Remainders and what they mean

When you reach for it

The camp is going out on the lake. There are 34 campers, and each canoe holds 4. How many canoes does the camp need?

Division asks how many groups of 4 fit in 34. Eight groups of 4 is 32, and nine groups of 4 is 36, which is more than 34. So 8 canoes can be filled, and 34 - 32 = 2 campers are still on the dock. Those 2 campers are the remainder: what is left over when no more full groups can be made.

Nobody is left behind, so the camp needs a ninth canoe for the last 2. The answer to the question is 9, even though the division “said” 8.

The same division can answer other questions about the same trip:

• How many canoes go out completely full? 8.
• How many canoes does the camp need? 9.
• How many campers ride in the canoe that is not full? 2.

One division, three different answers. Reach for this lesson whenever a division in a story does not come out even. The division always gives the same two numbers, the quotient and the remainder. The story decides what to do with them.

The procedure

Part 1: what a remainder is

Dividing 34 by 4 can ask how many groups of 4 fit in 34, or how many each person gets when 34 is shared among 4; What division means has both meanings. Either way, make equal groups until you cannot make another one. Here are the canoes, one dot for each camper.

Eight full groups of 4 use up 32 campers, and 2 are left. Two campers cannot fill a group of 4, so we stop. We write

and read it “34 divided by 4 is 8, remainder 2”. Each number has a name:

The remainder is always less than the divisor. If 4 or more campers were still on the dock, they could fill one more canoe, and we would not be done grouping. So when the divisor is 4, the only possible remainders are 0, 1, 2 and 3. A remainder of 0 means the division comes out even: 32 \div 4 = 8 \text{ R } 0, or just 32 \div 4 = 8.

How to find the two numbers. The quotient is the number of groups in the biggest multiple of the divisor that is not more than the dividend. Count by 4s: 4, 8, 12, \ldots, 28, 32, 36. The number 36 is past 34, so the biggest multiple that fits is 32 = 4 \times 8. The quotient is 8, and the remainder is what is left: 34 - 32 = 2. For small numbers a multiplication fact does this in one step. For a bigger dividend, Partial quotients takes away groups in large chunks, and Dividing by a single digit works place by place; the quotient and remainder mean exactly the same thing either way.

The remainder sentence. Putting the groups back together gives back all the campers: 8 groups of 4, and 2 more.

This holds for every division with a remainder:

with the remainder less than the divisor. Both parts matter. The sentence 34 = 4 \times 7 + 6 is true too, but 6 is not less than 4: four of those six campers could still fill an eighth canoe. Only 8 and 2 make the sentence true and keep the remainder smaller than the divisor.

Part 2: what the story asks

The division gives a quotient and a remainder. The answer to the story is one of three things, and the question says which.

Dropping the leftover. Some questions ask only about full groups. A part-filled group does not count, because it is not what the question wants: a ride you cannot pay for in full, a muffin tin that is not full, a team that is short of players. The answer is the quotient, and the leftover is simply not part of it.

Going up to the next whole number. Other questions need every item to have a place: every camper in a canoe, every cupcake in a box. Then the leftover needs one more group, even though that group is not full. The answer is the quotient plus 1. This happens only when the remainder is not 0: with 32 campers, 8 canoes are exactly enough.

Reporting the remainder. Some questions ask about the leftover itself: how many campers are in the last canoe, how many cupcakes stay behind, how much money is left. Then the answer is the remainder, and the quotient is only the step on the way to it.

Two habits keep the choice honest. First, give the answer with its unit, “9 canoes”, not “9” or “8 \text{ R } 2”. A remainder is not an answer to “how many canoes”; it is part of the work. Second, read the answer back into the story. “Eight canoes” would leave 2 campers on the dock, which the story does not allow, so the answer must be 9.

Part 3: sharing the leftover, and checking

So far the leftover has been campers, and campers cannot be cut into pieces. Some things can be. When 7 cupcakes are shared equally by Kavya, Ngaio and Kjell, each first gets 2 whole cupcakes, because 7 \div 3 = 2 \text{ R } 1. The last cupcake does not have to stay on the plate. Cut it into 3 equal pieces, and each camper gets one piece, which is \displaystyle \frac{1}{3} of a cupcake.

Each camper’s share is 2 cupcakes and \displaystyle \frac{1}{3} of a cupcake, written \displaystyle 2\frac{1}{3} cupcakes. The remainder 1 has been shared too: one whole cut into 3 equal parts. The pieces always take their size from the divisor, because there is one piece for each share: sharing among 3 cuts thirds, and sharing among 2 cuts halves. In A fraction as division this becomes a rule of its own: sharing 1 whole among 3 gives \displaystyle \frac{1}{3} each.

Money can be shared the same way, by changing the leftover dollars into cents. One dollar is 100 cents. If 4 campers share 6 dollars, each gets 1 dollar and 2 dollars are left over (6 \div 4 = 1 \text{ R } 2). Two dollars is 200 cents, and 200 \div 4 = 50, so each camper gets 1 dollar and 50 cents more: 1.50 dollars, since 50 cents is 50 hundredths of a dollar (Tenths and hundredths). Sharing a leftover as a decimal like this is taken further in Dividing decimals.

So there is a fourth possible answer to a story: share the leftover too. It is right only when the leftover can be cut into equal pieces and the question asks for an equal share. Campers, canoes and minivans cannot be cut; cupcakes, dollars and lengths of rope can.

Checking a division. Multiplication undoes division, so a division with a remainder is checked by multiplying and adding the remainder:

must give back the dividend, and the remainder must be less than the divisor. For 34 \div 4: 4 \times 8 + 2 = 32 + 2 = 34, and 2 < 4. Both parts pass, so 8 \text{ R } 2 is right. A shared answer is checked by putting the shares back together: 4 campers with 1.50 dollars each have 4 dollars and 4 \times 50 = 200 cents, which is 4 + 2 = 6 dollars.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3, 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Boundaries

• The divisor cannot be 0. Groups of 0 never use anything up, so no number of them makes 34, and there is no quotient or remainder. Division by 0 is not defined.
• Whole numbers only, here. The dividend, divisor, quotient and remainder in this lesson are whole numbers (0, 1, 2, and so on), and the divisor is at least 1.
• The remainder is at least 0 and less than the divisor. A larger “remainder” means one more group fits (Example 2). Dividing by 1 always leaves remainder 0.
• “8 \text{ R } 2” is two numbers, not one. It is not a single amount, so two “R” answers should not be joined with an equals sign. Both 34 \div 4 and 42 \div 5 give 8 \text{ R } 2, yet 34 and 42 are different amounts, and \displaystyle \frac{2}{4} of a group is not \displaystyle \frac{2}{5} of one. Write the remainder sentence instead: 34 = 4 \times 8 + 2 and 42 = 5 \times 8 + 2.
• Going up needs a remainder. Add a group only when the remainder is not 0. Thirty-two campers fill exactly 8 canoes.
• Only some leftovers can be shared. Campers, canoes and rides cannot be cut, so their leftover is dropped, rounded up or reported. Money can be shared only down to whole cents: 1 dollar shared by 3 is 33 cents each with 1 cent left, since 100 = 3 \times 33 + 1.
• The division alone cannot pick the answer. The same 34 \div 4 answers 8, 9 or 2 depending on the question. Only the story decides.

Where this goes next

Every division from here on can leave a remainder, and the methods for bigger numbers produce one in the same way. Long division and the The standard long division algorithm carry a remainder from place to place, and Dividing by a two-digit number leaves a remainder less than a two-digit divisor. The fact that one quotient and one remainder smaller than the divisor always exist, for any whole number divided by any whole number other than 0, is The division algorithm.

A remainder of 0 is a special event: it means the divisor fits exactly, which is how factors are found. Sharing the leftover leads to fractions and decimals: the fraction lesson linked in Part 3 reads 7 \div 3 as the number \displaystyle \frac{7}{3}, and a mixed number such as \displaystyle 2\frac{1}{3} is rewritten that way in Mixed numbers and improper fractions: rewriting and comparing.

Some later ideas care only about the remainder. On a clock, 3 hours after 11 o’clock is 2 o’clock, because only the remainder after dividing by 12 matters; that is Modular arithmetic. And going up to the next whole number has a twin: if 34 campers sit in just 8 canoes, some canoe must hold at least 5, because 8 canoes of 4 hold only 32. That is the pigeonhole principle.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 17 with Part 2, and 18 to 22 with Part 3. The reasoning group draws on all three.

Part 1: quotient and remainder

1. Find 23 \div 5 and check it.
2. Find 41 \div 6 and check it.
3. Find 58 \div 9 and check it.
4. Find 35 \div 7. What is the remainder, and what does it tell you?
5. Draw 27 dots in groups of 4. How many full groups are there, and how many dots are left? Write the division.
6. List every remainder that is possible when you divide by 5.
7. Kjell wrote 44 \div 6 = 6 \text{ R } 8. Is he right? If not, fix it.
8. Ngaio wrote 31 \div 4 = 7 \text{ R } 3. Is she right? Use the remainder sentence.

Part 2: what the story asks

9. There are 22 campers, and a rowboat holds 3. How many rowboats are needed for everyone to go out at once?
10. Kavya has batter for 33 cupcakes. A muffin tin has 6 cups. How many tins can she fill completely?
11. The camp has 17 paddles, and each canoe needs 2. How many canoes get both of their paddles, and how many paddles are left over?
12. There are 53 campers, and each minivan has seats for 8 campers. How many minivans does the camp need?
13. Ngaio has 25 dollars. A canoe costs 6 dollars an hour to rent. For how many whole hours can she rent a canoe, and how much money is left?
14. There are 43 campers, and a canoe holds 5. (a) How many canoes are needed? (b) How many canoes go out full? (c) How many campers ride in the canoe that is not full?
15. Kjell puts 28 cupcakes into boxes of 6. How many full boxes can he make, and how many cupcakes are left out of a box?
16. Kavya needs 40 cupcakes for the end-of-camp party. A muffin tin has 6 cups. How many tins of batter must she bake?
17. Twenty-nine campers and 3 counselors are going to town. Each minivan has seats for 7 people. How many minivans are needed?

Part 3: sharing the leftover, and checking

18. Kjell and Ngaio share 9 cupcakes equally, cutting any cupcake that is left over. How much does each of them get?
19. Four campers share 9 dollars equally. How much does each camper get?
20. Kavya says 52 \div 7 = 7 \text{ R } 3. Check her answer.
21. Kjell says 61 \div 8 = 7 \text{ R } 6. Check his answer, and fix it if it is wrong.
22. Five campers split the 17 dollar cost of a rowboat equally. How much does each camper pay? Check by putting the shares back together.

Reasoning

23. Ngaio divided a number by 6 and got 7 \text{ R } 4. What was the number? Explain.
24. Use canoes that hold 4 campers to explain why a remainder can never be 4 or more.
25. Twenty campers go out in rowboats that hold 3 each. Write three questions about this trip: one whose answer is 6, one whose answer is 7, and one whose answer is 2. Say why each answer is right.