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.

A base-ten diagram representing 62. 6 rectangles labeled, 6 tens. 2 squares labeled, 2 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 \(62 \div 5\).

5 groups of base-ten diagrams. Each group consists of 1 rectangle labeled, tens, 2 squares labeled, ones, and 4 small rectangles labeled, tenths. 

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 \(62 \div 5\).

Long division calculation of 62 divided by 5, 4 steps.
  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.

A base-ten diagram representing 53 point 8. 5 rectangles labeled, 5 tens. 3 squares labeled, 3 ones. 8 small rectangles labeled, 8 tenths.

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.

4 groups of base-ten diagrams, Group 1, Group 2, Group 3, Group 4. 
  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?


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.

4 groups of base-ten diagrams.

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 \(86 \div 4 = 21.5\).

Long division calculation of 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 \(86 \div 4 = 21.5\).

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 \(513 = 128 \times 4 + 1\).

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