Lesson 7
What Fraction of a Group?
Let’s think about dividing things into groups when we can’t even make one whole group.
7.1: Estimating a Fraction of a Number
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Estimate the quantities:
- What is \(\frac13\) of 7?
- What is \(\frac45\) of \(9\frac23\)?
- What is \(2\frac47\) of \(10\frac19\)?
- Write a multiplication expression for each of the previous questions.
7.2: 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.
- 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.
- Rope B is _______ times as long as rope A.
- Rope C is _______ times as long as rope A.
- Rope D is _______ times as long as rope A.
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Each equation can be used to answer a question about Ropes C and D. What could each question be?
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\({?} \boldcdot 3=9\) and \(9 \div 3={?}\)
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\({?} \boldcdot 9=3\) and \(3 \div 9= {?}\)
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7.3: 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: \(22 \frac12\) cups
- Thursday: 6 cups
- Friday: \(7 \frac12\) cups
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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.
- Monday
- Tuesday
- Monday
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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.
- Thursday
- Friday
- Thursday
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For each question, write a division equation, draw a tape diagram, and find the answer.
- What fraction of 9 is 3?
- What fraction of 5 is \(\frac 12\)?
- What fraction of 9 is 3?
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, \(\frac34\) of a bag would weigh (\(\frac34 \boldcdot 5)\) kg.
![Fraction bar diagram. 4 equal parts. 3 parts shaded.](https://cms-im.s3.amazonaws.com/kL8KerHZFCQ1uZXVacYNy1Y9?response-content-disposition=inline%3B%20filename%3D%226-6.4.B3.Image.09a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B3.Image.09a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162120Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5b3e01f396a6a0a4e9df50931d40e600a48075f94986d1ddd88c9d31125dac3d)
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.
\(\displaystyle {?} \boldcdot 6 = 3\) \(\displaystyle 3\div 6 = {?}\)
![A tape diagram.](https://cms-im.s3.amazonaws.com/6ccPsVR14TfZM6x1gGhGN3gp?response-content-disposition=inline%3B%20filename%3D%226-6.4.B3.Image.10a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B3.Image.10a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162120Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=91e00d7fe28546a454117e980a4bd0ab8b49a370416a6088bfbb9c65e2264cf8)
We can see from the diagram that 3 is \(\frac12\) of 6, and we can check this answer by multiplying: \(\frac12 \boldcdot 6 = 3\).
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 \(2\frac14\)?” can be expressed as \({?} \boldcdot 3 = 2\frac14\), which can also be written as \(2\frac14\div 3 = {?}\).
The answer to “What is \(2\frac14 \div 3\)?” is also the answer to the original question.
![Fraction bar diagram. 12 equal parts. 9 parts shaded.](https://cms-im.s3.amazonaws.com/USNzreNSc4ha4MMtCSxsQGah?response-content-disposition=inline%3B%20filename%3D%226-6.4.B3.Image.11a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B3.Image.11a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162120Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3848a503faa0d603c4d19526dec00f5f1dc6c0d3419245e0f249756fbafab71e)
The diagram shows that 3 wholes contain 12 fourths, and \(2\frac14\) contains 9 fourths, so the answer to this question is \(\frac{9}{12}\), which is equivalent to \(\frac34\).
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 \(\frac94\) is \(\frac32\)?,” which can be written as \(\frac32 \div \frac94 = {?}\).
![Fraction bar diagram. 9 equal parts. 6 parts shaded.](https://cms-im.s3.amazonaws.com/e9KfpzpnMc9SgTD3bB9Vyi2b?response-content-disposition=inline%3B%20filename%3D%226-6.4.B3.Image.12a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.4.B3.Image.12a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T162120Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e985fe8a5291e5891391d5a21b793c2c532414b0dbf650de653c4e1c23558b6f)
We can see that the quotient is \(\frac69\), which is equivalent to \(\frac23\). To check this, let’s multiply. \(\frac23 \boldcdot \frac94 = \frac{18}{12}\), and \(\frac{18}{12}\) is, indeed, equal to \(\frac32\).