Dividing by a decimal

When you reach for it

Every spring, Gunnhild, Ludvik and Jokull help their grandmother make maple syrup. They collect sap from the trees and boil it down for hours until it turns thick and sweet. At the end of one day, Gunnhild has 4.2 liters of syrup. Each bottle holds 0.7 liter. How many bottles can she fill?

That is a “how many groups” question. How many groups of 0.7 fit in 4.2? Questions like that are division (see what division means), so the answer is 4.2 \div 0.7.

You have already divided a decimal by a whole number, like 8.37 \div 3, in Dividing decimals. There the size of the group, or the number of groups, was always a whole number. Now the number you divide by has a point in it.

Three words help. In 4.2 \div 0.7:

• the dividend is 4.2, the amount being split up;
• the divisor is 0.7, the size of one group;
• the quotient is the answer, the number of groups.

You reach for this lesson whenever the size of one group is a decimal with tenths. Liters of syrup per bottle. Meters of ribbon per bow. Kilograms of flour per bag. In every one, you ask how many of the small amounts fit in the big one.

This lesson takes three days. Part 1 counts tenths on a strip. Part 2 uses an area model and breaks the dividend into friendly parts. Part 3 shows that 4.2 \div 0.7 has the same answer as 42 \div 7, and why, and it deals with half a group left over.

The procedure

Part 1: how many groups of tenths fit?

The key move is to count everything in tenths. A tenth of a liter is one small piece, and ten of those pieces make one liter (see Tenths and hundredths).

• 4.2 liters is 4 liters and 2 tenths. Each liter is 10 tenths, so 4 liters are 40 tenths. With the 2 more, that is 42 tenths.
• 0.7 liter is 7 tenths.

So the question becomes: how many groups of 7 tenths fit in 42 tenths? Lay the 42 tenths in a strip and mark off groups of 7.

Count the groups as they fill up: 7, 14, 21, 28, 35, 42. That is 6 groups with nothing left over. So

Gunnhild fills 6 bottles.

Check by multiplying. Division and multiplication undo each other (see Fact families). If 6 bottles each hold 0.7 liter, they hold 6 \times 0.7 liters in all (see Multiplying decimals). That is 6 \times 7 = 42 tenths, which is 4.2 liters. The answer is right.

The answer is bigger than 4.2. That can feel odd, since dividing a whole number by a whole number never gives more than you started with. But here each group is smaller than one liter. A bottle holds less than a liter, so more bottles than liters are needed. Whenever the divisor is less than 1, more than one group fits in each whole, and the quotient is bigger than the dividend.

Part 2: the area model

A strip of tenths works, but it gets long. For 9.1 liters you would need 91 cells. An area model is faster.

Draw a rectangle. Make one side the divisor, 0.7, and make the area the dividend, 4.2. A rectangle’s area is one side times the other (see The area model), so the missing side is the number that times 0.7 gives 4.2. That number is the quotient.

To find the missing side, break the area into parts you can divide in your head. A few groups of 0.7 are easy to know:

• 1 group of 0.7 is 0.7;
• 2 groups make 1.4;
• 5 groups make 3.5;
• 10 groups make 7.

For 4.2: ten groups make 7, which is too much. Five groups make 3.5, which fits. That leaves 4.2 - 3.5 = 0.7, and 0.7 is exactly one more group. So the area splits as 3.5 + 0.7, and the lengths are 5 and 1.

Why the split is allowed. Cutting the rectangle into two pieces cuts its area into two pieces, and the two lengths along the top add up to the whole length. In symbols, this is the distributive property:

Ten groups is the friendliest part. Ten groups of 0.7 is 0.7 \times 10 = 7. Multiplying by ten moves every digit one place to the left (see Multiplying a decimal by ten, one hundred or one thousand), so the 7 tenths become 7 ones. Whenever the dividend is bigger than ten groups, start with ten. This is how Example 4 finds 9.1 \div 0.7.

Part 3: the same count in whole numbers

Look back at Part 1. Once both numbers were counted in tenths, the question was “how many groups of 7 fit in 42?” That is just 42 \div 7. The word “tenths” was on both numbers, so it did not change the count. It is like asking how many 7-cent stickers you can buy with 42 cents: the answer is how many 7s fit in 42.

There is a second way to see it. Multiply both numbers by ten. Every digit moves one place to the left, so 4.2 \times 10 = 42 and 0.7 \times 10 = 7. The 42 tenths become 42 ones, and the 7 tenths become 7 ones. Every piece got ten times bigger, but there are still 42 pieces and still 7 in each group. The count of tenths does not change, so the quotient does not change.

The same works when the dividend is a whole number. Two liters is 20 tenths, so 2 \div 0.5 = 20 \div 5 = 4, both numbers again multiplied by ten.

Half a group left over. Not every division comes out in whole groups. Ludvik has 3.8 liters and 0.4-liter bottles. In tenths, that is 38 tenths in groups of 4 tenths, so 3.8 \div 0.4 = 38 \div 4.

Now 38 \div 4 is a whole number divided by a whole number, which you can carry past the ones place as in Dividing decimals. Write 38 as 38.0. The 38 ones give 9 in each group, using 36, with 2 ones left. Trade them for 20 tenths, and 20 tenths give 5 tenths each.

Read that in the story. Ludvik fills 9 bottles, which hold 9 \times 0.4 = 3.6 liters. The 3.8 - 3.6 = 0.2 liter left is half of a 0.4-liter bottle. That is the 0.5 in 9.5: nine bottles and half a bottle more.

Worked examples

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

Boundaries

• The divisor is never 0. The question “how many groups of 0 fit in 4.2?” has no answer, because no number of empty groups ever adds up to 4.2. Division by 0 is not defined.
• The divisors here have tenths only. A divisor such as 0.45 has hundredths. It is counted in hundredths and both numbers are multiplied by 100. That general rule, with long division for any decimal divisor, is in Dividing multi-digit decimals with the standard algorithm.
• Both numbers are multiplied by the same ten. Multiplying only one of them changes the question. 42 \div 0.7 = 60, ten times as many bottles, because there is ten times as much syrup. And 4.2 \div 7 = 0.6, because now each group is ten times bigger. Only 42 \div 7 keeps the count of 6.
• The groups do not always come out whole. 3.8 \div 0.4 = 9.5 is nine groups and half a group. In a story, read what the question wants: 9 full bottles, or 10 bottles to hold all the syrup.
• A divisor less than 1 gives a quotient bigger than the dividend, as long as the dividend is more than 0. And when the dividend is smaller than the divisor, less than one group fits: 0.3 \div 0.6 = 0.5, half a group.

Where this goes next

The next step lets the divisor have hundredths and more, like 3.15 \div 0.45. That is Dividing multi-digit decimals with the standard algorithm: multiply both numbers by the same power of ten until the divisor is a whole number, which is the same idea as Part 3, and then use long division.

The question “how many of these fit in that?” also comes back with fractions in Dividing fractions: how many fit. Since 0.7 is \displaystyle \frac{7}{10}, asking how many groups of 0.7 fit in 4.2 is the same kind of question as asking how many \displaystyle \frac{3}{4}-cup scoops fit in 6 cups. And stories that mix all four operations with decimals come back in Decimal problems in words.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 15 with Part 2, and 16 to 22 with Part 3. The reasoning group draws on all three.

Part 1: how many groups of tenths fit?

1. How many tenths are in each number? (a) 3.6, (b) 0.9, (c) 5.
2. Count tenths to find each quotient: (a) 1.6 \div 0.2, (b) 2.1 \div 0.3, (c) 3.2 \div 0.4.
3. Count tenths to find each quotient: (a) 3 \div 0.5, (b) 1 \div 0.1, (c) 2 \div 0.4.
4. Gunnhild pours 3.6 liters of syrup into 0.4-liter bottles. How many bottles does she fill? Check by multiplying.
5. Ludvik has 4.5 liters of syrup and jugs that hold 0.9 liter. How many jugs can he fill?
6. Describe a tenths strip for 1.2 \div 0.3: how many cells it has, how the groups are marked, and what the quotient is.
7. How many 0.2-liter cups of syrup can be poured from a 1.4-liter jug?
8. At the pancake breakfast, each plate gets 0.1 liter of syrup. How many plates does 2.5 liters serve?

Part 2: the area model

9. Use an area model with the area split as 4 + 2.4 to find 6.4 \div 0.8.
10. Use the split 8.4 = 7 + 1.4 to find 8.4 \div 0.7.
11. Ten groups of 0.6 make 6. Use that to find 7.8 \div 0.6.
12. Use an area model to find 9.6 \div 0.8. Say how you split the area.
13. A garden bed is shaped like a rectangle with area 5.4 square meters. One side is 0.6 meter. How long is the other side?
14. Gunnhild fills 0.3-liter bottles from 6.6 liters of syrup. Use an area model to find how many bottles she fills.
15. Use an area model to find 12.6 \div 0.9.

Part 3: the same count in whole numbers

16. Write each as a division of whole numbers with the same quotient, then find it: (a) 4.9 \div 0.7, (b) 8.1 \div 0.9, (c) 3 \div 0.6.
17. Find each quotient: (a) 6.3 \div 0.7, (b) 4.8 \div 0.6, (c) 7 \div 0.5.
18. Each quotient has a half: (a) 1.5 \div 0.2, (b) 3.3 \div 0.6.
19. Ludvik has 4.6 liters of syrup and 0.4-liter bottles. Find 4.6 \div 0.4. How many bottles does he fill all the way, and how much syrup is left?
20. Jokull’s family boils down 9 liters of syrup and pours it into 0.6-liter bottles. How many bottles do they fill?
21. The family makes 8 liters of syrup, keeps 0.8 liter for the kitchen, and pours the rest into 0.9-liter bottles. How many bottles do they fill?
22. Find 2.8 \div 0.7 and 28 \div 0.7. How do the two quotients compare, and why?

Reasoning

23. Explain, using tenths, why 5.4 \div 0.9 and 54 \div 9 have the same quotient.
24. Jokull says 0.8 \div 0.2 must be less than 0.8, because dividing always makes a number smaller. Find the quotient and explain what is wrong with his reason.
25. The same amount of syrup is poured into 0.6-liter bottles or into 0.8-liter bottles. Without dividing, say which size fills more bottles, and why. Then check with 2.4 liters.