// Numbas version: exam_results_page_options {"name": "Decimals: division (includes rounding the answer) - long or short division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Decimals: division (includes rounding the answer) - long or short division", "tags": [], "metadata": {"description": "

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

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

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

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

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.

", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dd2": {"name": "dd2", "group": "Ungrouped variables", "definition": "mod(floor(dividend1*10),10)", "description": "", "templateType": "anything", "can_override": false}, "prod2": {"name": "prod2", "group": "Ungrouped variables", "definition": "divisor1*qd2", "description": "", "templateType": "anything", "can_override": false}, "qd0": {"name": "qd0", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*1000),10)", "description": "", "templateType": "anything", "can_override": false}, "dd1": {"name": "dd1", "group": "Ungrouped variables", "definition": "0", "description": "

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

", "templateType": "anything", "can_override": false}, "divisor1": {"name": "divisor1", "group": "Ungrouped variables", "definition": "random(3..32 except [5,10,20,30])", "description": "

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

", "templateType": "anything", "can_override": false}, "dividend1": {"name": "dividend1", "group": "Ungrouped variables", "definition": "random(max(divisor1,25)..99 except list(divisor1/2..99#divisor1/2))/10\n//divisor1*quotient1+remainder/100", "description": "", "templateType": "anything", "can_override": false}, "divisorscaleorder": {"name": "divisorscaleorder", "group": "Ungrouped variables", "definition": "random(1,2,3,4)", "description": "", "templateType": "anything", "can_override": false}, "givendivisor": {"name": "givendivisor", "group": "Ungrouped variables", "definition": "divisor1/divisorscale", "description": "", "templateType": "anything", "can_override": false}, "dd0": {"name": "dd0", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything", "can_override": false}, "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", "can_override": false}, "qd1": {"name": "qd1", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*100),10)", "description": "", "templateType": "anything", "can_override": false}, "prod1": {"name": "prod1", "group": "Ungrouped variables", "definition": "divisor1*qd1", "description": "", "templateType": "anything", "can_override": false}, "prod3": {"name": "prod3", "group": "Ungrouped variables", "definition": "divisor1*qd3", "description": "", "templateType": "anything", "can_override": false}, "qd3": {"name": "qd3", "group": "Ungrouped variables", "definition": "mod(floor(quotient1),10)", "description": "", "templateType": "anything", "can_override": false}, "diff2": {"name": "diff2", "group": "Ungrouped variables", "definition": "b2-prod2", "description": "", "templateType": "anything", "can_override": false}, "remainder": {"name": "remainder", "group": "Ungrouped variables", "definition": "mod(dividend1,divisor1)", "description": "", "templateType": "anything", "can_override": false}, "b0": {"name": "b0", "group": "Ungrouped variables", "definition": "10*diff1+dd0", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "10*diff3+dd2", "description": "", "templateType": "anything", "can_override": false}, "divisorscale": {"name": "divisorscale", "group": "Ungrouped variables", "definition": "10^divisorscaleorder", "description": "", "templateType": "anything", "can_override": false}, "qd2": {"name": "qd2", "group": "Ungrouped variables", "definition": "mod(floor(quotient1*10),10)", "description": "

", "templateType": "anything", "can_override": false}, "diff3": {"name": "diff3", "group": "Ungrouped variables", "definition": "dd3-prod3", "description": "", "templateType": "anything", "can_override": false}, "dd3": {"name": "dd3", "group": "Ungrouped variables", "definition": "mod(floor(dividend1),10)", "description": "", "templateType": "anything", "can_override": false}, "diff0": {"name": "diff0", "group": "Ungrouped variables", "definition": "b0-prod0", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(quotient1,2)", "description": "", "templateType": "anything", "can_override": false}, "givendividend": {"name": "givendividend", "group": "Ungrouped variables", "definition": "dividend1/divisorscale", "description": "", "templateType": "anything", "can_override": false}, "prod0": {"name": "prod0", "group": "Ungrouped variables", "definition": "divisor1*qd0", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "10*diff2+dd1", "description": "", "templateType": "anything", "can_override": false}, "diff1": {"name": "diff1", "group": "Ungrouped variables", "definition": "b1-prod1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "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"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\var{givendividend}\\div\\var{givendivisor}=$[[0]] (2 decimal places)

", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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 $\\var{givendividend}$?\"

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 $\\var{dividend1}\\div\\var{divisor1}$.

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

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!

The following is the above working in terms of fractions:

\\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}

"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Short Division

We want to calculate $\\var{dividend1}\\div\\var{divisor1}$. By the way, this is the same thing as $\\frac{\\var{dividend1}}{\\var{divisor1}}$. Both of these expressions mean \"how many $\\var{divisor1}$s go into $\\var{dividend1}$?\"

The short 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:

$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$

Note the positions of the numbers!

Actually, since we want the answer to two decimal places we add as many zeroes after the decimal place to ensure we have three decimal places!

$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}}$

Why three? We use that extra digit to determine whether to round up or down.

We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.

\\[\\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}}\\]

The ones column

Step 1: The first thing we ask ourselves is, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{dd3}}$?\"

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:

$\\begin{array}{r} \\color{red}{\\var{qd3}\\,.\\phantom{^\\var{diff3}\\var{qd2}}\\,\\phantom{^{\\var{diff2}}\\var{qd1}^{\\var{diff1}}\\var{qd0}}} \\\\[-.5cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\color{green}{\\var{dd3}}\\,.^\\phantom{\\var{diff3}}\\var{dd2}\\,^\\phantom{\\var{diff2}}\\var{dd1}^\\phantom{\\var{diff1}}\\var{dd0}} \\end{array}$

Step 2: We only made it to $\\var{prod3}$ but we were aiming for $\\var{dd3}$, so we were $\\var{diff3}$ away. This is the remainder (what remains to be divided) in the ones column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff3}}\\var{dd2}$ in the tenths column. We made it to exactly $\\var{dd3}$, so there is no remainder in the ones column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd2}$ in the tenths column. You do not need to do this but you can if you wish.

$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}\\var{qd2}}\\,\\phantom{{^\\var{diff2}}\\var{qd1}\\,{^\\var{diff1}}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^\\color{red}{\\var{diff3}}\\var{dd2}\\,\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$

The tenths column

Step 1: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b2}}$?\" (note this $\\var{b2}$ actually represents $\\var{b2*0.1}$)

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:

$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}\\color{red}{\\var{qd2}}\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^\\color{green}{\\var{diff3}}\\color{green}{\\var{dd2}}\\phantom{^\\color{red}{\\var{diff2}}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$

Step 2: We only made it to $\\var{prod2}$ but we were aiming for $\\var{b2}$, so we were $\\var{diff2}$ away. This is the remainder (what remains to be divided) in the tenths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff2}}\\var{dd1}$ in the hundredths column. We made it to exactly $\\var{b2}$, so there is no remainder in the tenths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd1}$ in the hundredths column. You do not need to do this but you can if you wish.

$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}\\var{qd1}\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{red}{\\var{diff2}}\\var{dd1}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$

The hundredths column

Step 1: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b1}}$?\" (note this $\\var{b1}$ actually represents $\\var{b1*0.01}$)

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:

$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}\\color{red}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^{\\var{diff3}}{\\var{dd2}}\\,^\\color{green}{\\var{diff2}}\\color{green}{\\var{dd1}}\\,\\phantom{^\\color{red}{\\var{diff1}}}\\var{dd0}}\\end{array}$

Step 2: We only made it to $\\var{prod1}$ but we were aiming for $\\var{b1}$, so we were $\\var{diff1}$ away. This is the remainder (what remains to be divided) in the hundredths column and we write it in front of the next digit as a superscript in order to represent $\\color{red}{\\var{diff1}}\\var{dd0}$ in the thousandths column. We made it to exactly $\\var{b1}$, so there is no remainder in the hundredths column and you can continue to the next column. However, for the sake of my own sanity (writing this solution), I will put a remainder of $\\color{red}{0}$ in front of the next digit as a superscript in order to represent $\\color{red}{0}\\var{dd0}$ in the thousandths column. You do not need to do this but you can if you wish.

$\\begin{array}{r} {\\var{qd3}\\,.\\,\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}\\var{qd0}}} \\\\[-.7cm] {\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3} \\,.\\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{red}{\\var{diff1}}\\var{dd0}}\\end{array}$

The thousandths column

Step 1: Now we ask ourselves, \"How many $\\color{green}{\\var{divisor1}}$s go into $\\color{green}{\\var{b0}}$?\" (note this $\\var{b0}$ actually represents $\\var{b0*0.001}$)

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:

$\\begin{array}{r} {\\var{qd3}\\,\\,.\\phantom{^\\var{diff3}}{\\var{qd2}}\\,\\phantom{^\\var{diff2}}{\\var{qd1}}\\phantom{\\,^\\var{diff1}}\\color{red}{\\var{qd0}}} \\\\[-.7cm] \\color{green}{\\var{divisor1}} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dd3}\\,. \\,^{\\var{diff3}}{\\var{dd2}}\\,^{\\var{diff2}}{\\var{dd1}}\\,^\\color{green}{\\var{diff1}}\\color{green}{\\var{dd0}}}\\end{array}$

Step 2: We could work out the remainder, 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 the second decimal place and so we now stop the procedure.

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

Long Division

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 $\\var{dividend1}$?\"

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:

$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}\\var{dividend1}}$

Note the positions of the numbers!

Actually, since we want the answer to two decimal places we add as many zeroes after the decimal place to ensure we have three decimal places!

$\\var{divisor1} \\strut \\overline{\\smash{\\raise.09ex{)}}{\\var{dd3}}.\\!\\var{dd2}\\var{dd1}\\var{dd0}}$

Why three? We use that extra digit to determine whether to round up or down.

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

Divide
Multiply
Subtract
Bring down

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\".

We need to know the $\\var{divisor1}$ times tables or write the $\\var{divisor1}$ times tables out (by repeatedly adding $\\var{divisor1}$) so that we can refer to them.

The ones column

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

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

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:

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

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.

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

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

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

The tenths column

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)

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

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:

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

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.

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

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

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

The hundredths column

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)

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

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:

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

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.

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

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

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

The thousandths column

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)

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

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:

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

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.

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

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.

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.).

"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}