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)