Lesson 22: "Equal Groups" Problems with Fractions

New Concept

To find a fractional part of a number, we first divide the total into a number of equal groups. The denominator of the fraction tells us how many equal groups to create.

What’s next

This introduces the core idea of finding a fractional part. Next, you'll apply this two-step method to solve a variety of word problems involving money, percentages, and more.

Property

To find a fraction of a whole number, first divide the total by the denominator to find the size of one fractional part. Then, multiply that result by the numerator to find the value of the desired number of parts.

Examples

To find $\frac{2}{3}$ of 12 musicians: first $12 \div 3 = 4$, then $2 \times 4 = 8$ musicians.
To find $\frac{3}{4}$ of 28 problems: first $28 \div 4 = 7$, then $3 \times 7 = 21$ problems.
To find $\frac{5}{6}$ of 24 pizzas: first $24 \div 6 = 4$, then $5 \times 4 = 20$ pizzas.

Explanation

Imagine a bag of 12 cookies! To find $\frac{2}{3}$ of them, the '3' tells you to split the cookies into 3 equal groups, meaning each group has 4 cookies. The '2' then tells you to take 2 of those groups. So, you get $2 \times 4 = 8$ cookies. It's the secret to fairly sharing any treasure!

Property

When a problem asks for a fraction of a total, you must first find the size of one unit fraction. To find $\frac{n}{d}$ of a total, you must infer the first step is to divide the total by the denominator, $d$.

Examples

For $\frac{2}{3}$ of 12 musicians, you divide by 3 to find that one 'third' is 4 musicians.
For $\frac{3}{5}$ of 3.00 dollars, you divide by 5 to infer that one 'fifth' is 0.60 dollars.
For $\frac{9}{10}$ of 100 percent, you divide by 10 to find that one 'tenth' is 10 percent.

Explanation

Why divide first? Think of the denominator as a clue telling you how many equal shares a treasure is split into. By dividing the total by this number, you figure out the size of just one share. Once you know the value of one piece, you can easily calculate the value of any number of pieces, just like a detective solving a case!

Property

To find a fraction of an amount of money or a percentage, use the same two-step process: divide the total amount by the denominator, then multiply the result by the numerator.

Examples

How much is $\frac{3}{5}$ of 3.00 dollars? First, $3.00 \div 5 = 0.60$ dollars. Then, $3 \times 0.60 = 1.80$ dollars.
How much is $\frac{2}{3}$ of 4.50 dollars? First, $4.50 \div 3 = 1.50$ dollars. Then, $2 \times 1.50 = 3.00$ dollars.
What is $\frac{2}{5}$ of 100%? First, $100\% \div 5 = 20\%$. Then, $2 \times 20\% = 40\%$.

Explanation

This awesome math trick works on money and percents, too! To find $\frac{3}{5}$ of 3.00 dollars, just divide the three dollars into 5 equal piles (60 cents each). Then grab 3 piles for a total of 1.80 dollars. The same goes for 100%: finding $\frac{2}{5}$ of it means dividing 100% into 5 parts and taking 2!