Lesson 19Dividing Numbers that Result in Decimals

Learning Goal

Let’s find quotients that are not whole numbers.

Learning Targets

I can divide a decimal by a whole number.
I can use long division to find the quotient of two whole numbers when the quotient is not a whole number.

Lesson Terms

long division

Warm Up: Keep Dividing

Mai used base-ten diagrams to calculate $62 \div 5.$ She started by representing 62.

A base ten diagram of 62 showing 6 rectangles of tens and 2 squares of ones.

She then made 5 groups, each with 1 ten. There was 1 ten left. She unbundled it into 10 ones and distributed the ones across the 5 groups.

Here is Mai’s diagram for $62 \div 5$ .

A base ten diagram of 62 with 5 groups of a tens rectangle, 2 ones squares, and 5 tenths rectangles.

Discuss these questions with a partner and write down your answers:

Problem 1

Mai should have a total of 12 ones, but her diagram shows only 10. Why?

Problem 2

She did not originally have tenths, but in her diagram each group has 4 tenths. Why?

Problem 3

What value has Mai found for $62 \div 5$ ? Explain your reasoning.

Activity 1: Using Long Division to Calculate Quotients

Problem 1

Here is how Lin calculated $62 \div 5$ .

A long division solution of 62 divided by 5

Discuss with your partner:

Lin put a 0 after the remainder of 2. Why? Why does this 0 not change the value of the quotient?
Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?
What value did Lin find for $62 \div 5$ ?

Problem 2

Use long division to find the value of each expression. Then pause so your teacher can review your work.

$126 \div 8$
$90 \div 12$

Problem 3

Use long division to show that:

$5 \div 4$ , or $\frac{5}{4}$ , is 1.25.
$4 \div 5$ , or $\frac{4}{5}$ , is 0.8.
$1 \div 8$ , or $\frac{1}{8}$ , is 0.125.
$1 \div 25$ , or $\frac{1}{25}$ , is 0.04.

Problem 4

Noah said we cannot use long division to calculate $10 \div 3$ because there will always be a remainder.

What do you think Noah meant by “there will always be a remainder”?
Do you agree with his statement? Why or why not?

Activity 2: Using Diagrams to Represent Division

To find $53.8 \div 4$ using diagrams, Elena began by representing 53.8.

She placed 1 ten into each group, unbundled the remaining 1 ten into 10 ones, and went on distributing the units.

This diagram shows Elena’s initial placement of the units and the unbundling of 1 ten.
Complete the diagram by continuing the division process. How would you use the available units to make 4 equal groups?
As the units get placed into groups, show them accordingly and cross out those pieces from the bottom. If you unbundle a unit, draw the resulting pieces.
What value did you find for $53.8 \div 4$ ? Be prepared to explain your reasoning.
Use long division to find $53.8 \div 4$ . Check your answer by multiplying it by the divisor 4.
Use long division to find $77.4 \div 5$ . If you get stuck, you can draw diagrams or use another method.

Are you ready for more?

A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.

A group of 4 craftsmen are paid 1 of each jewel. If they split the jewels evenly amongst themselves, which jewels does each craftsman get?

Lesson Summary

Dividing a whole number by another whole number does not always produce a whole-number quotient. Let’s look at $86 \div 4$ , which we can think of as dividing 86 into 4 equal groups.

A base ten diagram of 86 divided by 4. 86 is shown in 4 groups with 2 ones unbundled into 20 tenths squares.

We can see in the base-ten diagram that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we can unbundle or decompose the 2 ones into 20 tenths, which enables us to put 5 tenths in each group.

Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so $86 \div 4 = 21.5$ .

A long division solution to 86 divided by 4.

We can also calculate $86 \div 4$ using long division.

The calculation shows that, after removing 4 groups of 21, there are 2 ones remaining. We can continue dividing by writing a 0 to the right of the 2 and thinking of that remainder as 20 tenths, which can then be divided into 4 groups.

To show that the quotient we are working with now is in the tenth place, we put a decimal point to the right of the 1 (which is in the ones place) at the top. It may also be helpful to draw a vertical line to separate the ones and the tenths.

There are 4 groups of 5 tenths in 20 tenths, so we write 5 in the tenths place at the top. The calculation likewise shows $86 \div 4 = 21.5$ .