Lesson 71: Finding the Whole Group When a Fraction is Known

New Concept

We can find the total number in a group when we only know a fraction of it. This relationship can be expressed with an equation:

What’s next

This card is just the foundation. Next, you'll work through examples using diagrams and equations to solve for the 'whole group' in different situations.

Property

To find a total when a fraction of it is known, like if $\frac{3}{5}$ of a group is 45, you must first determine the value of one single part.

Examples

• If $\frac{3}{5}$ of fish in a pond are 45 bluegills, then one-fifth is $45 \div 3 = 15$ fish. The total number of fish is $5 \times 15 = 75$.
• If $\frac{3}{8}$ of a book is 51 pages, then one-eighth is $51 \div 3 = 17$ pages. The total number of pages is $8 \times 17 = 136$.

Explanation

Imagine the whole group is a big chocolate bar broken into equal parts (the denominator). If you know the value of a few of those parts (the numerator), you can figure out the value of just one part by dividing. Then, multiply to find the total! It’s that simple.

Property

A fraction problem can be written as an equation.

Examples

• To solve $\frac{3}{8}P = 51$, multiply both sides by its reciprocal, $\frac{8}{3}$. This gives $1P = \frac{8}{3} \cdot 51$, so $P = 136$.
• To solve $\frac{3}{4}H = 12$, multiply both sides by $\frac{4}{3}$. This gives $1H = \frac{4}{3} \cdot 12$, so $H = 16$.

Explanation

Let algebra do the heavy lifting! To find the total (P), you need to get it all by itself. You can zap the fraction by multiplying both sides of the equation by its reciprocal. This cool trick isolates the variable you want to solve for, giving you the answer.

Property

If a fraction of a group has a certain status (like $\frac{3}{5}$ of lights are on), the rest of the group is the remaining fraction ($1 - \frac{3}{5} = \frac{2}{5}$ of lights are off).

Examples

• $\frac{3}{5}$ of lights are on, and 30 are off. This means $\frac{2}{5}$ of the lights equals 30. So, one-fifth is $30 \div 2 = 15$ lights. The number of lights on is $3 \times 15 = 45$.
• $\frac{5}{8}$ of clowns have happy faces, while 15 do not. This means $\frac{3}{8}$ of the clowns is 15. So, one-eighth is $15 \div 3 = 5$ clowns. The total is $8 \times 5 = 40$ clowns.

Explanation

Sometimes a problem tells you about the 'leftover' part. No sweat! First, find what fraction the leftovers represent. If $\frac{5}{8}$ of the clowns are happy, then you know the other $\frac{3}{8}$ are not. From there, you can easily find the value of one part and solve for the total.