# Lesson 11

Dividing Numbers that Result in Decimals

Let’s find quotients that are not whole numbers.

### 11.1: Number Talk: Evaluating Quotients

Find the quotients mentally.

$$400\div8$$

$$80\div8$$

$$16\div8$$

$$496\div8$$

### 11.2: 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 $$62 \div 5$$.

1. 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.
2. Find the quotient of $$511 \div 5$$ by drawing base-ten diagrams or by using the partial quotients method. Show your reasoning. If you get stuck, work with your partner to find a solution.
3. Four students share a \$271 prize from a science competition. How much does each student get if the prize is shared equally? Show your reasoning.

### 11.3: Using Long Division to Calculate Quotients

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

• 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.

### 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 a way to show the steps for dividing numbers in decimal form. It finds the quotient one digit at a time, from left to right.

For example, here is the long division for $$57 \div 4$$.

$$\displaystyle \require{enclose} \begin{array}{r} 14.25 \\[-3pt] 4 \enclose{longdiv}{57.00}\kern-.2ex \\[-3pt] \underline{-4\phantom {0}}\phantom{.00} \\[-3pt] 17\phantom {.00} \\[-3pt]\underline{-16}\phantom {.00}\\[-3pt]{10\phantom{.0}} \\[-3pt]\underline{-8}\phantom{.0}\\ \phantom{0}20 \\[-3pt] \underline{-20} \\[-3pt] \phantom{00}0 \end{array}$$