22.4: What Fraction of a Group?

Lesson

Let's think about dividing things into groups when we can't even make one whole group.

Exercise \(\PageIndex{1}\): Estimating a Fraction of a Number

1. Estimate the quantities:
   1. What is \(\frac{1}{3}\) of \(7\)?
   2. What is \(\frac{4}{5}\) of \(9\frac{2}{3}\)?
   3. What is \(2\frac{4}{7}\) of \(10\frac{1}{9}\)?
2. Write a multiplication expression for each of the previous questions.

Exercise \(\PageIndex{2}\): Fractions of Ropes

The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.

1. Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.
   1. Rope B is _______ times as long as rope A.
   2. Rope C is _______ times as long as rope A.
   3. Rope D is _______ times as long as rope A.
2. Each equation can be used to answer a question about Ropes C and D. What could each question be?
   1. \(?\cdot 3=9\) and \(9\div 3=?\)
   2. \(?\cdot 9=3\) and \(3\div 9=?\)

Exercise \(\PageIndex{3}\): Fractional Batches of Ice Cream

One batch of an ice cream recipe uses 9 cups of milk. A chef makes different amounts of ice cream on different days. Here are the amounts of milk she used:

• Monday: \(12\) cups
• Tuesday: \(22\frac{1}{2}\) cups
• Thursday: \(6\) cups
• Friday: \(7\frac{1}{2}\) cups

1. How many batches of ice cream did she make on these days? For each day, write a division equation, draw a tape diagram, and find the answer.

1. Monday

Figure \(\PageIndex{1}\)

2. Tuesday

Figure \(\PageIndex{2}\)

2. What fraction of a batch of ice cream did she make on these days? For each day, write a division equation, draw a tape diagram, and find the answer.

1. Thursday

Figure \(\PageIndex{3}\)

2. Friday

Figure \(\PageIndex{4}\)

3. For each question, write a division equation, draw a tape diagram, and find the answer.

1. What fraction of \(9\) is \(3\)?

Figure \(\PageIndex{5}\)

2. What fraction of \(5\) is \(\frac{1}{2}\)?

Figure \(\PageIndex{6}\)

Summary

It is natural to think about groups when we have more than one group, but we can also have a fraction of a group.

To find the amount in a fraction of a group, we can multiply the fraction by the amount in the whole group. If a bag of rice weighs 5 kg, \(\frac{3}{4}\) of a bag would weigh \(\left(\frac{3}{4}\cdot 5\right)\) kg.

Figure \(\PageIndex{7}\): Fraction bar diagram. 4 equal parts. 3 parts shaded. Total labeled 1 bag and 5 kilograms. 3 parts labeled the fraction 3 over 4 bag and parentheses the fraction 3 over 4 times 5 kilograms.

Sometimes we need to find what fraction of a group an amount is. Suppose a full bag of flour weighs 6 kg. A chef used 3 kg of flour. What fraction of a full bag was used? In other words, what fraction of 6 kg is 3 kg?

This question can be represented by a multiplication equation and a division equation, as well as by a diagram.

\(?\cdot 6=3\)

\(3\div 6=?\)

Figure \(\PageIndex{8}\): A tape diagram of 6 equal parts. Above the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled 6 kilograms. Below the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled 1 bag. A third brace that contains the first three parts is labeled three kilograms. Below the diagram, a fourth brace which also contains the first three parts is labeled question mark bag.

We can see from the diagram that 3 is \(\frac{1}{2}\) of 6, and we can check this answer by multiplying: \(\frac{1}{2}\cdot 6=3\).

In any situation where we want to know what fraction one number is of another number, we can write a division equation to help us find the answer.

For example, “What fraction of 3 is \(2\frac{1}{4}\)?” can be expressed as \(?\cdot 3=2\frac{1}{4}\), which can also be written as \(2\frac{1}{4}\div 3=?\).

The answer to “What is \(2\frac{1}{4}\div 3\)?” is also the answer to the original question.

Figure \(\PageIndex{9}\): Fraction bar diagram. 12 equal parts. 9 parts shaded. Total labeled 1 group and 3 cups. 9 parts labeled unknown quantity group and 2 and the fraction 1 over 4 or the fraction 9 over 4.

The diagram shows that 3 wholes contain 12 fourths, and \(2\frac{1}{4}\) contains 9 fourths, so the answer to this question is \(\frac{9}{12}\), which is equivalent to \(\frac{3}{4}\).

We can use diagrams to help us solve other division problems that require finding a fraction of a group. For example, here is a diagram to help us answer the question: “What fraction of \(\frac{9}{4}\) is \(\frac{3}{2}\)?,” which can be written as \(\frac{3}{2}\div\frac{9}{4}=?\).

Figure \(\PageIndex{10}\): Fraction bar diagram. 9 equal parts. 6 parts shaded. Total labeled 1 group and 2 and the fraction 1 over 4 or the fraction 9 over 4. 6 parts labeled unknown quantity group and the fraction 3 over 2 or the fraction 6 over 4.

We can see that the quotient is \(\frac{6}{9}\), which is equivalent to \(\frac{2}{3}\). To check this, let’s multiply. \(\frac{2}{3}\cdot\frac{9}{4}=\frac{18}{12}\), and \(\frac{18}{12}\) is, indeed, equal to \(\frac{3}{2}\).

Practice

Exercise \(\PageIndex{4}\)

A recipe calls for \(\frac{1}{2}\) lb of flour for 1 batch. How many batches can be made with each of these amounts?

1. \(1\) lb
2. \(\frac{3}{4}\) lb
3. \(\frac{1}{4}\) lb

Exercise \(\PageIndex{5}\)

Whiskers the cat weighs \(2\frac{2}{3}\) kg. Piglio weighs \(4\) kg. For each question, write a multiplication equation and a division equation, decide whether the answer is greater than 1 or less than 1, and then find the answer.

1. How many times as heavy as Piglio is Whiskers?
2. How many times as heavy as Whiskers is Piglio?

Exercise \(\PageIndex{6}\)

Andre is walking from his home to a festival that is \(1\frac{5}{8}\) kilometers away. He walks \(\frac{1}{3}\) kilometer and then takes a quick rest. Which question can be represented by the equation \(?\cdot 1\frac{5}{8}=\frac{1}{3}\) in this situation?

1. What fraction of the trip has Andre completed?
2. What fraction of the trip is left?
3. How many more kilometers does Andre have to walk to get to the festival?
4. How many kilometers is it from home to the festival and back home?

Exercise \(\PageIndex{7}\)

Draw a tape diagram to represent the question: What fraction of \(2\frac{1}{2}\) is \(\frac{4}{5}\)?
Then find the answer.

Exercise \(\PageIndex{8}\)

How many groups of \(\frac{3}{4}\) are in each of these quantities?

1. \(\frac{11}{4}\)
2. \(6\frac{1}{2}\)

(From Unit 4.2.3)

Exercise \(\PageIndex{9}\)

Which question can be represented by the equation \(4\div\frac{2}{7}=?\)

1. What is \(4\) groups of \(\frac{2}{7}\)?
2. How many \(\frac{2}{7}\)s are in \(4\)?
3. What is \(\frac{2}{7}\) of \(4\)?
4. How many \(4\)s are in \(\frac{2}{7}\)?

(From Unit 4.2.1)