Lesson 45: Ratio Problems Involving Totals

New Concept

Some ratio problems require us to consider the total to solve the problem. For these problems we add a third row for the total to our ratio table.

What’s next

This card lays the foundation. Soon, we'll master this skill through worked examples that visually break down how to build and use the three-row ratio table.

Property

When a problem involves a total, add the ratio numbers to create a ratio for the total. Use a three-row table to organize the parts and the total amount.

Examples

Ratio of wins to losses is 5:2. In 21 games, find wins (w): $\frac{5}{7} = \frac{w}{21} \rightarrow w=15$.
Ratio of ducks to geese is 4:3. With 16 ducks, find the total birds (b): $\frac{4}{7} = \frac{16}{b} \rightarrow b=28$.

Explanation

Think of it as a power-up! Adding ratio parts creates a new part-to-total ratio, which is the secret key you need to solve for any missing number in the problem.

Property

To solve ratio word problems, you must formulate a correct proportion. This means writing an equation relating the ratios to the actual counts of the items involved.

Examples

Acrobats to clowns is 3:5. Total is 24. Formulate to find clowns (c): $\frac{5}{8} = \frac{c}{24}$.
Small to large buses is 2:7. With 84 large buses, formulate for the total (t): $\frac{7}{9} = \frac{84}{t}$.

Explanation

This is your translator skill! You turn the story about clowns and buses into a clean math proportion. Once you translate it into an equation, the math is easy.

Property

Find the scale factor by dividing an actual count by its ratio number. Multiply other ratio parts by this factor to find their actual values.

Examples

Total ratio 8, actual 24. Factor: $24 \div 8=3$. A part ratio of 5 is $5 \times 3=15$.
Part ratio 7, actual 84. Factor: $84 \div 7=12$. A total ratio of 9 is $9 \times 12=108$.

Explanation

This shortcut finds the 'growth' number from ratio to reality. Use this single factor on any ratio part to find its real amount, skipping the proportion.