Use the BODMAS rule to determine the order in which to evaluate some arithmetic expressions.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Wrong order of solving operations can often lead to incorrect answers. Therefore, the order in which we carry out a calculation is important.
\nBODMAS is a mnemonic which tells us the correct order in which operations should be carried out:
\n\n\nBrackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction
\n
Apply BODMAS and try to solve these calculations.
", "advice": "The correct order of carrying out operations can be remembered by the mnemonic BODMAS:
\n\n\nBrackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction
\n
It is important to notice that division and multiplication have the same priority - division does not have a priority over multiplication. Similarly, adition and subtraction also have the same priority. When the order is unclear, we work from left to right.
\nNote that brackets have the highest priority, but when we evaluate them, we still need to follow BODMAS inside them.
\nSometimes, an alternative acronym BIDMAS (Brackets, Indices, ...) is also used.
\na)
\nDivision and multiplication have the same priority, so we just work from left to right. $\\var{int*int} ÷ \\var{int} = \\var{int}$ and hence
\n\\[\\begin{align} \\var{int*int} ÷ \\var{int} \\times \\var{int} &= \\var{int} \\times \\var{int} \\\\&= \\var{int*int} \\text{.} \\end{align}\\]
\n\nb)
\nApplying BODMAS, multiplication has a priority over addition. $\\var{sint + 2} \\times \\var{sint} = \\var{(sint + 2)*sint}$ and hence
\n\\[\\begin{align} \\var{sint} + \\var{sint + 2} \\times \\var{sint} &= \\var{sint} + \\var{(sint + 2)*sint} \\\\&= \\var{sint + (sint + 2)*sint}\\text{.} \\end{align}\\]
\n\nc)
\nApplying BODMAS, multiplication and division have priority over addition and subtraction. $1 \\times 0 = 0$ and $\\var{bint}\\div\\var{bint} = 1$ so
\n\\[\\begin{align} \\var{bint - 15} - 1 \\times 0 + \\var{bint}\\div\\var{bint} &= \\var{bint - 15} - 0 + 1 \\\\&= \\var{bint - 14}\\text{.} \\end{align}\\]
\n", "rulesets": {}, "variables": {"int": {"name": "int", "group": "Ungrouped variables", "definition": "random(2..10 except sint)", "description": "Random integer from 2 to 10.
", "templateType": "anything"}, "oint": {"name": "oint", "group": "Ungrouped variables", "definition": "random(1..9 #2 except int except sint)", "description": "Random odd integer from 1 to 9.
", "templateType": "anything"}, "bint": {"name": "bint", "group": "Ungrouped variables", "definition": "random(20..50)", "description": "A random slightly bigger integer.
", "templateType": "anything"}, "pint": {"name": "pint", "group": "Ungrouped variables", "definition": "random(1..4 except 3)", "description": "1, 2 or 4.
", "templateType": "anything"}, "eint": {"name": "eint", "group": "Ungrouped variables", "definition": "random(1..9 #2 except int except sint)", "description": "Random even integer from 2 to 10.
", "templateType": "anything"}, "sint": {"name": "sint", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "Random integer from 1 to 5.
", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["int", "sint", "eint", "oint", "pint", "bint"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\var{int*int} ÷ \\var{int} \\times \\var{int} =$ $?$
", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["{int*int}
", "{int}
", "1
"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\var{sint} + \\var{sint + 2} \\times \\var{sint} =$ $?$
", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["{(sint + sint + 2)*sint}
", "{sint + (sint + 2)*sint}
", "{(((sint + sint + 2)*sint) + (sint + (sint + 2)*sint))/2}
"], "matrix": [0, "1", 0], "distractors": ["", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\var{bint - 15} - 1 \\times 0 + \\var{bint}\\div\\var{bint} =$ $?$
", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["1
", "{bint - 14}
"], "matrix": ["0", "1"], "distractors": ["", ""]}], "type": "question"}, {"name": "Musa's copy of 3 MA1002 fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Kieran Mulchrone", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1243/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "This is a set of questions designed to help you practice adding, subtracting, multiplying and dividing fractions.
\nAll of these can be done without a calculator.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "These are basic questions to help you practice adding, subtracting, multiplying and dividing fractions.
\nAttempt the questions without a calculator.
\nGive your answer as a fraction.
", "advice": "When adding/subtracting fractions, you must first find a common denominator between the fractions. If they already have the same denominator then you only need to worry about adding/subtracting the numerators and dividing the result by the common denominator.
\nFor example:
To find a common denominator of $\\frac{2}{5} + \\frac{7}{15}$, the most obvious would be $15$, because $5\\times3=15$. Therefore, you must multiply both sides of the fraction $\\frac{2}{5}$ by $3$ to obtain a new fraction $\\frac{6}{15}$. This is known as 'scaling up'.
Now you can add the two fractions together (by adding the numerators) because they have the same denominator:
$\\frac{6}{15}+\\frac{7}{15}=\\frac{13}{15}$.
The same applies with subtraction as well as addition.
\n\nWhen multiplying fractions, you can simply multiply the two numerators and divide this by the multiplication of the two denominators.
\nFor example:
$\\frac{a}{b}\\times\\frac{c}{d}$ = $\\frac{a\\times{c}}{b\\times{d}}$
When dividing fractions, you firstly need to reciprocate (flip) the second fraction, then multiply the numerators and denominators as you would a normal multiplication question.
\nFor example:
$\\frac{a}{b} \\div \\frac{c}{d}$ would be flipped to become $\\frac{a}{b} \\div \\frac{d}{c}$ and then treated as a normal multiplication question (as explained above).
What is the answer to $\\frac{\\var{a}}{\\var{b}} \\times \\frac{\\var{c}}{\\var{b}}$?
", "stepsPenalty": 0, "steps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Multiply the numerators
", "minValue": "a*c", "maxValue": "a*c", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Multiply the denominators
", "minValue": "b*b", "maxValue": "b*b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Put into a fraction, with the new numerator over the new denominator
", "minValue": "(a*c)/(b*b)", "maxValue": "(a*c)/(b*b)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "(a*c)/(b*b)", "maxValue": "(a*c)/(b*b)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "What is the answer to $\\frac{\\var{h}}{\\var{j}} + \\frac{\\var{k}}{\\var{j}}$?
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Check to see if the denominators are the same. If they are, you only need to add the numerators together and leave the denominator as it is for the final answer.
"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Add the numerators
", "minValue": "h+k", "maxValue": "h+k", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Write as a fraction over the similar denominator; cancel down if you can.
", "minValue": "(h+k)/j", "maxValue": "(h+k)/j", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "(h+k)/j", "maxValue": "(h+k)/j", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "What is the answer to $\\frac{\\var{o}}{\\var{p}} \\times \\frac{\\var{q}}{\\var{r}}$?
", "stepsPenalty": "0", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Multiply the numerators to make the top of the fraction
\nMultiply the denominators to make the bottom of the fraction
\nCancel down if you can
"}], "minValue": "(o*q)/(p*r)", "maxValue": "(o*q)/(p*r)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": false, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "What is the answer to $\\frac{\\var{w}}{\\var{x}} + \\frac{\\var{y}}{\\var{z}}$?
", "stepsPenalty": "0", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Check to see if the denominators are the same
\nIf they are not - multiply each fraction up to equivalent fractions with equal denominators
\nOnce they are equal add the numerators and put over the equal denominator
\nCancel down if needed
"}], "minValue": "((w*z)+(y*x))/(x*z)", "maxValue": "((w*z)+(y*x))/(x*z)", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "type": "question"}, {"name": "Musa's copy of 3 algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}, {"name": "Anna Strzelecka", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2945/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Simplify the following algebraic expressions.
\nNote: Although the question may accept coefficients in their decimal forms, it would be more appropriate to keep them in their most simplified fraction forms.
", "advice": "Click 'Try another question like this one' if you need more practice.
", "rulesets": {}, "variables": {"e2": {"name": "e2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "co2": {"name": "co2", "group": "Ungrouped variables", "definition": "random(2..5 except co1)", "description": "", "templateType": "anything"}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(1..9 except n1)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "g2": {"name": "g2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except g1)", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(1..9 except b1)", "description": "", "templateType": "anything"}, "co4": {"name": "co4", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "random(1..9 except b3)", "description": "", "templateType": "anything"}, "co3": {"name": "co3", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "random(-5..5 except f1)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..9 except a1)", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "g1": {"name": "g1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "n1/d1", "description": "", "templateType": "anything"}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "co1": {"name": "co1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "e3": {"name": "e3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except e2)", "description": "", "templateType": "anything"}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n1", "d1", "ans", "a1", "a2", "n2", "p2", "b1", "b2", "b3", "b4", "c1", "c2", "e1", "e2", "e3", "co1", "co2", "f1", "f2", "co3", "co4", "p3", "g1", "g2"], "variable_groups": [], "functions": {"": {"parameters": [], "type": "number", "language": "jme", "definition": ""}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\simplify{({n1}{a1}x^2+{n1}{a2}x)/({d1}{a1}x+{d1}{a2})}$
\n", "stepsPenalty": "0", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "${\\simplify{({n1}{a1}x^2+{n1}{a2}x)/({d1}{a1}x+{d1}{a2})}}$
\nFactorise the numerator and denominator so that the binomials in both are the same.
\n${\\big(\\frac{\\var{n1}x}{\\var{d1}}\\big)\\big(\\frac{\\var{a1}x+\\var{a2}}{\\var{a1}x+\\var{a2}}\\big)}$
\nThe binomials cancel, leaving $x$ and its coefficient:
\n$\\big({\\simplify{{n1}/{d1}}}\\big)x$
"}], "answer": "{{n1}/{d1}}*x", "answerSimplification": "!basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "maxlength": {"length": "0", "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["^", "x*x"], "showStrings": false, "partialCredit": 0, "message": "Please simplify further.
"}, "valuegenerators": [{"name": "x", "value": ""}]}], "type": "question"}, {"name": "Musa's copy of 3 Expand brackets", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "For each expression below, collect like terms and expand brackets.
\nThe * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.
When simplifying expressions, only terms of the same type or like terms can be added together.
\nAlgebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.
\n\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]
random variables for part 1
", "templateType": "anything"}, "b": {"name": "b", "group": "Part a", "definition": "repeat(random(2..10),5)", "description": "", "templateType": "anything"}, "g": {"name": "g", "group": "Part a", "definition": "repeat(random(2..15),7)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "B group", "variables": ["a1", "b1", "c1"]}, {"name": "Part a", "variables": ["a", "b", "c", "d", "f", "g", "h", "j"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({c[1]}+{c[0]}+{c[2]})x", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "maxlength": {"length": "0", "partialCredit": 0, "message": "You must collect like terms to fully simplify.
"}, "mustmatchpattern": {"pattern": "$n*x", "partialCredit": 0, "message": "You haven't simplified: you still have two or more like terms that should be collected together.", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Fractions are division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "Divisor is single digit. There is a remainder which we express as a decimal by continuing the long division process. Rounding is required to one decimal place. The working suggests determining the second decimal place so the student knows whether to round up or down.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write the following question down on paper and evaluate it without using a calculator.
\nIf 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.
", "advice": "", "rulesets": {}, "variables": {"prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything"}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything"}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything"}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(quotient1,1)", "description": "", "templateType": "anything"}, "dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything"}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "mod(dividend1,divisor1)", "description": "", "templateType": "anything"}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "random(25..99 except list(divisor1/2..99#divisor1/2))\n//divisor1*quotient1+remainder/100", "description": "", "templateType": "anything"}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything"}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything"}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..9 except 5)", "description": "excluded 5 so that the decimal part is longer than 1 place.
", "templateType": "anything"}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything"}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1/10),10)", "description": "", "templateType": "anything"}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1/10),10)", "description": "", "templateType": "anything"}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything"}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything"}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything"}, "quotient1": {"name": "quotient1", "group": "Ungrouped variables", "definition": "dividend1/divisor1\n//random(ceil(1001/divisor1)..floor(9999/divisor1) except list(100..10000#10))/100", "description": "", "templateType": "anything"}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "", "templateType": "anything"}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "mod(dividend1*100,10)", "description": "", "templateType": "anything"}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything"}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything"}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "qd2
", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["divisor1", "remainder", "quotient1", "dividend1", "dd0", "dd1", "dd2", "dd3", "qd3", "qd2", "qd1", "qd0", "prod3", "prod2", "prod1", "prod0", "diff3", "b2", "diff2", "b1", "diff1", "b0", "diff0", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\displaystyle\\frac{\\var{dividend1}}{\\var{divisor1}}=$[[0]] (1 decimal place)
\n", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "We want to calculate $\\frac{\\var{dividend1}}{\\var{divisor1}}$, which is just the same as $\\var{dividend1}\\div\\var{divisor1}$, since both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"
\n\nThe 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$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$
\nNote the positions of the numbers!
\nActually, since we want the answer to one decimal place we add as many zeroes after the decimal place to ensure we have two decimal places!
\n$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}}$
\nWhy two? We use that extra digit to determine whether to round up or down.
\nThe 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
\nand 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\nWe 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 tens column
\nD: The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\" (note this $\\var{dd3}$ actually represents $\\var{dd3*10}$ since it is in the tens column)
\nWell, none! $\\var{divisor1}$ is too big to fit into $\\var{dd3}$. So we write $\\color{red}0$ above the $\\var{dd3}$ in the tens 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 tens 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 tens column:
\n$\\begin{array}{r} \\color{red}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\end{array}$
\nM: Now since $\\color{green}{\\var{qd3}}\\times \\color{green}{\\var{divisor1}}=\\var{prod3}$ we write $\\color{red}{\\var{prod3}}$ underneath in the tens column:
\n$\\begin{array}{r} \\color{green}{\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-.5cm] \\color{red}{\\var{prod3}}\\phantom{5.55}\\end{array}$
\nS: We now do the subtraction, $\\color{green}{\\var{dd3}-\\var{prod3}}$, to determine the remainder (what remains to be divided) in the tens column.
\n$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\var{dd2}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{\\color{green}{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{red}{\\var{diff3}}\\phantom{5.55}\\end{array}$
\nB: Now we bring the $\\color{green}{\\var{dd2}}$ in the ones column down next to the remainder so that it forms $\\var{diff3}\\var{dd2}$.
\n$\\begin{array}{r} {\\var{qd3}\\phantom{\\var{qd2}.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}\\color{green}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}}\\color{red}{\\var{dd2}}\\phantom{.55}\\end{array}$
\n\nThe ones column
\nD: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this $\\var{b2}$ does actually represent $\\var{b2}$ since it is in the ones column)
\nWell, none! $\\var{divisor1}$ is too big to fit into $\\var{b2}$. So we write $\\color{red}0$ above the $\\var{dd2}$ in the ones 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 ones 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 ones column:
\n$\\begin{array}{r} {\\var{qd3}\\color{red}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\end{array}$
\nM: Now since $\\color{green}{\\var{qd2}}\\times \\color{green}{\\var{divisor1}}=\\var{prod2}$ we write $\\color{red}{\\var{prod2}}$ underneath in the ones column:
\n$\\begin{array}{r} {\\var{qd3}\\color{green}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{prod2}}\\phantom{.55}\\end{array}$
\nS: We now do the subtraction, $\\color{green}{\\var{b2}-\\var{prod2}}$, to determine the remainder (what remains to be divided) in the ones column.
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\var{dd1}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]\\color{green}{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{red}{\\var{diff2}}\\phantom{.55}\\end{array}$
\nB: Now we bring the $\\color{green}{\\var{dd1}}$ in the tenths column down next to the remainder so that it forms $\\var{diff2}\\var{dd1}$.
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}\\phantom{.\\!\\var{qd1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!\\color{green}{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\var{diff2}\\phantom{.}\\color{red}{\\var{dd1}}\\phantom{5}\\end{array}$
\n\nThe tenths column
\nD: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1/10}$ since it is in the tenths column)
\nWell, none! $\\var{divisor1}$ is too big to fit into $\\var{b1}$. So we write $\\color{red}0$ above the $\\var{dd1}$ in the tenths 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 tenths 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 tenths column:
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{red}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\end{array}$
\nM: Now since $\\color{green}{\\var{qd1}}\\times \\color{green}{\\var{divisor1}}=\\var{prod1}$ we write $\\color{red}{\\var{prod1}}$ underneath in the tenths column:
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!\\color{green}{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\color{red}{\\var{prod1}}\\phantom{5}\\end{array}$
\nS: We now do the subtraction, $\\color{green}{\\var{b1}-\\var{prod1}}$, to determine the remainder (what remains to be divided) in the tenths column.
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\var{dd0}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]\\color{green}{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{red}{\\var{diff1}}\\phantom{5}\\end{array}$
\nB: Now we bring the $\\color{green}{\\var{dd0}}$ in the hundredths column down next to the remainder so that it forms $\\var{diff1}\\var{dd0}$.
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\phantom{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}\\color{green}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] {\\var{diff1}}\\color{red}{\\var{dd0}}\\end{array}$
\n\nThe hundredths column
\nD: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0/100}$ since it is in the hundredths column)
\nWell, none! $\\var{divisor1}$ is too big to fit into $\\var{b0}$. So we write $\\color{red}0$ above the $\\var{dd0}$ in the hundredths 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 hundredths 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 hundredths column:
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm] \\color{green}{\\var{diff1}\\var{dd0}}\\end{array}$
\nM: Now since $\\color{green}{\\var{qd0}}\\times \\color{green}{\\var{divisor1}}=\\var{prod0}$ we write $\\color{red}{\\var{prod0}}$ underneath in the hundredths column:
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}\\color{green}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\color{red}{\\var{prod0}}\\end{array}$
\nS: We now do the subtraction, $\\color{green}{\\var{b0}-\\var{prod0}}$, to determine the remainder (what remains to be divided) in the hundredths column.
\n$\\begin{array}{r} {\\var{qd3}{\\var{qd2}}.\\!{\\var{qd1}}{\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}{\\var{dd2}}.\\!{\\var{dd1}}{\\var{dd0}}} \\\\[-0.5cm] \\underline{{\\var{prod3}}}\\phantom{5.55}\\\\[-.7cm]{\\var{diff3}\\var{dd2}}\\phantom{.55}\\\\[-0.5cm]\\underline{{\\var{prod2}}}\\phantom{.55}\\\\[-0.5cm]{\\var{diff2}\\phantom{.}\\var{dd1}}\\phantom{5}\\\\[-0.5cm]\\underline{{\\var{prod1}}}\\phantom{5}\\\\[-0.5cm]\\color{green}{\\var{diff1}\\var{dd0}}\\\\[-0.5cm]\\underline{\\color{green}{\\var{prod0}}}\\\\[-0.5cm]\\color{red}{\\var{diff0}}\\end{array}$
\nNow we could keep adding zeros and continue the procedure but we only needed to determine the second decimal place in order to correctly round to one decimal place and so we now stop the procedure.
\nSince the second decimal place was $\\var{qd0}$ we round up down to $\\var{ans}$. Therefore, $\\frac{\\var{dividend1}}{\\var{divisor1}}=\\var{ans}$ (1 dec. pl.).
"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Matrix sum", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "Addition, subtraction and multiplication by a scalar for 2 x 2 matrices.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Find the answers to the following matrix calculations, filling in your answers in the spaces provided.", "advice": "
$\\var{a}+\\var{b}=\\var{u}$
\nJust add corresponding elements together.
\n", "rulesets": {"ruleset0": []}, "variables": {"at": {"name": "at", "group": "Ungrouped variables", "definition": "transpose(a)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "a+b", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "c21": {"name": "c21", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "bat": {"name": "bat", "group": "Ungrouped variables", "definition": "b*at", "description": "", "templateType": "anything"}, "b22": {"name": "b22", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-10..10#1 except 0 except 1 except q)", "description": "", "templateType": "anything"}, "deta": {"name": "deta", "group": "Ungrouped variables", "definition": "det(a)", "description": "", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "pc-qd", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "if(test*a11=a21*a12,test+1,test)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "a+b", "description": "", "templateType": "anything"}, "d21": {"name": "d21", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "c12": {"name": "c12", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "matrix([b11,b12],[b21,b22])", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(2..10#1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "matrix([d11,d12],[d21,d22])", "description": "", "templateType": "anything"}, "det": {"name": "det", "group": "Ungrouped variables", "definition": "a11*a22-a21*a12", "description": "", "templateType": "anything"}, "b11": {"name": "b11", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "qd": {"name": "qd", "group": "Ungrouped variables", "definition": "q*d", "description": "", "templateType": "anything"}, "d12": {"name": "d12", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "b21": {"name": "b21", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "matrix([a11,a12],[a21,a22])", "description": "", "templateType": "anything"}, "c11": {"name": "c11", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "b12": {"name": "b12", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "adj": {"name": "adj", "group": "Ungrouped variables", "definition": "matrix([a22,-a12],[-a21,a11])", "description": "", "templateType": "anything"}, "d11": {"name": "d11", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "matrix([c11,c12],[c21,c22])", "description": "", "templateType": "anything"}, "c22": {"name": "c22", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "d22": {"name": "d22", "group": "Ungrouped variables", "definition": "random(-10..10#1)", "description": "", "templateType": "anything"}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "c-d", "description": "", "templateType": "anything"}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "p*c", "description": "", "templateType": "anything"}, "ab": {"name": "ab", "group": "Ungrouped variables", "definition": "a*b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a21", "a22", "ab", "ans2", "b22", "b21", "at", "ans", "d11", "d12", "p", "pc", "b12", "b11", "test", "c12", "c11", "adj", "a", "a11", "a12", "c", "b", "c22", "bat", "c21", "d", "deta", "d22", "det", "q", "d21", "u", "qd", "v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\var{a}+\\var{b}=$[[0]]
\n\n\n", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "a+b", "marks": "4", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "correctAnswer": "{a}+{b}", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 L1 - Matrix multiplication (2x2)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "Multiplication of $2 \\times 2$ matrices.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Calculate the following products of these matrices:
", "advice": "\\begin{align}
\\mathbf{AB} &= \\var{A}\\var{B} \\\\
&= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{b[0][0]}+{a[0][1]}*{b[1][0]} } & \\simplify[]{ {a[0][0]}*{b[0][1]} + {a[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{b[0][0]} + {a[1][1]}*{b[1][0]} } & \\simplify[]{ {a[1][0]}*{b[0][1]} + {a[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\
&= \\var{a*b}
\\end{align}
$\\mathbf{AB} = \\var{A}\\var{B} = $ [[0]]
", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "a*b", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "contributors": [{"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}