# Lesson 19

Dividing Numbers that Result in Decimals

Let’s find quotients that are not whole numbers.

### 19.1: Keep Dividing

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

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 $$62 \div 5$$.

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

1. Mai should have a total of 12 ones, but her diagram shows only 10. Why?
2. She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
3. What value has Mai found for $$62 \div 5$$? Explain your reasoning.

### 19.2: Using Long Division to Calculate Quotients

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

1. 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 $$62 \div 5$$?
2. Use long division to find the value of each expression. Then pause so your teacher can review your work.

1. $$126 \div 8$$
2. $$90 \div 12$$

3. Use long division to show that:

1. $$5 \div 4$$, or $$\frac 54$$, is 1.25.

2. $$4 \div 5$$, or $$\frac 45$$, is 0.8.

3. $$1 \div 8$$, or $$\frac 18$$, is 0.125.

4. $$1 \div 25$$, or $$\frac {1}{25}$$, is 0.04.

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

1. What do you think Noah meant by “there will always be a remainder”?
2. Do you agree with him? Explain your reasoning.

### 19.3: 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.

1. 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.

2. What value did you find for $$53.8 \div 4$$? Be prepared to explain your reasoning.

3. Use long division to find $$53.8 \div 4$$. Check your answer by multiplying it by the divisor 4.

4. Use long division to find $$77.4 \div 5$$. If you get stuck, you can draw diagrams or use another method.

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?

### 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.

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 $$86 \div 4 = 21.5$$.

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 $$86 \div 4 = 21.5$$.

### Glossary Entries

• long division

Long division is an algorithm for finding the quotient of two numbers expressed in decimal form. It works by building up the quotient one digit at a time, from left to right. Each time you get a new digit, you multiply the divisor by the corresponding base ten value and subtract that from the dividend.

Using long division we see that $$513 \div 4 = 128 \frac14$$. We can also write this as $$513 = 128 \times 4 + 1$$.

$$\displaystyle \require{enclose} \begin{array}{r} 128 \\[-3pt] 4 \enclose{longdiv}{513}\kern-.2ex \\[-3pt] \underline{4{00}} \\[-3pt] 113 \\[-3pt] \underline{\phantom{0}8{0}} \\[-3pt] \phantom{0}33 \\[-3pt] \underline{\phantom{0}32} \\[-3pt] \phantom{00}1 \end{array}$$