Chapter 2 · Fraction review

Fraction Products and Scaling in Context

Solve word problems by multiplying fractions and mixed numbers. Then predict how a product compares with the starting amount by comparing the factor with one.

13 min read

You already know how to multiply two fractions and how to multiply mixed numbers. Here you use those skills to solve word problems. Then you learn to predict how a product compares with the amount you started with, before you multiply.

Start with what the word of means. Taking $\frac{1}{2}$ of 6 apples means splitting the apples into 2 equal groups and keeping 1 group. That gives 3 apples, which matches $\frac{1}{2} \times 6 = 3$ . Taking $\frac{2}{3}$ of 6 apples means splitting them into 3 equal groups of 2 apples and keeping 2 groups. That gives 4 apples, which matches $\frac{2}{3} \times 6 = 4$ .

A fraction of a quantity means that fraction times the quantity. This is why so many fraction word problems are multiplication problems.

In a multiplication, the numbers you multiply are factors, and the result is the product. To set up a word problem, decide what each factor stands for.

The starting quantity is the amount you take a part of, or make copies of. It has a unit, such as liters or miles.
The other number tells how much of the starting quantity you need, such as $\frac{2}{5}$ of it or $2\frac{2}{3}$ times it. In this lesson, the factor means this number.

For every fraction in a problem, ask: a fraction of what? The answer is that fraction's whole. A fraction with a unit, such as $\frac{3}{4}$ liter, has 1 liter as its whole. A fraction of an amount in the story has that amount as its whole.

When the factor counts parts or copies of the starting quantity, the product has the same unit as the starting quantity.

Worked example

A fraction of a fraction

A pitcher holds $\frac{3}{4}$ liter of juice. Maya pours out $\frac{2}{5}$ of the juice. How many liters of juice does Maya pour out?

Find the starting quantity and the factor. The pitcher holds 3/4 liter, so the whole for that fraction is 1 liter. The 2/5 has a different whole: all of the juice in the pitcher. So the starting quantity is 3/4 liter and the factor is 2/5.
Write the multiplication. The problem asks for 2/5 of the juice, and the juice is 3/4 liter. A fraction of a quantity means that fraction times the quantity, so you multiply 2/5 by 3/4.
$\frac{2}{5} \times \frac{3}{4}$
Multiply the numerators and multiply the denominators. Here is why that fits the story. Cut each fourth of a liter into 5 equal pieces. Then 5 × 4 = 20 pieces fill 1 liter, so each piece is 1/20 liter. Taking 2/5 of the juice means taking 2 pieces from each of the 3 fourths. That is 2 × 3 = 6 pieces, or 6/20 liter.
$\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}$
Simplify. You can divide both 6 and 20 by 2. Dividing the numerator and the denominator by 2 takes out 2/2, which equals 1, so the amount stays the same.
$\frac{6}{20} = \frac{3 \times 2}{10 \times 2} = \frac{3}{10} \times \frac{2}{2} = \frac{3}{10} \times 1 = \frac{3}{10}$
Give the unit. The factor 2/5 counts parts of the juice, so the product is in liters, like the starting quantity.
$\frac{2}{5} \times \frac{3}{4}\text{ liter} = \frac{3}{10}\text{ liter}$
Check against the story. Maya pours out part of the juice, not all of it: the factor 2/5 is between 0 and 1, and the starting amount is greater than 0. So the answer must be less than 3/4 liter. Rewrite 3/4 in twentieths to compare. Multiplying by 5/5 does not change it, because 5/5 equals 1.
$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}, \qquad \frac{3}{10} = \frac{6}{20} < \frac{15}{20}$

Answer

Maya pours out $\frac{3}{10}$ liter of juice.

Worked example

Mixed numbers on a trail

One lap of a park trail is $1\frac{1}{2}$ miles long. Jordan walks $2\frac{2}{3}$ laps. How many miles does Jordan walk?

Find the starting quantity and the factor. One lap is 1 1/2 miles, so that is the starting quantity. Jordan walks that distance 2 2/3 times, so 2 2/3 is the factor. The product will be in miles.
$2\frac{2}{3} \times 1\frac{1}{2}$
Write each mixed number as one fraction, so you can multiply numerators and denominators. The 2 wholes are 6 thirds, and the 1 whole is 2 halves. Each amount stays the same. Only the way you write it changes.
$2\frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}, \qquad 1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$
Multiply the numerators and multiply the denominators. The rule from the juice problem works here because you have written both numbers as fractions.
$\frac{8}{3} \times \frac{3}{2} = \frac{8 \times 3}{3 \times 2} = \frac{24}{6}$
Simplify. The fraction bar means division, and 24 divided by 6 is 4. This makes sense because 6 sixths make 1 whole, so 24 sixths make 4 wholes.
$\frac{24}{6} = 24 \div 6 = 4$
Check by building the distance in parts. Jordan walks 2 full laps and 2/3 of another lap. The 2 full laps are 1 1/2 miles twice, which is 3 miles. The extra 2/3 of a lap is 2/3 of 1 1/2 miles, which is 1 mile. The parts add up to the same total.
$1\frac{1}{2} + 1\frac{1}{2} = 3, \qquad \frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1, \qquad 3 + 1 = 4$

Answer

Jordan walks $4$ miles in $2\frac{2}{3}$ laps.

Look back at the two answers. Maya's $\frac{3}{10}$ liter is less than the $\frac{3}{4}$ liter in the pitcher. Jordan's 4 miles is more than the $1\frac{1}{2}$ miles in one lap. Both problems used multiplication. So why is one product less than its starting quantity and the other greater?

The reason is scaling: multiplying a quantity by a factor resizes it. To see how, take 12 as the starting quantity and think in fourths. One fourth of 12 is 3.

$\frac{3}{4} \times 12$ keeps 3 of the 4 fourths: $3 \times 3 = 9$ . It leaves out one fourth, so the product is less than 12.
$\frac{4}{4} \times 12$ keeps all 4 fourths: $4 \times 3 = 12$ . Since $\frac{4}{4} = 1$ , nothing changes.
$\frac{5}{4} \times 12$ keeps all 4 fourths and adds 1 more: $5 \times 3 = 15$ . The extra fourth makes the product greater than 12.

The same reasoning works for any positive starting quantity. Positive means greater than 0. The quantity can be a whole number, a fraction or a mixed number. For a positive starting quantity and a positive factor:

A factor between 0 and 1 is more than 0 but less than 1. It keeps only part of the quantity, so the product is less than the starting quantity.
A factor greater than 1 keeps the whole quantity and adds more, so the product is greater than the starting quantity.
A factor equal to 1 keeps exactly the whole quantity, so the product equals the starting quantity.

A starting quantity of 0 behaves differently. Any part of nothing is nothing, and any number of copies of nothing is nothing. Zero stays zero for every factor.

The table shows each case. It includes the two word problems you solved. It also shows why equivalent fractions are equal: multiplying by $\frac{2}{2}$ is multiplying by 1.

How the factor changes the size of the product

Starting quantity	Factor	Factor compared with 1	Product	Product compared with the starting quantity
$12$	$\frac{3}{4}$	between 0 and 1	$9$	less than $12$
$12$	$\frac{4}{4}$	equal to 1	$12$	equal to $12$
$12$	$\frac{5}{4}$	greater than 1	$15$	greater than $12$
$\frac{3}{4}$ liter	$\frac{2}{5}$	between 0 and 1	$\frac{3}{10}$ liter	less than $\frac{3}{4}$ liter
$1\frac{1}{2}$ miles	$2\frac{2}{3}$	greater than 1	$4$ miles	greater than $1\frac{1}{2}$ miles
$\frac{3}{10}$	$\frac{2}{2}$	equal to 1	$\frac{6}{20}$	equal to $\frac{3}{10}$ , because the fractions are equivalent
$0$	$\frac{5}{4}$	greater than 1	$0$	equal to $0$ , because zero stays zero

Key concept

Compare the factor with 1

A fraction of a quantity means that fraction times the quantity. The factor scales the quantity, which means it resizes it. For a positive starting quantity and a positive factor:

a factor greater than 1 gives a product greater than the starting quantity
a factor between 0 and 1 gives a product less than the starting quantity
a factor equal to 1 gives a product equal to the starting quantity

If the starting quantity is 0, the product is 0 for every factor. Zero stays zero.

\text{product} = \text{factor} \times \text{starting quantity}

Mistake: multiplying always makes a number bigger. In the juice problem, this idea says the answer should be more than $\frac{3}{4}$ liter. A learner who trusts it may reject $\frac{3}{10}$ and add instead: $\frac{2}{5} + \frac{3}{4} = \frac{8}{20} + \frac{15}{20} = \frac{23}{20}$ liters. That is more juice than the pitcher holds.

The idea comes from whole-number factors greater than 1, such as $3 \times 4 = 12$ . It fails here because the factor $\frac{2}{5}$ is between 0 and 1. It keeps only part of a positive quantity, so the product is less than $\frac{3}{4}$ liter. Division can surprise you too: dividing a positive quantity by a positive number below 1 can increase it. For example, $3 \div \frac{1}{2}$ asks how many halves are in 3, and there are 6.

Worked example

Predict first, then confirm

A photo is $2\frac{1}{2}$ inches wide. A small print is $\frac{3}{5}$ as wide as the photo. A large print is $1\frac{1}{5}$ times as wide as the photo. Without multiplying, decide whether each print is narrower or wider than the photo. Then find both widths to confirm.

Name the starting quantity and the factors. The photo's width, 2 1/2 inches, is the starting quantity, and it is positive. Each print's width is its factor times 2 1/2 inches. The factors are 3/5 and 1 1/5.
Compare each factor with 1. The fraction 3/5 is 3 of 5 equal parts of a whole, so it is more than 0 and less than 1. The mixed number 1 1/5 is 1 whole plus another 1/5, so it is greater than 1.
$0 < \frac{3}{5} < 1 < 1\frac{1}{5}$
Predict. The starting width is positive. The factor 3/5 is between 0 and 1, so the small print keeps only part of the width and is narrower than 2 1/2 inches. The factor 1 1/5 is greater than 1, so the large print has the whole width and more, and is wider than 2 1/2 inches.
Confirm the small print. Write 2 1/2 as 4/2 + 1/2 = 5/2, because 2 wholes are 4 halves. Multiply numerators and denominators to get 15/10. Dividing 15 and 10 by 5 takes out 5/5, which equals 1, so 15/10 equals 3/2. Then 3/2 is 2/2 + 1/2, which is 1 1/2.
$\frac{3}{5} \times 2\frac{1}{2} = \frac{3}{5} \times \frac{5}{2} = \frac{3 \times 5}{5 \times 2} = \frac{15}{10} = \frac{3}{2} = 1\frac{1}{2}$
Confirm the large print. Write 1 1/5 as 5/5 + 1/5 = 6/5, because 1 whole is 5 fifths. Multiply numerators and denominators to get 30/10. The fraction bar means division, and 30 divided by 10 is 3.
$1\frac{1}{5} \times 2\frac{1}{2} = \frac{6}{5} \times \frac{5}{2} = \frac{6 \times 5}{5 \times 2} = \frac{30}{10} = 3$
Compare each width with the photo's width. Both predictions hold. The small print's 1 1/2 inches is less than 2 1/2 inches, and the large print's 3 inches is greater than 2 1/2 inches.
$1\frac{1}{2} < 2\frac{1}{2} < 3$

Answer

The small print is narrower than the photo, at $1\frac{1}{2}$ inches wide. The large print is wider, at $3$ inches wide.

Takeaway. To find a fraction of a quantity, multiply. Find the starting quantity, find the factor, multiply, and give the unit. Then compare the factor with 1 to check the size of your answer. For a positive starting quantity and a positive factor: a factor greater than 1 gives a product greater than the starting quantity, a factor between 0 and 1 gives a product less than it, and a factor of 1 leaves it unchanged. A starting quantity of 0 stays 0.

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