is their face value and place value).

", "variableReplacements": [], "type": "information", "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "matrix": [0, 0, "1"], "displayType": "dropdownlist", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "shuffleChoices": false, "type": "1_n_2", "minMarks": 0, "choices": ["greater than", "less than", "equal to"], "maxMarks": 0, "showFeedbackIcon": true, "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "distractors": ["", "", ""], "showCellAnswerState": true, "displayColumns": 0}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

The number {trail} is [] {trail}

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "steps": [{"unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "prompt": "

You may have suspected that $\\var{fnz}$ was greater than $\\var{fnz}$ simply because $\\var{fnzdigbig}$ was greater than $\\var{fnzdigsmall}$, however, $\\var{fnzdigbig}$ is in a column with a smaller place value!

\n

You may have suspected that $\\var{fnz}$ was less than $\\var{fnz}$ simply because $\\var{fnzdigsmall}$ was less than $\\var{fnzdigbig}$, however, $\\var{fnzdigsmall}$ is in a column with a larger place value!

\n

In general, the first non-zero digit does not tell us anything about how big the decimal is. The only things that affect the actual value of a decimal are the non-zero digits and their placement relative to the decimal point (that is their face value and place value).

\n

You can add zeros so that the decimals have the same number of decimals places, and then, comparing them might be easier. That is, by appending a zero (which doesn't affect the value) onto the end of $\\var{fnzdigsmall/10}$ it might be clearer that $\\var{fnzdigsmall/10}0$ is greater than $\\var{fnzdigbig/100}$. Note that $\\var{fnzdigsmall/10}0$ is $\\var{fnzdigsmall}0$ hundredths whereas $\\var{fnzdigbig/100}$ is $\\var{fnzdigbig}$ hundredths.

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The number $\\var{fnz}$ is [] $\\var{fnz}$

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "tags": []}, {"name": "Decimals: Addition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"threedigit2": {"definition": "cdigs/1000+cdigs/100+cdigs/10", "group": "Ungrouped variables", "templateType": "anything", "name": "threedigit2", "description": ""}, "chuncarry": {"definition": "floor(chunsum/10)", "group": "Ungrouped variables", "templateType": "anything", "name": "chuncarry", "description": ""}, "threedigit1": {"definition": "cdigs/1000+cdigs/100+cdigs/10", "group": "Ungrouped variables", "templateType": "anything", "name": "threedigit1", "description": ""}, "cunitcarry": {"definition": "floor(cunitsum/10)", "group": "Ungrouped variables", "templateType": "anything", "name": "cunitcarry", "description": ""}, "cunitsumlastdigit": {"definition": "mod(cunitsum,10)", "group": "Ungrouped variables", "templateType": "anything", "name": "cunitsumlastdigit", "description": ""}, "cans": {"definition": "threedigit1+threedigit2", "group": "Ungrouped variables", "templateType": "anything", "name": "cans", "description": ""}, "ctensumlastdigit": {"definition": "mod(ctensum,10)", "group": "Ungrouped variables", "templateType": "anything", "name": "ctensumlastdigit", "description": ""}, "cdigs": {"definition": "+shuffle(3..9)", "group": "Ungrouped variables", "templateType": "anything", "name": "cdigs", "description": ""}, "chunsumlastdigit": {"definition": "mod(chunsum,10)", "group": "Ungrouped variables", "templateType": "anything", "name": "chunsumlastdigit", "description": ""}, "cunitsum": {"definition": "cdigs+cdigs", "group": "Ungrouped variables", "templateType": "anything", "name": "cunitsum", "description": ""}, "chunsum": {"definition": "ctencarry+cdigs+cdigs", "group": "Ungrouped variables", "templateType": "anything", "name": "chunsum", "description": ""}, "ctencarry": {"definition": "floor(ctensum/10)", "group": "Ungrouped variables", "templateType": "anything", "name": "ctencarry", "description": ""}, "ctensum": {"definition": "cdigs+cdigs+cunitcarry", "group": "Ungrouped variables", "templateType": "anything", "name": "ctensum", "description": ""}}, "variable_groups": [], "functions": {}, "type": "question", "preamble": {"js": "", "css": ""}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "scripts": {}, "prompt": "

Generally we set up $\\var{threedigit1}+\\var{threedigit2}$ with the decimal points lined up vertically so that the columns with the same place value are also lined up vertically:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $0$ . $\\var{cdigs}$ $\\var{cdigs}$ $+$ $0$ . $\\var{cdigs}$ $\\var{cdigs}$ $\\var{cdigs}$ $\\phantom{0}$
\n

\n

Note that we can pad out the decimal with zeros if we prefer:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $0$ . $\\var{cdigs}$ $\\var{cdigs}$ $\\color{red}{\\var{cdigs}}$ $+$ $0$ . $\\var{cdigs}$ $\\var{cdigs}$ $\\var{cdigs}$ $\\phantom{0}$
\n

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Now we add the digits in the column to the far right (in this case, the thousandths column).

\n

This results in $\\var{cunitsum}$ and so we place $\\var{cunitsumlastdigit}$ under the line in this column.

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This results in $\\var{cunitsum}$ and so we place $\\var{cunitsumlastdigit}$ under the line in this column and carry the $1$ into the next column to the left.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{0}$ . $\\overset{\\phantom{1}}{\\var{cdigs}}$ $\\overset{\\color{red}1}{\\var{cdigs}}$ $\\overset{\\phantom{0}}{\\var{cdigs}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{cdigs}}}$ $+$ $0$ . $\\var{cdigs}$ $\\var{cdigs}$ $\\color{green}{\\var{cdigs}}$ $\\color{red}{\\var{cunitSumLastDigit}}$
\n

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Now we add the digits in the next column to the left (in this case, the hundredths column).

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This results in $\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in this column.

\n

This results in $\\var{ctenSum}$ and so we place $\\var{ctenSumlastdigit}$ under the line in this column and carry the $1$ into the next column to the left.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{0}$ . $\\overset{\\color{red}{1}}{\\var{cdigs}}$ $\\overset{\\phantom{1}}{\\var{cdigs}}$ $\\color{green}{\\overset{1}{\\var{cdigs}}}$ $\\color{green}{\\overset{\\phantom{0}}{\\var{cdigs}}}$ $\\overset{\\phantom{1}}{\\var{cdigs}}$ $+$ $0$ . $\\var{cdigs}$ $\\color{green}{\\var{cdigs}}$ $\\var{cdigs}$ $\\color{red}{\\var{ctenSumlastdigit}}$ ${\\var{cunitSumLastDigit}}$
\n

\n

\n

Now we add the digits in the next column to the left (in this case, the tenths column).

\n

This is $\\var{chunsum}$ so we place $\\var{chunsum}$ under the line in this column.

\n

This is $\\var{chunsum}$ so we place $\\var{chunsumlastdigit}$ under the line in this column and carry $\\var{chuncarry}$ into the next column to the left (which in this case is the ones column on the other side of the decimal point).

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\color{red}1}{0}$ $\\overset{\\phantom{1}}{0}$ . $\\color{green}{\\overset{1}{\\var{cdigs}}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{cdigs}}}$ $\\overset{1}{\\var{cdigs}}$ $\\overset{\\phantom{0}}{\\var{cdigs}}$ $\\overset{\\phantom{1}}{\\var{cdigs}}$ $+$ $0$ . $\\color{green}{\\var{cdigs}}$ $\\var{cdigs}$ $\\var{cdigs}$ . $\\color{red}{\\var{chunsumlastdigit}}$ $\\var{ctenSumlastdigit}$ ${\\var{cunitSumLastDigit}}$
\n

\n

Now we add the digits in the next column to the left (in this case, the ones column).

\n

This is just $0$ so we place $0$ under the line in this column.

\n

This is just $\\var{chuncarry}$ so we place $\\var{chuncarry}$ under the line in this column.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\color{green}{\\overset{1}{0}}$ $\\color{green}{\\overset{\\phantom{1}}{0}}$ . $\\overset{1}{\\var{cdigs}}$ $\\overset{\\phantom{1}}{\\var{cdigs}}$ $\\overset{1}{\\var{cdigs}}$ $\\overset{\\phantom{0}}{\\var{cdigs}}$ $\\overset{\\phantom{1}}{\\var{cdigs}}$ $+$ $\\color{green}{0}$ . $\\var{cdigs}$ $\\var{cdigs}$ $\\var{cdigs}$ $\\color{red}{\\var{chuncarry}}$ . $\\var{chunsumlastdigit}$ $\\var{ctenSumlastdigit}$ ${\\var{cunitSumLastDigit}}$
\n

The answer is therefore $\\var{cans}$.

", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "mustBeReduced": false, "unitTests": [], "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "maxValue": "cans", "marks": 1, "minValue": "cans"}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{threedigit1}+\\var{threedigit2} =$ []

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "ungrouped_variables": ["cdigs", "threedigit1", "threedigit2", "cans", "cunitsum", "cunitsumlastdigit", "cunitcarry", "ctensum", "ctensumlastdigit", "ctencarry", "chunsum", "chunsumlastdigit", "chuncarry"], "statement": "

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

Decimals addition algorithm. 2 and 3 digit numbers. Carrying.

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "tags": []}, {"name": "Decimals: Subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"anshun": {"definition": "mod(floor(ans*10),10)", "group": "c", "templateType": "anything", "name": "anshun", "description": ""}, "tendiff": {"definition": "if(unitdiff>=0,top-bot,top-1-bot)", "group": "c", "templateType": "anything", "name": "tendiff", "description": ""}, "botnum": {"definition": "bot/1000+bot/100+bot/10", "group": "c", "templateType": "anything", "name": "botnum", "description": ""}, "top": {"definition": "random([random(0..9),random(1..8),random(2..9)],[0,random(1..9),random(2..9)])", "group": "c", "templateType": "anything", "name": "top", "description": ""}, "unitdiff": {"definition": "top-bot", "group": "c", "templateType": "anything", "name": "unitdiff", "description": ""}, "topnum": {"definition": "top/1000+top/100+top/10", "group": "c", "templateType": "anything", "name": "topnum", "description": ""}, "newtophun": {"definition": "if(tendiff>=0,top,top-1)", "group": "c", "templateType": "anything", "name": "newtophun", "description": ""}, "hundiff": {"definition": "if(tendiff>=0,top-bot,top-1-bot)", "group": "c", "templateType": "anything", "name": "hundiff", "description": ""}, "ansunit": {"definition": "mod(ans*1000,10)", "group": "c", "templateType": "anything", "name": "ansunit", "description": ""}, "ans": {"definition": "topnum-botnum", "group": "c", "templateType": "anything", "name": "ans", "description": ""}, "ansten": {"definition": "mod(floor(ans*100),10)", "group": "c", "templateType": "anything", "name": "ansten", "description": ""}, "bot": {"definition": "random(\nif(top<9,[random(top+1..9), random(top..9), random(1..top-1)],if(top<9,[random(top..9), random(top+1..9), random(1..top-1)]),\"error\"),\nif(top<9,[random(0..top), random(top+1..9), random(1..top-1)],if(top=9,[random(top..9), random(0..9), random(1..top-1)]),\"error\")\n)\n", "group": "c", "templateType": "anything", "name": "bot", "description": "

This should force some borrowing and paying back, and that the final answer is positive.

"}, "newtopten": {"definition": "if(unitdiff>=0,top,top-1)", "group": "c", "templateType": "anything", "name": "newtopten", "description": ""}}, "ungrouped_variables": [], "type": "question", "preamble": {"js": "", "css": ""}, "variable_groups": [{"variables": ["top", "bot", "topnum", "botnum", "ans", "unitdiff", "tendiff", "hundiff", "ansunit", "ansten", "anshun", "newtopten", "newtophun"], "name": "c"}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "statement": "

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Subtracting a decimal with 3 decimal places from a decimal with 2 or 3 decimal places. borrowing is necessary. This was modified from a subtraction question using integers with each number divided by 1000 so the variables have names referring to ones, tens, hundreds etc.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "

Generally we set up $\\var{topnum}-\\var{botnum}$ with the decimal points lined up vertically so that the columns with the same place value are also lined up vertically. We also pad out the decimals with zeros:

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $0$ . $\\var{top}$ $\\var{top}$ $\\color{red}{\\var{top}}$ $\\var{top}$ $-$ $0$ . $\\var{bot}$ $\\var{bot}$ $\\var{bot}$ . $\\phantom{0}$
\n

\n

Now we try to subtract the digits in the column to the far right (in this case, the thousandths column).

\n

Since this is $\\var{ansunit}$ we write $\\var{ansunit}$ under the line in this column.

\n

Since we can't take $\\var{bot}$ away from $\\var{top}$ (without using negative numbers) we borrow from the next column to the left (in this case, the hundredths column). This means we cross out the $\\var{top}$ in the hundredths column and replace it with a $\\var{top-1}$, and the $\\var{top}$ becomes a $\\var{10+top}$. Now we can do $\\var{10+top}-\\var{bot}$, and write the result, $\\var{ansunit}$, under the line in the thousandths column.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{0}$ . $\\overset{\\phantom{1}}{\\var{top}}$ $\\overset{\\color{red}{\\var{newtopten}}}{\\var{top}\\mkern-7.5mu\\color{red}/}$ $\\overset{\\phantom{1}}{\\var{top}}$ $\\color{red}{^1}\\overset{\\phantom{1}}{\\color{green}{\\var{top}}}$ $\\overset{\\phantom{1}}{\\color{green}{\\var{top}}}$ $-$ $0$ . $\\var{bot}$ $\\var{bot}$ $\\color{green}{\\var{bot}}$ . $\\color{red}{\\var{ansunit}}$
\n

\n

Now we try to subtract the digits in the hundredths column.

\n

Since this is $\\var{ansten}$ we write $\\var{ansten}$ under the line in this column.

\n

Since we can't take $\\var{bot}$ away from $\\var{newtopten}$ (without using negative numbers) we borrow from the next column to the left (in this case, the tenths column). This means we cross out the $\\var{top}$ in the tenths column and replace it with a $\\var{top-1}$, and the $\\var{newtopten}$ in the hundredths column becomes a $\\var{10+newtopten}$. Now we can do $\\var{10+newtopten}-\\var{bot}$, and write the result, $\\var{ansten}$, under the line in the hundredths.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{0}$ . \n$\\overset{\\color{red}{\\var{newtophun}}}{\\var{top}\\mkern-7.5mu\\color{red}{/}}$ $\\overset{\\phantom{1}}{\\var{top}}$\n \n$\\overset{\\color{red}{1}\\color{green}{\\var{newtopten}}}{\\var{top}\\mkern-7.5mu/}$ $\\overset{\\color{green}{\\var{newtopten}}}{\\var{top}\\mkern-7.5mu/}$ $\\color{red}{^1}\\overset{\\phantom{1}}{\\color{green}{\\var{top}}}$ $\\overset{\\phantom{1}}{\\color{green}{\\var{top}}}$\n ${^1}\\overset{\\phantom{1}}{\\var{top}}$ $\\overset{\\phantom{1}}{\\var{top}}$ $-$ $0$ . $\\var{bot}$ $\\color{green}{\\var{bot}}$ $\\var{bot}$ . $\\color{red}{\\var{ansten}}$ $\\var{ansunit}$
\n

\n

Now we try to subtract the digits in the tenths column and then subtract the digits in the ones column.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{\\color{green}{0}}$ . \n$\\overset{\\color{green}{\\var{newtophun}}}{\\var{top}\\mkern-7.5mu/}$ $\\overset{\\phantom{1}}{\\color{green}{\\var{top}}}$\n \n$\\overset{{1}\\var{newtopten}}{\\var{top}\\mkern-7.5mu/}$ $\\overset{\\var{newtopten}}{\\var{top}\\mkern-7.5mu/}$ ${^1}\\overset{\\phantom{1}}{\\var{top}}$ $\\overset{\\phantom{1}}{\\var{top}}$\n ${^1}\\overset{\\phantom{1}}{\\var{top}}$ $\\overset{\\phantom{1}}{\\var{top}}$ $-$ $\\color{green}0$ . $\\color{green}{\\var{bot}}$ $\\var{bot}$ $\\var{bot}$ $\\color{red}0$ . $\\color{red}{\\var{anshun}}$ $\\var{ansten}$ $\\var{ansunit}$
\n

\n

\n

The answer is therefore $\\var{ans}$.

", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "information", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "marks": 1, "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "mustBeReduced": false, "unitTests": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "ans", "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "minValue": "ans"}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{topnum}-\\var{botnum} =$ []

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "tags": []}, {"name": "Decimals: Multiplying and dividing by powers of ten", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"poweroften": {"definition": "map(10^n,n,power)", "group": "Ungrouped variables", "templateType": "anything", "name": "poweroften", "description": ""}, "ans3": {"definition": "dec3/poweroften", "group": "Ungrouped variables", "templateType": "anything", "name": "ans3", "description": "

ans3

"}, "pronpower": {"definition": "[switch(power=2,'two',power=3,'three',power=4,'four','one'),switch(power=2,'two',power=3,'three',power=4,'four','one'),switch(power=2,'two',power=3,'three',power=4,'four','one'),switch(power=2,'two',power=3,'three',power=4,'four','one')]", "group": "Ungrouped variables", "templateType": "anything", "name": "pronpower", "description": ""}, "ans2": {"definition": "dec2*poweroften", "group": "Ungrouped variables", "templateType": "anything", "name": "ans2", "description": ""}, "ans4": {"definition": "dec4/poweroften", "group": "Ungrouped variables", "templateType": "anything", "name": "ans4", "description": ""}, "dec2": {"definition": "random(list(0.011..0.099#0.0001))", "group": "Ungrouped variables", "templateType": "anything", "name": "dec2", "description": ""}, "dec1": {"definition": "random(list(0.111..0.999#0.001))*10", "group": "Ungrouped variables", "templateType": "anything", "name": "dec1", "description": ""}, "dec4": {"definition": "random(list(0.111..0.499#0.001))", "group": "Ungrouped variables", "templateType": "anything", "name": "dec4", "description": ""}, "dec3": {"definition": "random(list(0.500..0.999#0.001))*10", "group": "Ungrouped variables", "templateType": "anything", "name": "dec3", "description": ""}, "power": {"definition": "shuffle([1,2,3,4])", "group": "Ungrouped variables", "templateType": "anything", "name": "power", "description": ""}, "ans1": {"definition": "dec1*poweroften", "group": "Ungrouped variables", "templateType": "anything", "name": "ans1", "description": ""}}, "functions": {}, "type": "question", "preamble": {"js": "", "css": ""}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "ungrouped_variables": ["dec1", "dec2", "dec3", "dec4", "power", "poweroften", "ans1", "ans2", "ans3", "ans4", "pronpower"], "statement": "

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

By powers of ten I mean a 1 followed by some 0s. The scientific notation questions will take care of the power of ten notation.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "scripts": {}, "prompt": "

$\\var{poweroften}$ has {pronpower} $0$s after the leading $1$. This means to evaluate $\\var{dec1}\\times \\var{poweroften}$ we just move the decimal point in $\\var{dec1}$ {pronpower} decimal places to the right (to make the decimal $\\var{poweroften}$ times bigger) and get $\\var{ans1}$.

\n

", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "ans1", "unitTests": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "maxValue": "ans1", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{dec1}\\times \\var{poweroften}=$ []

\n

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "scripts": {}, "prompt": "

$\\var{poweroften}$ has {pronpower} $0$s after the leading $1$. This means to evaluate $\\var{dec2}\\times \\var{poweroften}$ we just move the decimal point in $\\var{dec2}$ {pronpower} decimal places to the right (to make the decimal $\\var{poweroften}$ times bigger) and get $\\var{ans2}$.

", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "ans2", "unitTests": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "maxValue": "ans2", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{dec2}\\times\\var{poweroften}=$ []

\n

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "scripts": {}, "prompt": "

$\\var{poweroften}$ has {pronpower} $0$s after the leading $1$. This means to evaluate $\\var{dec3}\\div \\var{poweroften}$ we just move the decimal point in $\\var{dec3}$ {pronpower} decimal places to the left (to make the decimal smaller) and get $\\var{ans3}$.

", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "ans3", "unitTests": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "maxValue": "ans3", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{dec3}\\div\\var{poweroften}=$ []

\n

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "sortAnswers": false}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "unitTests": [], "scripts": {}, "prompt": "

Recall, the fraction bar simply denotes division.

\n

$\\var{poweroften}$ has {pronpower} $0$s after the leading $1$. This means to evaluate $\\var{dec4}\\div \\var{poweroften}$ we just move the decimal point in $\\var{dec4}$ {pronpower} decimal places to the left (to make the decimal smaller) and get $\\var{ans4}$.

", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 0, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "customMarkingAlgorithm": "", "minValue": "ans4", "unitTests": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "maxValue": "ans4", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\displaystyle \\frac{\\var{dec4}}{\\var{poweroften}}=$ []

Multiplying or dividing by a power of ten (such as a $1$ followed by some $0$s) moves the decimal point. Multiplying moves the decimal point to make the number bigger (that is, to the right). Dividing moves the decimal to make the number smaller (that is, to the left). The number of $0$s indicates the number of places you should move the decimal place.

", "tags": []}, {"name": "Decimals: Multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"afact1": {"definition": "random(10,100,1000,10000)", "group": "2digit", "description": "", "name": "afact1", "templateType": "anything"}, "adec1": {"definition": "atopnum/afact1", "group": "2digit", "description": "", "name": "adec1", "templateType": "anything"}, "atop": {"definition": "if(adigs<>0,[adigs,adigs],[adigs,adigs])", "group": "2digit", "description": "", "name": "atop", "templateType": "anything"}, "aansone": {"definition": "mod(aans,10)", "group": "2digit", "description": "", "name": "aansone", "templateType": "anything"}, "ab0t0": {"definition": "atop*abot", "group": "2digit", "description": "", "name": "ab0t0", "templateType": "anything"}, "dps": {"definition": "log(easyfactprod)", "group": "Ungrouped variables", "description": "", "name": "dps", "templateType": "anything"}, "ab0t1": {"definition": "abot*atop", "group": "2digit", "description": "", "name": "ab0t1", "templateType": "anything"}, "easy2": {"definition": "easydig2/easyfact2", "group": "Ungrouped variables", "description": "", "name": "easy2", "templateType": "anything"}, "ab1t0": {"definition": "abot*atop", "group": "2digit", "description": "", "name": "ab1t0", "templateType": "anything"}, "afactprod": {"definition": "afact1*afact2", "group": "2digit", "description": "", "name": "afactprod", "templateType": "anything"}, "aanshun": {"definition": "mod((aans-aansone-aansten*10)/100,10)", "group": "2digit", "description": "", "name": "aanshun", "templateType": "anything"}, "aans": {"definition": "atopnum*abotnum", "group": "2digit", "description": "", "name": "aans", "templateType": "anything"}, "atopnum": {"definition": "atop*10+atop", "group": "2digit", "description": "", "name": "atopnum", "templateType": "anything"}, "asum2": {"definition": "10*abot*atopnum", "group": "2digit", "description": "

sum2

", "name": "asum2", "templateType": "anything"}, "dpsword": {"definition": "switch(dps=6,\"six\", dps=5, \"five\", dps=4,\"four\", dps=3,\"three\", dps=2, \"two\", dps=1,\"one\",dps)", "group": "Ungrouped variables", "description": "", "name": "dpsword", "templateType": "anything"}, "easyfact1": {"definition": "random(1,10,100,1000)", "group": "Ungrouped variables", "description": "", "name": "easyfact1", "templateType": "anything"}, "abotnum": {"definition": "abot*10+abot", "group": "2digit", "description": "

botnum

", "name": "abotnum", "templateType": "anything"}, "ab1t1last": {"definition": "mod(ab1t1pluscarry,10)", "group": "2digit", "description": "", "name": "ab1t1last", "templateType": "anything"}, "adecans": {"definition": "aans/afactprod", "group": "2digit", "description": "", "name": "adecans", "templateType": "anything"}, "easydig1": {"definition": "random(3..9)", "group": "Ungrouped variables", "description": "", "name": "easydig1", "templateType": "anything"}, "ab1t1": {"definition": "abot*atop", "group": "2digit", "description": "", "name": "ab1t1", "templateType": "anything"}, "easyans": {"definition": "easydigprod/(easyfactprod)", "group": "Ungrouped variables", "description": "

eas

", "name": "easyans", "templateType": "anything"}, "ab1t1pluscarry": {"definition": "ab1t1+ab1t0carry", "group": "2digit", "description": "", "name": "ab1t1pluscarry", "templateType": "anything"}, "ab0t0last": {"definition": "mod(ab0t0,10)", "group": "2digit", "description": "", "name": "ab0t0last", "templateType": "anything"}, "ab1t0last": {"definition": "mod(ab1t0,10)", "group": "2digit", "description": "", "name": "ab1t0last", "templateType": "anything"}, "adec2": {"definition": "abotnum/afact2", "group": "2digit", "description": "", "name": "adec2", "templateType": "anything"}, "asum1": {"definition": "abot*atopnum", "group": "2digit", "description": "", "name": "asum1", "templateType": "anything"}, "ab1t1carry": {"definition": "(ab1t1pluscarry-ab1t1last)/10", "group": "2digit", "description": "", "name": "ab1t1carry", "templateType": "anything"}, "afact2": {"definition": "if(afact1=10000,random(10,100,1000),random(10,100,1000,10000))", "group": "2digit", "description": "", "name": "afact2", "templateType": "anything"}, "ab0t1last": {"definition": "mod(ab0t1pluscarry,10)", "group": "2digit", "description": "", "name": "ab0t1last", "templateType": "anything"}, "easydig2": {"definition": "if(easydig1=3,random(4..9),random(3..9))", "group": "Ungrouped variables", "description": "", "name": "easydig2", "templateType": "anything"}, "adps": {"definition": "log(afactprod)", "group": "2digit", "description": "", "name": "adps", "templateType": "anything"}, "easy1": {"definition": "easydig1/easyfact1", "group": "Ungrouped variables", "description": "", "name": "easy1", "templateType": "anything"}, "ab0t1pluscarry": {"definition": "ab0t1+ab0t0carry", "group": "2digit", "description": "

ab

", "name": "ab0t1pluscarry", "templateType": "anything"}, "adigs": {"definition": "shuffle(1..9)[0..4]", "group": "2digit", "description": "

we want distinct digits so it is easier to refer to digits unambiguously.

abot

", "name": "abot", "templateType": "anything"}, "ab1t0carry": {"definition": "(ab1t0-ab1t0last)/10", "group": "2digit", "description": "", "name": "ab1t0carry", "templateType": "anything"}, "easydigprod": {"definition": "easydig1*easydig2", "group": "Ungrouped variables", "description": "", "name": "easydigprod", "templateType": "anything"}, "easyfact2": {"definition": "if(easyfact1=1,random(10,100,1000),random(1,10,100,1000))", "group": "Ungrouped variables", "description": "", "name": "easyfact2", "templateType": "anything"}, "aanstho": {"definition": "mod((aans-aansone-aansten*10-aanshun*100)/1000,10)", "group": "2digit", "description": "", "name": "aanstho", "templateType": "anything"}, "easyfactprod": {"definition": "easyfact1*easyfact2", "group": "Ungrouped variables", "description": "", "name": "easyfactprod", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "type": "question", "ungrouped_variables": ["easydig1", "easydig2", "easyfact1", "easyfact2", "easydigprod", "easyfactprod", "easy1", "easy2", "easyans", "dps", "dpsword"], "variable_groups": [{"variables": ["adigs", "atop", "abot", "atopnum", "abotnum", "aans", "ab0t0", "ab0t0last", "ab0t0carry", "ab0t1", "ab0t1pluscarry", "ab0t1last", "ab0t1carry", "ab1t0", "ab1t0last", "ab1t0carry", "ab1t1", "ab1t1pluscarry", "ab1t1last", "ab1t1carry", "asum1", "asum2", "aansone", "aansten", "aanshun", "aanstho", "afact1", "afact2", "afactprod", "adps", "adpsword", "adec1", "adec2", "adecans"], "name": "2digit"}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "statement": "

Write the following questions down on paper and evaluate them without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

a) Multiplying decimals with a single non-zero digit. Students are told to preserve the number of decimal places (from the question to the answer).

\n

b) Multiplying decimals requiring the multiplication algorithm.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "

Remove the decimal points, do the multiplication of whole numbers, then put the decimal place in the answer so that the number of decimal places in the question and the answer are the same.

\n

\n

That is,

\n
\n
• since $\\var{easydig1}\\times\\var{easydig2}=\\var{easydigprod}$, the answer will involve the digits $\\var{easydigprod}$,
• \n
• since there are {dpsword} decimal places in the question, there will be {dpsword} decimal places in the answer, is only one decimal place in the question, there will be one decimal place in the question,
• \n
\n

and therefore $\\var{easy1}\\times\\var{easy2}=\\var{easyans}0$. But note, we don't need to write the last zero so we could also write $\\var{easy1}\\times\\var{easy2}=\\var{easyans}$.

\n

and therefore $\\var{easy1}\\times\\var{easy2}=\\var{easyans}$.

\n

\n

This procedure works because it is the following in disguise:

\n

\\begin{align}\\var{easy1}\\times\\var{easy2}&=\\frac{\\var{easydig1}}{\\var{easyfact1}}\\times\\frac{\\var{easydig2}}{\\var{easyfact2}}&&\\text{(convert the decimals to fractions)}\\\\&=\\frac{\\var{easydig1}\\times\\var{easydig2}}{\\var{easyfact1}\\times\\var{easyfact2}}&&\\text{(multiply the fractions)}\\\\&=\\frac{\\var{easydigprod}}{\\var{easyfactprod}}\\\\&=\\var{easyans}&&\\text{(convert back to a decimal)}\\end{align}

", "showCorrectAnswer": true, "unitTests": [], "showFeedbackIcon": true, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "information", "customMarkingAlgorithm": ""}], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "marks": 1, "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "unitTests": [], "maxValue": "easyans", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "minValue": "easyans"}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{easy1}\\times \\var{easy2}=$ []

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "showFeedbackIcon": true}, {"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "

Remove the decimal points, do the multiplication of whole numbers, then put the decimal place in the answer so that the number of decimal places in the question and the answer are the same.

\n

\n

That is,

\n
\n
• since $\\var{atopnum}\\times\\var{abotnum}=\\var{aans}$ (see the working below), the answer will involve the digits $\\var{aans}$,
• \n
• since there are {adpsword} decimal places in the question, there will be {adpsword} decimal places in the answer, is only one decimal place in the question, there will be one decimal place in the question,
• \n
\n

and therefore $\\var{adec1}\\times\\var{adec2}=\\var{adecans}0$. But note, we don't need to write the last zero so we could also write $\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.

\n

and therefore $\\var{adec1}\\times\\var{adec2}=\\var{adecans}00$. But note, we don't need to write the two trailing zeros so we could also write $\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.

\n

and therefore $\\var{adec1}\\times\\var{adec2}=\\var{adecans}$.

\n

\n
\n

\n

This procedure works because it is the following in disguise:

\n

\\begin{align}\\var{adec1}\\times\\var{adec2}&=\\frac{\\var{atopnum}}{\\var{afact1}}\\times\\frac{\\var{abotnum}}{\\var{afact2}}&&\\text{(convert the decimals to fractions)}\\\\&=\\frac{\\var{atopnum}\\times\\var{abotnum}}{\\var{afact1}\\times\\var{afact2}}&&\\text{(multiply the fractions)}\\\\&=\\frac{\\var{aans}}{\\var{afactprod}}\\\\&=\\var{adecans}&&\\text{(convert back to a decimal)}\\end{align}

\n

\n
\n

", "showCorrectAnswer": true, "unitTests": [], "showFeedbackIcon": true, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "type": "information", "customMarkingAlgorithm": ""}, {"variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "

### How to calculate $\\var{atopnum}\\times\\var{abotnum}$

\n

Generally we set up $\\var{atopnum}\\times\\var{abotnum}$ with the ones and tens columns lined up vertically:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\var{atop}$ $\\var{atop}$ $\\times$ $\\var{abot}$ $\\var{abot}$ $\\phantom{0}$
\n

\n

We need to multiply each digit in the bottom number by each digit in the top number whilst respecting their place values.

\n

\n

We multiply the digits in the ones column, that is, $\\color{green}{\\var{abot}\\times \\var{atop}}$.

\n

Since this is just $\\var{ab0t0}$ we write $\\var{ab0t0}$ under the line in the ones column.

\n

Since this is $\\var{ab0t0}$ we write the $\\var{ab0t0last}$ under the line in the ones column and carry the $\\var{ab0t0carry}$ into the tens column to be dealt with in the next step.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\color{red}{\\var{ab0t0carry}}}{\\var{atop}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{atop}}}$ $\\times$ $\\var{abot}$ $\\color{green}{\\var{abot}}$ $\\color{red}{\\var{ab0t0last}}$
\n

\n

\n

We now multiply diagonally, $\\color{green}{\\var{abot}\\times \\var{atop}}$.

\n

This just gives us $\\var{ab0t1}$ so we write $\\var{ab0t1}$ under the line in the tens column.

\n

This gives us $\\var{ab0t1}$ so we write this under the line with the $\\var{ab0t1last}$ in the tens column.

\n

This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write $\\var{ab0t1pluscarry}$ under the line with the $\\var{ab0t1last}$ in the tens column.

\n

This gives us $\\var{ab0t1}$ but we have to add the $\\var{ab0t0carry}$ we carried earlier and so we write $\\var{ab0t1pluscarry}$ under the line in the tens column.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\color{green}{\\overset{{\\var{ab0t0carry}}}{\\var{atop}}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{atop}}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\times$ $\\var{abot}$ $\\color{green}{\\var{abot}}$ $\\color{red}{\\var{ab0t1carry}}$ $\\color{red}{\\var{ab0t1last}}$ ${\\var{ab0t0last}}$
\n

\n

We are now finished with the digit $\\var{abot}$ and move on to work with the $\\var{abot}$ in the tens column. Since this is really a $\\var{abot*10}$ we place a zero in the ones column on the next line to pad our numbers out. We also crossout or erase any carry marks that we have used.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\phantom{1}}{\\var{atop}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\times$ $\\var{abot}$ $\\var{abot}$ ${\\var{ab0t1carry}}$ ${\\var{ab0t1last}}$ ${\\var{ab0t0last}}$ $\\color{red}0$
\n

\n

We now multiply along the other diagonal, that is, $\\color{green}{\\var{abot}\\times\\var{atop}}$.

\n

Since this is just $\\var{ab1t0}$ we write $\\var{ab1t0}$ under the line in the tens column.

\n

Since this is $\\var{ab1t0}$ we write the $\\var{ab1t0last}$ under the line in the tens column and carry the $\\var{ab1t0carry}$ into the tens column to be dealt with in the next step.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{\\color{red}{\\var{ab1t0carry}}}{\\var{atop}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{atop}}}$ $\\times$ $\\color{green}{\\var{abot}}$ $\\var{abot}$ ${\\var{ab0t1carry}}$ ${\\var{ab0t1last}}$ ${\\var{ab0t0last}}$ $\\color{red}{\\var{ab1t0last}}$ ${0}$
\n

\n

We now multiply the digits in the tens column, that is, $\\color{green}{\\var{abot}\\times \\var{atop}}$.

\n

This just gives us $\\var{ab1t1}$ so we write $\\var{ab1t1}$ under the line in the hundreds column.

\n

This gives us $\\var{ab1t1}$ so we write this under the line with the $\\var{ab1t1last}$ in the hundreds column.

\n

This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write $\\var{ab1t1pluscarry}$ under the line with the $\\var{ab1t1last}$ in the hundreds column.

\n

This gives us $\\var{ab1t1}$ but we have to add the $\\var{ab1t0carry}$ we carried earlier and so we write $\\var{ab1t1pluscarry}$ under the line in the hundreds column.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\color{green}{\\overset{{\\var{ab1t0carry}}}{\\var{atop}}}$ $\\color{green}{\\overset{\\phantom{1}}{\\var{atop}}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\times$ $\\color{green}{\\var{abot}}$ $\\var{abot}$ ${\\var{ab0t1carry}}$ ${\\var{ab0t1last}}$ ${\\var{ab0t0last}}$ $\\color{red}{\\var{ab1t1carry}}$ $\\color{red}{\\var{ab1t1last}}$ ${\\var{ab1t0last}}$ ${0}$
\n

\n

We now add the two results to get the total, that is, $\\color{green}{\\var{asum1}+\\var{asum2}}$.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\overset{{\\var{ab1t0carry}}}{\\var{atop}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\overset{\\phantom{1}}{\\var{atop}}$ $\\times$ $\\var{abot}$ $\\var{abot}$ $\\color{green}{\\var{ab0t1carry}}$ $\\color{green}{\\var{ab0t1last}}$ $\\color{green}{\\var{ab0t0last}}$ $+$ $\\color{green}{\\var{ab1t1carry}}$ $\\color{green}{\\var{ab1t1last}}$ $\\color{green}{\\var{ab1t0last}}$ $\\color{green}{0}$ $\\color{red}{\\var{aanstho}}$ $\\color{red}{\\var{aanshun}}$ $\\color{red}{\\var{aansten}}$ $\\color{red}{\\var{aansone}}$
\n

\n

\n

$\\var{atopnum}\\times\\var{abotnum}$ is therefore $\\var{aans}$.

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$\\var{adec1}\\times\\var{adec2} =$ []

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "tags": []}, {"name": "Decimals: division (includes rounding the answer)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/", "name": "Ben Brawn"}], "variables": {"dd0": {"definition": "0", "group": "Ungrouped variables", "templateType": "anything", "name": "dd0", "description": ""}, "diff2": {"definition": "b2-prod2", "group": "Ungrouped variables", "templateType": "anything", "name": "diff2", "description": ""}, "qd0": {"definition": "mod(floor(quotient1*1000),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "qd0", "description": ""}, "quotient1": {"definition": "dividend1/divisor1\n//random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "group": "Ungrouped variables", "templateType": "anything", "name": "quotient1", "description": ""}, "prod0": {"definition": "divisor1*qd0", "group": "Ungrouped variables", "templateType": "anything", "name": "prod0", "description": ""}, "divisorscale": {"definition": "10^divisorscaleorder", "group": "Ungrouped variables", "templateType": "anything", "name": "divisorscale", "description": ""}, "givendividend": {"definition": "dividend1/divisorscale", "group": "Ungrouped variables", "templateType": "anything", "name": "givendividend", "description": ""}, "givendivisor": {"definition": "divisor1/divisorscale", "group": "Ungrouped variables", "templateType": "anything", "name": "givendivisor", "description": ""}, "qd2": {"definition": "mod(floor(quotient1*10),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "qd2", "description": "

\\

"}, "b0": {"definition": "10*diff1+dd0", "group": "Ungrouped variables", "templateType": "anything", "name": "b0", "description": ""}, "ans": {"definition": "precround(quotient1,2)", "group": "Ungrouped variables", "templateType": "anything", "name": "ans", "description": ""}, "prod1": {"definition": "divisor1*qd1", "group": "Ungrouped variables", "templateType": "anything", "name": "prod1", "description": ""}, "diff1": {"definition": "b1-prod1", "group": "Ungrouped variables", "templateType": "anything", "name": "diff1", "description": ""}, "b1": {"definition": "10*diff2+dd1", "group": "Ungrouped variables", "templateType": "anything", "name": "b1", "description": ""}, "dd3": {"definition": "mod(floor(dividend1),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "dd3", "description": ""}, "diff0": {"definition": "b0-prod0", "group": "Ungrouped variables", "templateType": "anything", "name": "diff0", "description": ""}, "remainder": {"definition": "mod(dividend1,divisor1)", "group": "Ungrouped variables", "templateType": "anything", "name": "remainder", "description": ""}, "qd1": {"definition": "mod(floor(quotient1*100),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "qd1", "description": ""}, "dividend1": {"definition": "random(max(divisor1,25)..99 except list(divisor1/2..99#divisor1/2))/10\n//divisor1*quotient1+remainder/100", "group": "Ungrouped variables", "templateType": "anything", "name": "dividend1", "description": ""}, "dd2": {"definition": "mod(floor(dividend1*10),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "dd2", "description": ""}, "diff3": {"definition": "dd3-prod3", "group": "Ungrouped variables", "templateType": "anything", "name": "diff3", "description": ""}, "divisor1": {"definition": "random(3..32 except [5,10,20,30])", "group": "Ungrouped variables", "templateType": "anything", "name": "divisor1", "description": "

excluded 5 so that the decimal part is longer than 1 place.

"}, "prod3": {"definition": "divisor1*qd3", "group": "Ungrouped variables", "templateType": "anything", "name": "prod3", "description": ""}, "b2": {"definition": "10*diff3+dd2", "group": "Ungrouped variables", "templateType": "anything", "name": "b2", "description": ""}, "divisorscaleorder": {"definition": "random(1,2,3,4)", "group": "Ungrouped variables", "templateType": "anything", "name": "divisorscaleorder", "description": ""}, "qd3": {"definition": "mod(floor(quotient1),10)", "group": "Ungrouped variables", "templateType": "anything", "name": "qd3", "description": ""}, "prod2": {"definition": "divisor1*qd2", "group": "Ungrouped variables", "templateType": "anything", "name": "prod2", "description": ""}, "dd1": {"definition": "0", "group": "Ungrouped variables", "templateType": "anything", "name": "dd1", "description": "

Was mod(floor(dividend1*100),10) but is now hard coded because there were rounding errors

"}}, "variable_groups": [], "statement": "

Write the following question down on paper and evaluate it without using a calculator.

\n

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

", "type": "question", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "scripts": {}, "prompt": "

We want to calculate $\\var{givendividend}\\div\\var{givendivisor}$. By the way, this is the same thing as $\\frac{\\var{givendividend}}{\\var{givendivisor}}$. Both of these expressions mean \"how many $\\var{givendivisor}$s go into $\\var{givendividend}$?\"

\n

\n

To work out such a thing we normally convert the division/fraction into an equivalent division/fraction where we are dividing by a whole number. We do this by multiplying both $\\var{givendividend}$ (the dividend) and $\\var{givendivisor}$ (the divisor) by $\\var{divisorscale}$ so that the decimal points are moved $\\var{divisorscaleorder}$ places to the right and so we get the division $\\var{dividend1}\\div\\var{divisor1}$.

\n

Note we want the divisor to be a whole number but we don't need the dividend to be whole.

\n

Why is this division equivalent? Think of division as sharing some amount equally amoungst some number of people. Now consider the scenario where you have $\\var{divisorscale}$ times the original amount to share but also $\\var{divisorscale}$ times the original number people, everyone will get the same amount as in the original scenario!

\n

\n

\n

The following is the above working in terms of fractions:

\n

\\begin{align}\\var{givendividend}\\div\\var{givendivisor}&=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\\0.1cm]&=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\times 1\\\\[0.1cm]&=\\frac{\\var{givendividend}}{\\var{givendivisor}}\\times\\frac{\\var{divisorscale}}{\\var{divisorscale}}\\\\[0.1cm]&=\\frac{\\var{givendividend}\\times \\var{divisorscale}}{\\var{givendivisor}\\times\\var{divisorscale}}\\\\[0.1cm]&=\\frac{\\var{dividend1}}{\\var{divisor1}}\\end{align} \n \n \n We want to calculate \\var{dividend1}\\div\\var{divisor1}. Note, this is the same thing as \\frac{\\var{dividend1}}{\\var{divisor1}} and both expressions mean \"how many \\var{divisor1}s go into \\var{dividend1}?\" \n \n The long division algorithm allows you to work this out by working from the left to the right of \\var{dividend1} whilst respecting place value. We normally set up the division in the following way: \n \\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}} \n Note the positions of the numbers! \n Actually, since we want the answer to two decimal place we add as many zeroes after the decimal place to ensure we have three decimal places! \n \\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \n Why three? We use that extra digit to determine whether to round up or down. \n The algorithm (or procedure) seems complicated at first but you might find a mnemonic helps to remember the steps. We work left to right doing the following steps \n \n 1. Divide 2. \n 3. Multiply 4. \n 5. Subtract 6. \n 7. Bring down 8. \n \n and repeating until we run out of digits. The steps form the acronym DMSB. Popular mnemonics include \"Does McDonalds Sell Burgers?\", \"Dracula Must Suck Blood\" and \"Dead Mice Smell Bad\". \n \n We need to know the \\var{divisor1} times tables or write the \\var{divisor1} times tables out (be repeatedly adding \\var{divisor1}) so that we can refer to them. \n \\[\\boxed{\\begin{align}1\\times\\var{divisor1}&=\\var{divisor1}\\\\2\\times\\var{divisor1}&=\\var{2*divisor1}\\\\3\\times\\var{divisor1}&=\\var{3*divisor1}\\\\4\\times\\var{divisor1}&=\\var{4*divisor1}\\\\5\\times\\var{divisor1}&=\\var{5*divisor1}\\\\6\\times\\var{divisor1}&=\\var{6*divisor1}\\\\7\\times\\var{divisor1}&=\\var{7*divisor1}\\\\8\\times\\var{divisor1}&=\\var{8*divisor1}\\\\9\\times\\var{divisor1}&=\\var{9*divisor1}\\end{align}}\

\n

\n

The ones column

\n

D: The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\"

\n

Well, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the ones column: Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fit and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the ones column:  Well, $\\var{qd3}\\times \\var{divisor1}=\\var{prod3}$ so $\\var{qd3}$ fits and we write $\\color{red}{\\var{qd3}}$ above the $\\var{dd3}$ in the ones column:

\n

$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\end{array}$

\n

M: Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write $\\color{red}{\\var{prod3}}$ underneath in the ones column:

\n

$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{.000}\\end{array}$

\n

S: We now do the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the remainder (what remains to be divided) in the ones column.

\n

$\\begin{array}{r} {\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{\\color{green}{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{.000}\\end{array}$

\n

B: Now we bring the $\\color{green}{\\var{dd2}}$ in the tenths column down next to the remainder so that it forms $\\var{diff3}\\var{dd2}$.

\n

$\\begin{array}{r} {\\var{qd3}\\phantom{.\\!\\var{qd2}\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\color{green}{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}}\\phantom{.}\\color{red}{\\var{dd2}}\\phantom{00}\\end{array}$

\n

\n

The tenths column

\n

D: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this $\\var{b2}$ actually represents $\\var{b2/10}$ since it is in the tenths column)

\n

Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the tenths column: Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fit and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the tenths column:  Well, $\\var{qd2}\\times \\var{divisor1}=\\var{prod2}$ so $\\var{qd2}$ fits and we write $\\color{red}{\\var{qd2}}$ above the $\\var{dd2}$ in the tenths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!\\color{red}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{green}{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\end{array}$

\n

M: Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write $\\color{red}{\\var{prod2}}$ underneath in the tenths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!\\color{green}{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{00}\\end{array}$

\n

S: We now do the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the remainder (what remains to be divided) in the tenths column.

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]\\color{green}{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{00}\\end{array}$

\n

B: Now we bring the $\\color{green}{\\var{dd1}}$ in the hundredths column down next to the remainder so that it forms $\\var{diff2}\\var{dd1}$.

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\phantom{\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\var{diff2}\\color{red}{\\var{dd1}}\\phantom{0}\\end{array}$

\n

\n

The hundredths column

\n

D: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1/100}$ since it is in the hundredths column)

\n

Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the hundredths column: Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fit and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the hundredths column:  Well, $\\var{qd1}\\times \\var{divisor1}=\\var{prod1}$ so $\\var{qd1}$ fits and we write $\\color{red}{\\var{qd1}}$ above the $\\var{dd1}$ in the hundredths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{0}\\end{array}$

\n

M: Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write $\\color{red}{\\var{prod1}}$ underneath in the hundredths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{0}\\end{array}$

\n

S: We now do the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the remainder (what remains to be divided) in the hundredths column.

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{0}\\end{array}$

\n

B: Now we bring the $\\color{green}{\\var{dd0}}$ in the thousandths column down next to the remainder so that it forms $\\var{diff1}\\var{dd0}$.

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$

\n

\n

The thousandths column

\n

D: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0/1000}$ since it is in the thousandths column)

\n

Well, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the thousandths column: Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fit and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the thousandths column:  Well, $\\var{qd0}\\times \\var{divisor1}=\\var{prod0}$ so $\\var{qd0}$ fits and we write $\\color{red}{\\var{qd0}}$ above the $\\var{dd0}$ in the thousandths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$

\n

M: Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write $\\color{red}{\\var{prod0}}$ underneath in the thousandths column:

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$

\n

S: We now do the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the remainder (what remains to be divided) in the thousandths column.

\n

$\\begin{array}{r} {\\var{qd3}.\\!{\\var{qd2}}{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!{\\var{dd2}}{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{.000}\\\\[-.7cm]{\\var{diff3}\\phantom{.}\\var{dd2}}\\phantom{00}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{00}\\\\[-0.5cm]{\\var{diff2}\\var{dd1}}\\phantom{0}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{0}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$

\n

\n

Now we could keep adding zeros and continue the procedure but we only needed to determine the third decimal place in order to correctly round to two decimal places and so we now stop the procedure.

\n

Since the third decimal place was $\\var{qd0}$ we round up down to $\\var{ans}$. Therefore, $\\var{dividend1}\\div\\var{divisor1}=\\var{ans}$ (2 dec. pl.).

", "variableReplacements": [], "type": "information", "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "customMarkingAlgorithm": ""}], "variableReplacements": [], "type": "gapfill", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0, "precision": "2", "precisionType": "dp", "marks": 1, "showPrecisionHint": true, "minValue": "ans", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry", "customMarkingAlgorithm": "", "strictPrecision": false, "unitTests": [], "maxValue": "ans", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "mustBeReduced": false}], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "prompt": "

$\\var{givendividend}\\div\\var{givendivisor}=$[] (2 decimal places)

\n

", "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "sortAnswers": false}], "rulesets": {}, "ungrouped_variables": ["divisor1", "remainder", "dividend1", "quotient1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "ans", "divisorscaleorder", "divisorscale", "givendivisor", "givendividend"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Issues: alignment in columns in the working - not sure what to do about it

\n

\n

Decimal divided by a decimal. Multiply by a power of ten to get an integer divisor. Long division process. There is a remainder which we express as a decimal by continuing the long division process. Rounding is required to some number of decimal places.

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Face, Place and actual value, adding, subtracting, multiplying, dividing and rounding decimals.

"}}