Lesson 66: Ratio Problems Involving Totals

New Concept

Welcome to Saxon Math! This course builds a strong foundation by connecting arithmetic, geometry, and early algebra, showing how math works as a whole.

What’s next

We'll begin by applying these ideas to ratio problems involving totals. Next, you’ll see worked examples using proportion boxes to find unknown values.

Property

To solve ratio problems with a given total, first add the individual ratio parts to find the 'total ratio'. This special number connects the ratio to the real-world total count you were given.

Examples

The ratio of cats to dogs is 2 to 3, with 50 pets total. To find cats (C), find the total ratio: $2+3=5$. Then solve: $\frac{2}{5} = \frac{C}{50} \rightarrow C = 20$.
The ratio of red to blue marbles is 7 to 4, with 121 marbles total. To find blue (B), find the total ratio: $7+4=11$. Then solve: $\frac{4}{11} = \frac{B}{121} \rightarrow B = 44$.
At a bike store, the ratio of mountain bikes to racing bikes is 3 to 5, with 72 bikes in all. To find racing bikes (R): $\frac{5}{8} = \frac{R}{72} \rightarrow R = 45$.

Explanation

It’s like making a party punch! If the recipe is 2 parts orange juice to 3 parts soda, the total ratio is 5 parts. This lets you figure out exactly how much juice you need for a 50-liter cooler. It connects the small ratio parts to the big total.

Property

To find an unknown amount, build a proportion using two rows from your ratio box: the row for the item you need to find and the 'Total' row. This gives you a solvable equation like $\frac{\text{part}}{\text{total}} = \frac{\text{actual part}}{\text{actual total}}$.

Examples

Given a girls-to-total ratio of $\frac{4}{9}$ and 180 total students, find girls (G): $\frac{4}{9} = \frac{G}{180} \rightarrow 9G = 4 \cdot 180 \rightarrow G=80$.
The ratio of vans to cars is 2 to 9, with 77 vehicles total. To find vans (V), use the total ratio of 11: $\frac{2}{11} = \frac{V}{77} \rightarrow 11V = 2 \cdot 77 \rightarrow V=14$.
A drink recipe is 3 parts juice to 5 parts soda, making 160 ounces total. To find juice (J): $\frac{3}{8} = \frac{J}{160} \rightarrow 8J = 3 \cdot 160 \rightarrow J=60$.

Explanation

This is the magic step! You're setting a small-scale ratio (like part-to-total) equal to the big-scale version (actual amount-to-actual total). A quick cross-multiplication and division reveals the secret number you've been looking for. Poof!

Property

Once you solve the proportion to find one of the actual counts, you can easily find the remaining count. Simply subtract the number you found from the given total to complete the ratio box and see the full picture.

Examples

In a group of 180 students with a boy-to-girl ratio of 5 to 4, we found there are 80 girls. The number of boys is $180 - 80 = 100$.
With 48 total players and a football-to-soccer ratio of 5 to 7, we found 20 football players. The number of soccer players is $48 - 20 = 28$.
A shelter has 60 cats and dogs. After finding there are 28 cats, the number of dogs must be $60 - 28 = 32$ dogs.

Explanation

You've found one piece of the puzzle, so the rest is a snap! Subtracting the known part from the whole is the final, easy step to see all the actual numbers and confirm your work. It's the satisfying final step!