Interpreting Quotients and Finding Division Mistakes | Grade 6 Math | LevelKite

You already know how to divide multi-digit numbers with the standard algorithm, and how to check a quotient by multiplying. This lesson is about the end of a division. What does the leftover number mean, and how should you report it?

First, the words. In $13 \div 4$ , the dividend is $13$ , the number being divided. The divisor is $4$ , the number you divide by. The quotient is the result of the division. The whole-number quotient is $3$ , because $3$ full groups of $4$ fit in $13$ . The remainder is $1$ , the whole-number amount left over after those full groups. You will often see this written with the letter R, as $3 \text{ R } 1$ .

In this lesson, every dividend and every divisor is a whole number. The divisor is never zero, because you cannot divide by zero. When a whole-number division is finished, the remainder is at least $0$ and less than the divisor.

Here is that same division told as two stories.

You cut $13$ meters of ribbon into $4$ equal pieces. Ribbon can be cut anywhere, so the $1$ leftover meter is cut into $4$ equal parts too. Each piece is $3\tfrac{1}{4}$ meters long. A number like $3\tfrac{1}{4}$ , a whole number plus a fraction, is called a mixed number. As a decimal, each piece is $3.25$ meters.
You share $13$ marbles equally among $4$ friends. A marble cannot be split. Each friend gets $3$ marbles, and $1$ marble is left over.

Both stories begin with the same arithmetic: $4 \times 3 = 12$ , and $13 - 12 = 1$ is left. The difference is what happens to that $1$ . A remainder is a division that is not finished yet. In the ribbon story you finish it by dividing the $1$ by $4$ as well. In the marble story you cannot, so the $1$ stays a remainder.

Here is the ribbon story in symbols.

13 \div 4 = 3 + \frac{1}{4} = 3\tfrac{1}{4} = 3.25

Key concept

A remainder is a division that is not finished

Divide by a divisor that is not zero. If the remainder is not $0$ , the story decides what happens to it.

One unit can be split: divide the remainder by the divisor too. It becomes the fraction $\frac{\text{remainder}}{\text{divisor}}$ , or the matching decimal. This quotient is exact (nothing is rounded), so $\text{quotient} \times \text{divisor} = \text{dividend}$ . A rounded decimal gives a product that is only close to the dividend.
Items cannot be split: keep a whole-number quotient and a remainder. Check that $\text{divisor} \times \text{whole-number quotient} + \text{remainder} = \text{dividend}$ .

If the remainder is $0$ , nothing is left over. Report the whole-number quotient alone, with no remainder and no fraction.

\text{dividend} \div \text{divisor} = \text{whole-number quotient} + \frac{\text{remainder}}{\text{divisor}}

A mixed number and a decimal are two ways to write a finished quotient. Both need a divisor that is not zero.

Mixed number. Write the whole-number quotient. Next to it, write a fraction with the remainder as the numerator (the top number) and the divisor as the denominator (the bottom number). Simplify the fraction if you can.

Decimal. Keep dividing past the ones place. One whole equals $10$ tenths, so trade the remainder for tenths and divide again. Trade any leftover tenths for hundredths, and keep going until nothing is left. In the ribbon story, the $1$ leftover meter is $10$ tenths. Dividing by $4$ gives $2$ tenths, because $4 \times 2 = 8$ , and $2$ tenths are left. Those are $20$ hundredths, and $20 \div 4 = 5$ , so you get $5$ hundredths. That is why $\frac{1}{4} = 0.25$ .

Some divisions never reach a remainder of $0$ . Then you round to the place the problem asks for. A rounded quotient is close to the exact quotient, not equal to it.

Worked example

Write a remainder as a fraction and as a decimal

Divide $1{,}794 \div 24$ . Write the quotient as a mixed number and as a decimal.

Divide with the standard algorithm. 24 × 7 = 168, and 179 − 168 = 11. Bring down the 4 to make 114. 24 × 4 = 96, and 114 − 96 = 18. The whole-number quotient is 74. The remainder is 18, which is less than 24, so no more full groups fit. The line below confirms it: 24 groups of 74, plus the 18 left over, rebuild the dividend.
$24 \times 74 + 18 = 1{,}776 + 18 = 1{,}794$
Finish the division. The 18 leftover ones must be divided by 24 as well. 18 divided by 24 is the fraction with numerator 18 and denominator 24. Add that fraction to the whole-number quotient to get a mixed number.
$1{,}794 \div 24 = 74 + \frac{18}{24} = 74\tfrac{18}{24}$
Simplify the fraction. Divide the numerator and the denominator by the same nonzero number, 6. The value does not change, because that is the same as dividing the fraction by 6 over 6, which equals 1.
$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$
Now the decimal. Trade the 18 leftover ones for 180 tenths, because each one equals 10 tenths. 24 × 7 = 168, so the quotient gets 7 tenths, and 12 tenths are left.
$180 - 24 \times 7 = 180 - 168 = 12$
Trade the 12 tenths for 120 hundredths. 24 × 5 = 120, so the quotient gets 5 hundredths and nothing is left. The division is finished. 74 ones, 7 tenths and 5 hundredths make 74.75. This agrees with the fraction, because three fourths is 0.75.
$120 - 24 \times 5 = 120 - 120 = 0$
Check. Multiplying by 24 undoes dividing by 24, because 24 is not zero. Both answers are exact, so the product must equal the dividend exactly. The mixed number passes the same check, since three fourths of 24 is 18.
$74.75 \times 24 = 74 \times 24 + 0.75 \times 24 = 1{,}776 + 18 = 1{,}794$

Answer

$1{,}794 \div 24$ equals $74\tfrac{3}{4}$ as a mixed number and $74.75$ as a decimal.

In a word problem, the numbers have units, and the story tells you how to report the quotient. Before you divide, ask two questions.

What does the quotient count or measure? Name its unit, such as bookmarks per classroom or liters per tank.
Can one of those units be split and still make sense? A liter of water can. A bookmark cannot.

Then divide, and choose the form that fits the story. The table shows the three cases. In every case the divisor is a whole number that is not zero.

Let the story choose how to report the quotient

What the quotient counts or measures	Can one unit be split?	How to report the result
A measured amount for each share, such as meters, liters or hours	Yes	A mixed number or a decimal. The remainder becomes a fractional share.
Whole items for each share, such as bookmarks, marbles or people	No	A whole-number quotient and a remainder. The leftover items stay whole.
Either kind, when the remainder is $0$	Nothing is left to split	The whole-number quotient alone. There is no remainder and no fractional share.

Worked example

Share items that cannot be split

A librarian shares $3{,}125$ bookmarks equally among $48$ classrooms. Each classroom gets the same whole number of bookmarks. How many bookmarks does each classroom get, and how many are left over?

Name the quantities. The dividend is 3,125 bookmarks. The divisor is 48 classrooms, which is not zero. The quotient is measured in bookmarks per classroom. A bookmark cannot be split, so the answer needs a whole-number quotient and a remainder.
$3{,}125 \div 48$
Divide with the standard algorithm. 48 × 6 = 288, and 312 − 288 = 24. Bring down the 5 to make 245. 48 × 5 = 240, and 245 − 240 = 5. No digits are left to bring down, so the whole-number quotient is 65 and the remainder is 5.
$312 - 288 = 24, \qquad 245 - 240 = 5$
Test the remainder. It must be less than the divisor, or another full round of sharing would fit. 5 is less than 48, so there are not enough bookmarks left to give every classroom one more.
$5 \lt 48$
Check. These are whole items with leftovers, so multiply the divisor by the whole-number quotient and add the remainder. 48 classrooms with 65 bookmarks each use 3,120 bookmarks. The 5 left over bring the total back to the dividend.
$48 \times 65 + 5 = 3{,}120 + 5 = 3{,}125$
Leave the remainder as a remainder. Writing 5 over 48 would give each classroom a fraction of a bookmark, and that makes no sense in this story. The 5 bookmarks stay whole.

Answer

Each classroom gets $65$ bookmarks, and $5$ bookmarks are left over.

Worked example

Share an amount that can be split

A farmer pours $2{,}150$ liters of water equally into $12$ tanks. How many liters go into each tank? Give the exact amount as a mixed number. Then give a decimal rounded to the nearest hundredth of a liter.

Name the quantities. The dividend is 2,150 liters. The divisor is 12 tanks, which is not zero. The quotient is measured in liters per tank. Water can be split into parts of a liter, so the remainder gets shared too.
$2{,}150 \div 12$
Divide with the standard algorithm. 12 × 1 = 12, and 21 − 12 = 9. Bring down the 5 to make 95. 12 × 7 = 84, and 95 − 84 = 11. Bring down the 0 to make 110. 12 × 9 = 108, and 110 − 108 = 2. The whole-number quotient is 179 and the remainder is 2, which is less than 12.
$12 \times 179 + 2 = 2{,}148 + 2 = 2{,}150$
Share the remainder. 2 leftover liters divided among 12 tanks gives each tank 2 over 12 of a liter. Divide the numerator and the denominator by 2 to simplify. The value stays the same, because 2 over 2 equals 1.
$179 + \frac{2}{12} = 179\tfrac{2}{12} = 179\tfrac{1}{6}$
Check the exact answer. This mixed number is an exact quotient, so multiplying it by the divisor, which is not zero, must give the dividend exactly. One sixth of 12 is 2.
$179\tfrac{1}{6} \times 12 = 179 \times 12 + \tfrac{1}{6} \times 12 = 2{,}148 + 2 = 2{,}150$
Write the decimal. Trade 2 ones for 20 tenths: 12 × 1 = 12, so 1 tenth, and 8 tenths are left. Trade for 80 hundredths: 12 × 6 = 72, so 6 hundredths, and 8 are left. The same 8 comes back every time, so the digit 6 repeats without end. Round to the nearest hundredth. The thousandths digit is 6, so the hundredths digit goes up from 6 to 7.
$2{,}150 \div 12 = 179.166\ldots \approx 179.17$
Check the rounded answer. A rounded quotient is not exact, so its product with the divisor is only close to the dividend. Here the product is 0.04 more than 2,150. That small gap comes from rounding up, not from a mistake.
$179.17 \times 12 = 2{,}148 + 2.04 = 2{,}150.04 \approx 2{,}150$

Answer

Each tank gets exactly $179\tfrac{1}{6}$ liters, which is about $179.17$ liters.

Match the check to the form of the answer (the divisor is never zero)

Form of the answer	Example	Check	What the check gives
Whole-number quotient and remainder, for items that cannot be split	$3{,}125 \div 48$ gives $65 \text{ R } 5$	$48 \times 65 + 5 = 3{,}125$ , and $5$ is less than $48$	Exactly the dividend
Mixed number	$2{,}150 \div 12 = 179\tfrac{1}{6}$	$179\tfrac{1}{6} \times 12 = 2{,}150$	Exactly the dividend
Decimal that ends	$1{,}794 \div 24 = 74.75$	$74.75 \times 24 = 1{,}794$	Exactly the dividend
Rounded decimal	$2{,}150 \div 12 \approx 179.17$	$179.17 \times 12 = 2{,}150.04$	A number close to the dividend, not equal to it
Whole number, when the remainder is $0$	$1{,}776 \div 24 = 74$	$74 \times 24 = 1{,}776$	Exactly the dividend, with no remainder and no fraction

A failed check tells you that something is wrong. It does not tell you where. To find a mistake in long division, test the work one step at a time. Each step has a fact that must be true when the work is correct. The table lists these facts. They apply when the dividend and the divisor are whole numbers and the divisor is not zero.

Facts that must be true at each step of long division

Step	What must be true	Sign of a mistake
Multiply	The product equals the new quotient digit times the divisor.	A wrong product, such as $24 \times 7$ written as $158$ instead of $168$ .
Subtract	The difference is correct, at least $0$ , and less than the divisor.	A difference that is not less than the divisor. The quotient digit is too small.
Bring down	After the first quotient digit, each digit you bring down gets one quotient digit, even if that digit is $0$ .	Two digits brought down, but only one quotient digit written.
Report the remainder	A remainder that is not $0$ goes over the divisor before it becomes a decimal.	The digits of the remainder copied after the decimal point.

Worked example

Find the mistake in a long division

A student divides $7{,}344 \div 36$ and gets $24$ . The student's work: $36$ goes into $73$ two times; $73 - 72 = 1$ ; bring down $4$ to make $14$ ; bring down $4$ to make $144$ ; $36$ goes into $144$ four times; $144 - 144 = 0$ . Find the mistake and give the correct quotient.

Check the answer first. The work ends with nothing left, so 24 is meant to be an exact quotient. An exact quotient times the divisor, which is not zero, equals the dividend. This product is far too small, so the work has a mistake.
$24 \times 36 = 720 + 144 = 864 \neq 7{,}344$
Test each multiply and subtract step. 36 × 2 = 72 and 73 − 72 = 1 are correct, and 1 is less than 36. 36 × 4 = 144 and 144 − 144 = 0 are correct too. The arithmetic is fine, so look at the bring-down steps.
Test each bring-down. After writing the 2, the student brought down a 4 to make 14. That digit needs its own quotient digit. 36 goes into 14 zero times, so a 0 belongs in the quotient. The student skipped it and brought down a second digit.
$36 \times 0 = 0, \qquad 14 - 0 = 14$
Fix the place values. 73 means 73 hundreds, so the 2 stands for 2 hundreds. The 14 means 14 tens, which gives 0 tens. The 144 means 144 ones, which gives 4 ones.
$2 \text{ hundreds} + 0 \text{ tens} + 4 \text{ ones} = 204$
Check the corrected quotient. The remainder is 0, so 204 is exact, and its product with the divisor must equal the dividend.
$204 \times 36 = 7{,}200 + 144 = 7{,}344$

Answer

The student left out the $0$ in the tens place. The correct quotient is $204$ , with no remainder.

One more mistake is harder to catch, because part of the check still works. Suppose someone divides $1{,}794 \div 24$ and, at the last step, says $24$ goes into $114$ three times. They subtract $114 - 72 = 42$ and report $73 \text{ R } 42$ . The numbers do rebuild the dividend: $24 \times 73 + 42 = 1{,}752 + 42 = 1{,}794$ . But $42$ is not less than $24$ , so the division stopped too soon. One more group of $24$ fits, since $42 - 24 = 18$ . The whole-number quotient goes up by $1$ to $74$ , and the remainder is $18$ .

So the check for a whole-number quotient with a remainder has two parts. The divisor, which is not zero, times the whole-number quotient, plus the remainder, must equal the dividend. And the remainder must be less than the divisor.

The takeaway: a remainder is a division that is not finished, and the story tells you whether to finish it.

When one unit can be split, write the remainder over the divisor and report a mixed number or a decimal.
When the items must stay whole, keep the whole-number quotient and the remainder.
When the remainder is $0$ , report the whole-number quotient alone.

Match the check to the form, with a divisor that is not zero. An exact quotient times the divisor equals the dividend. A rounded quotient only comes close. For whole items with leftovers, divisor times whole-number quotient plus remainder equals the dividend, and the remainder is less than the divisor. When a check fails, test each multiply, subtract and bring-down step to find the mistake.