22.3: Using Diagrams to Find the Number of Groups

Lesson

Let's draw tape diagrams to think about division with fractions.

Exercise $\PageIndex{1}$: How Many of These in That?

1. We can think of the division expression $10\div 2\frac{1}{2}$ as the question: “How many groups of $2\frac{1}{2}$ are in 10?” Complete the tape diagram to represent this question. Then find the answer.

Figure $\PageIndex{1}$

2. Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.

Figure $\PageIndex{2}$

Exercise $\PageIndex{2}$: Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many $\frac{2}{3}$s are in 1?,” Andre wrote equations and drew a tape diagram.

$?\cdot\frac{2}{3}=1$

$1\div\frac{2}{3}=?$

Figure $\PageIndex{3}$: A tape diagram with three equal parts. The first two parts are shaded and are each labeled one third, total 1. A bracket is labeled 1 group of two thirds, and contains the first two parts.

1. In an earlier task, we used pattern blocks to help us solve the equation $1\div\frac{2}{3}=?$. Explain how Andre’s tape diagram can also help us solve the equation.
2. Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.

1. How many $\frac{3}{4}$s are in $1$?

Figure $\PageIndex{4}$

2. How many $\frac{2}{3}$s are in $3$?

Figure $\PageIndex{5}$

3. How many $\frac{3}{2}$s are in $5$?

Figure $\PageIndex{6}$

Exercise $\PageIndex{3}$: Finding Number of Groups

1. Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.
   1. How many $\frac{3}{8}$-inch thick books make a stack that is 6 inches tall?
   2. How many groups of $\frac{1}{2}$ pound are in $2\frac{3}{4}$ pounds?
2. Write a question that can be represented by the division equation $5\div 1\frac{1}{2}=?$. Then, find the answer and explain or show your reasoning.

Summary

A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for $\frac{2}{5}$ kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of $\frac{2}{5}$ kilogram make 2 kilograms?” and represent that question with the equations:

$?\cdot\frac{2}{5}=2$

$2\div\frac{2}{5}=?$

To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.

Figure $\PageIndex{7}$: Fraction bar diagram. 10 equal parts. Each part labeled the fraction 1 over 5. Total labeled 2 kilograms and question mark batches. Every two parts labeled 1 batch.

We can see there are 5 groups of $\frac{2}{5}$ in 2. Multiplying 5 and $\frac{2}{5}$ allows us to check this answer: $5\cdot\frac{2}{5}=\frac{10}{5}$ and $\frac{10}{5}=2$, so the answer is correct.

Notice the number of groups that result from $2\div\frac{2}{5}$ is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:

Suppose one serving of rice is $\frac{3}{4}$ cup. How many servings are there in $3\frac{1}{2}$ cups?

$?\cdot\frac{3}{4}=3\frac{1}{2}$

$3\frac{1}{2}\div\frac{3}{4}=?$

Figure $\PageIndex{8}$: Fraction bar diagram. 16 equal parts. 14 parts shaded. 14 parts labeled unknown number of groups and 3 and one half cups. Each part labeled the fraction 1 over 4. First 3 parts labeled 1 serving.

Looking at the diagram, we can see there are 4 full groups of $\frac{3}{4}$, plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make $\frac{2}{3}$ of a group. So the number of servings (the “?” in each equation) is $4\frac{2}{3}$. We can check this by multiplying $4\frac{2}{3}$ and $\frac{3}{4}$.

$4\frac{2}{3}\cdot\frac{3}{4}=\frac{14}{3}\cdot\frac{3}{4}$, and $\frac{14}{3}\cdot\frac{3}{4}=\frac{14}{4}$, which is indeed equivalent to $3\frac{1}{2}$.

Practice

Exercise $\PageIndex{4}$

We can think of $3\div\frac{1}{4}$ as the question “How many groups of $\frac{1}{4}$ are in $3$?” Draw a tape diagram to represent this question. Then find the answer.

Exercise $\PageIndex{5}$

Describe how to draw a tape diagram to represent and answer $3\div\frac{3}{5}=?$ for a friend who was absent.

Exercise $\PageIndex{6}$

How many groups of $\frac{1}{2}$ day are in 1 week?

1. Write a multiplication equation or a division equation to represent the question.
2. Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.

Exercise $\PageIndex{7}$

Diego said that the answer to the question “How many groups of $\frac{5}{6}$ are in $1$?” is $\frac{6}{5}$ or $1\frac{1}{5}$. Do you agree with him? Explain or show your reasoning.

Exercise $\PageIndex{8}$

Select all the equations that can represent the question: “How many groups of $\frac{4}{5}$ are in $1$?”

1. $?\cdot 1=\frac{4}{5}$
2. $1\cdot\frac{4}{5}=?$
3. $\frac{4}{5}\div 1=?$
4. $?\cdot\frac{4}{5}=1$
5. $1\div\frac{4}{5}=?$

(From Unit 4.2.2)

Exercise $\PageIndex{9}$

Calculate each percentage mentally.

1. What is $10$% of $70$?
2. What is $10$% of $110$?
3. What is $25$% of $160$?
4. What is $25$% of $48$?
5. What is $50$% of $90$?
6. What is $50$% of $350$?
7. What is $75$% of $300$?
8. What is $75$% of $48$?

(From Unit 3.4.5)