Ratio Word Problems

Ratios are a great way to compare quantities in real-world situations. To solve ratio word problems, you need to figure out what kind of relationship is being described and use equivalent ratios to find the missing value.

Part-to-Part vs. Part-to-Whole

Before solving a problem, identify what the ratio compares:

• Part-to-Part: Compares one part of a group to another part (e.g., cats to dogs).
• Part-to-Whole: Compares one part of a group to the entire group (e.g., cats to all pets).

If the ratio of cats to dogs is $3:5$, the total number of "parts" is $3 + 5 = 8$. The part-to-whole ratio of cats to total pets is $3:8$.

Solving Part-to-Part Problems

Example: The ratio of cats to dogs at an animal shelter is $3:5$. If there are $15$ dogs, how many cats are there?

1. Set up equivalent ratios. Let $c$ be the unknown number of cats. $\frac{3 \text{ (cats)}}{5 \text{ (dogs)}} = \frac{c \text{ (cats)}}{15 \text{ (dogs)}}$
2. Find the multiplier. How do you get from $5$ to $15$? You multiply by $3$.
3. Multiply the top number by the same amount: $3 \times 3 = 9$.

There are $9$ cats.

Solving Part-to-Whole Problems

Sometimes you are given a part-to-part ratio but need to find an amount based on the total.

Example: Blue and red marbles are in a ratio of $2:3$. There are $30$ marbles in total. How many are blue?

1. Find the total number of parts: $2 \text{ (blue)} + 3 \text{ (red)} = 5 \text{ total parts}$.
2. Find the value of one part: Divide the actual total number of marbles by the total ratio parts. $30 \div 5 = 6 \text{ marbles per part}$
3. Multiply to find the specific part: The blue marbles make up $2$ parts in the ratio. $2 \times 6 = 12$

There are $12$ blue marbles.

Using Ratio Tables

A ratio table is a helpful visual tool for organizing equivalent ratios, especially for recipes or scaling up.

Example: A recipe uses flour and sugar in a $4:1$ ratio. How much sugar do you need for $12$ cups of flour?

Set up a table and scale up both sides until you reach the target number:

Flour	Sugar
$4$	$1$
$8$	$2$
$12$	$3$

By multiplying the original ratio by $3$ (since $4 \times 3 = 12$), we can easily see that you need $3$ cups of sugar.