Ratios & Proportions

A ratio compares two quantities, and a proportion says two ratios are equal.

Reading is good — doing is better. Practice Ratios & proportions as an interactive lesson.

Definition

A ratio is a way to compare two numbers or amounts by showing how many times one contains the other. For example, if there are 3 red marbles and 5 blue marbles, the ratio of red to blue is 3 to 5. A proportion is a statement that two ratios are equal, like saying 3/5 = 6/10.

Remember the rule

If a/b = c/d, then a × d = b × c (cross products are equal). To find a missing value, cross multiply and divide.

Key words

Ratio: A comparison of two numbers using division, written as 3 to 5, 3:5, or 3/5.
Proportion: An equation that shows two ratios are equal, like 2/3 = 4/6.
Terms: The two numbers being compared in a ratio. In 3:5, the terms are 3 and 5.
Equivalent ratios: Ratios that have the same value, like 1:2 and 3:6 and 5:10.
Cross multiplication: A way to check if two ratios are equal by multiplying diagonally: in a/b = c/d, check if a×d equals b×c.
Unit rate: A ratio where the second number is 1, like 60 miles per 1 hour.
Scaling up: Multiplying both terms of a ratio by the same number to make an equivalent ratio.
Scaling down: Dividing both terms of a ratio by the same number to simplify it.

Worked examples

A recipe uses 2 cups of flour for every 3 cups of sugar. How many cups of flour are needed for 9 cups of sugar?

→ Set up the proportion: 2/3 = ?/9. Cross multiply: 2 × 9 = 3 × ?, so 18 = 3 × ?, so ? = 6. You need 6 cups of flour. · Always keep the same order in both ratios: flour on top, sugar on bottom.

A car travels 150 miles in 3 hours. How far will it travel in 5 hours at the same speed?

→ Set up the proportion: 150/3 = ?/5. Cross multiply: 150 × 5 = 3 × ?, so 750 = 3 × ?, so ? = 250. The car travels 250 miles. · Finding the unit rate first (50 miles per hour) is another way to solve this.

Are the ratios 4:6 and 10:15 equivalent?

→ Write them as fractions: 4/6 and 10/15. Cross multiply: 4 × 15 = 60 and 6 × 10 = 60. Since both equal 60, yes, the ratios are equivalent.

In a class of 28 students, the ratio of boys to girls is 3:4. How many boys are there?

→ Boys + girls = 3 + 4 = 7 parts total. Each part = 28 ÷ 7 = 4 students. Boys = 3 × 4 = 12 boys. · This method of dividing into equal parts works great when you know the total.

A map uses a scale of 1 inch = 50 miles. How many miles is 3.5 inches on the map?

→ Set up the proportion: 1/50 = 3.5/?. Cross multiply: 1 × ? = 50 × 3.5, so ? = 175. The distance is 175 miles.

Common mistakes

Flipping the order of a ratio mid-problem, like writing flour:sugar for one ratio but sugar:flour for the other.
Forgetting that both numbers in a ratio must be multiplied or divided by the same number to stay equivalent.
Adding the same number to both terms instead of multiplying, for example thinking 2:3 and 4:5 are equivalent because you added 2.
Setting up a proportion with mismatched units, like putting miles on top for one ratio and hours on top for the other.
Stopping after cross multiplying without finishing the division to find the unknown value.

FAQs

What is the difference between a ratio and a fraction?

They look the same but mean different things. A fraction shows part of a whole, like 1/4 of a pizza. A ratio compares any two quantities, like 1 cat to 4 dogs, where nothing has to be a whole.

Can a ratio have a zero in it?

The first number can be zero (0:5 means none of the first thing), but the second number cannot be zero because you cannot divide by zero.

Do ratios always have to be in lowest terms?

Not always, but simplifying makes ratios easier to compare. Always check if your teacher wants the answer in simplest form.

How do I know which number goes first in a ratio?

Follow the order of the words in the problem. If it says 'boys to girls,' boys go first. If it says 'girls to boys,' girls go first. Order matters!

Is a proportion the same as an equation?

Yes! A proportion is a special type of equation that says two ratios are equal. You can use algebra to solve for a missing number in a proportion.

When will I use ratios and proportions in real life?

All the time! Cooking recipes, reading maps, comparing prices at the store, mixing paint colors, and understanding sports statistics all use ratios and proportions.

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