Proportions and Ratios

What Are Ratios?

A ratio is a comparison of two quantities. If you have 3 apples and 5 oranges, the ratio of apples to oranges is 3 to 5. We can write this in several ways:

Using the word "to": 3 to 5
Using a colon: 3:5
As a fraction: $\frac{3}{5}$

All three mean the same thing. The fraction form is often most useful for calculations.

Ratios show up constantly in everyday life. When you're mixing paint colors, following a recipe, or comparing prices, you're working with ratios. A recipe might call for a 2:1 ratio of flour to sugar, meaning for every 2 cups of flour, you use 1 cup of sugar.

Example 1: A classroom has 12 boys and 15 girls. What is the ratio of boys to girls?

The ratio is $12:15$, which simplifies to $4:5$ (dividing both by 3).

This means for every 4 boys, there are 5 girls.

Example 2: A paint mixture uses 3 parts blue to 2 parts yellow. If you have 6 gallons of blue paint, how much yellow paint do you need?

The ratio is $3:2$. If you're using 6 gallons of blue (which is $3 \times 2$), you need $2 \times 2 = 4$ gallons of yellow.

What Are Proportions?

A proportion is an equation that says two ratios are equal. If $\frac{2}{3} = \frac{4}{6}$, that's a proportion—two equivalent ratios set equal to each other.

Proportions let you solve problems where you know three values and need to find the fourth. This is incredibly useful for scaling recipes, converting units, working with maps, and countless other situations.

Example 3: Is $\frac{2}{5} = \frac{6}{15}$ a true proportion?

Check by simplifying $\frac{6}{15}$: $$\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}$$

Yes, it's a true proportion.

Example 4: Is $\frac{3}{4} = \frac{9}{16}$ a true proportion?

Simplify $\frac{9}{16}$—it doesn't reduce to $\frac{3}{4}$. Or check: $\frac{3}{4} = 0.75$ and $\frac{9}{16} = 0.5625$.

No, these ratios are not equal.

Cross-Multiplication

The most powerful method for solving proportions is cross-multiplication. When you have $\frac{a}{b} = \frac{c}{d}$, cross-multiplication says:

$$a \cdot d = b \cdot c$$

You multiply diagonally across the equal sign. This works because you're essentially multiplying both sides of the equation by $bd$ to clear the fractions.

Example 5: Solve for $x$: $\frac{3}{5} = \frac{x}{20}$

Cross-multiply: $$3 \cdot 20 = 5 \cdot x$$ $$60 = 5x$$ $$x = 12$$

Check: Does $\frac{3}{5} = \frac{12}{20}$? Yes, both equal 0.6.

Example 6: Solve for $y$: $\frac{7}{y} = \frac{21}{15}$

Cross-multiply: $$7 \cdot 15 = y \cdot 21$$ $$105 = 21y$$ $$y = 5$$

Example 7: Solve for $n$: $\frac{4}{9} = \frac{12}{n}$

Cross-multiply: $$4 \cdot n = 9 \cdot 12$$ $$4n = 108$$ $$n = 27$$

Real-World Applications

Proportions are everywhere once you start looking for them. Here are some common situations:

Recipes

If a recipe serves 4 people but you need to serve 6, you use proportions to scale the ingredients.

Example 8: A cookie recipe for 24 cookies calls for 2 cups of flour. How much flour do you need for 36 cookies?

Set up the proportion: $$\frac{2 \text{ cups}}{24 \text{ cookies}} = \frac{x \text{ cups}}{36 \text{ cookies}}$$

Cross-multiply: $$2 \cdot 36 = 24 \cdot x$$ $$72 = 24x$$ $$x = 3$$

You need 3 cups of flour.

Unit Conversions

Example 9: If 2.54 centimeters equal 1 inch, how many centimeters are in 5 inches?

$$\frac{2.54 \text{ cm}}{1 \text{ inch}} = \frac{x \text{ cm}}{5 \text{ inches}}$$

Cross-multiply: $$2.54 \cdot 5 = 1 \cdot x$$ $$x = 12.7$$

There are 12.7 centimeters in 5 inches.

Shopping and Unit Prices

Example 10: If 3 pounds of apples cost $4.50, how much would 5 pounds cost?

$$\frac{$4.50}{3 \text{ lbs}} = \frac{x}{5 \text{ lbs}}$$

Cross-multiply: $$4.50 \cdot 5 = 3 \cdot x$$ $$22.50 = 3x$$ $$x = 7.50$$

Five pounds would cost $7.50.

Scale and Map Problems

Maps use proportions constantly. A map might have a scale of 1 inch = 50 miles, which is really the proportion $\frac{1 \text{ inch}}{50 \text{ miles}}$.

Example 11: On a map with a scale of 1 inch = 25 miles, two cities are 3.5 inches apart. What is the actual distance?

$$\frac{1 \text{ inch}}{25 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}$$

Cross-multiply: $$1 \cdot x = 25 \cdot 3.5$$ $$x = 87.5$$

The actual distance is 87.5 miles.

Example 12: An architect's scale drawing shows 1 centimeter = 2 meters. If a room is 15 meters long in real life, how long is it on the drawing?

$$\frac{1 \text{ cm}}{2 \text{ m}} = \frac{x \text{ cm}}{15 \text{ m}}$$

Cross-multiply: $$1 \cdot 15 = 2 \cdot x$$ $$15 = 2x$$ $$x = 7.5$$

The room is 7.5 centimeters on the drawing.

Rates and Unit Rates

A rate is a ratio that compares two different units, like miles per hour or dollars per pound. A unit rate has 1 in the denominator.

Example 13: If a car travels 180 miles in 3 hours, what is the unit rate (speed)?

$$\frac{180 \text{ miles}}{3 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}}$$

The unit rate is 60 miles per hour.

Example 14: A 12-ounce can of soda costs $1.20. What is the unit price per ounce?

$$\frac{$1.20}{12 \text{ oz}} = \frac{$0.10}{1 \text{ oz}}$$

The unit price is $0.10 per ounce (or 10 cents per ounce).

Unit rates help you compare prices. If one store sells 5 apples for $2 and another sells 8 apples for $3, which is the better deal?

Store 1: $\frac{$2}{5} = $0.40$ per apple
Store 2: $\frac{$3}{8} = $0.375$ per apple

Store 2 is slightly cheaper per apple.

Distance, Rate, and Time

The formula $d = rt$ (distance = rate × time) is really just a proportion in disguise.

Example 15: If you drive at 55 mph for 2.5 hours, how far do you travel?

$$d = 55 \times 2.5 = 137.5 \text{ miles}$$

Example 16: You need to travel 300 miles and you're driving at 60 mph. How long will it take?

$$300 = 60 \cdot t$$ $$t = 5 \text{ hours}$$

Or set it up as a proportion: $$\frac{60 \text{ miles}}{1 \text{ hour}} = \frac{300 \text{ miles}}{x \text{ hours}}$$

Same answer: 5 hours.

Practice Problems

Work through these, then check your answers below.

Simplify the ratio 15:25
Is $\frac{4}{7} = \frac{12}{21}$ a true proportion?
Solve for $x$: $\frac{5}{8} = \frac{x}{32}$
If 6 pencils cost $2.40, how much do 10 pencils cost?
On a map, 2 inches represent 75 miles. How many miles do 5 inches represent?
A recipe for 8 servings uses 3 cups of rice. How much rice is needed for 12 servings?
Find the unit rate: 240 miles in 4 hours
Which is the better deal: 16 oz for $3.20 or 24 oz for $4.32?

Check Your Answers

3:5
Divide both by 5: $15 \div 5 = 3$, $25 \div 5 = 5$
Yes
$\frac{12}{21} = \frac{4}{7}$ when simplified (divide by 3)
$x = 20$
Cross-multiply: $5 \cdot 32 = 8x$, so $160 = 8x$, thus $x = 20$
$4.00
$\frac{2.40}{6} = \frac{x}{10}$. Cross-multiply: $2.40 \cdot 10 = 6x$, so $24 = 6x$, thus $x = 4$
187.5 miles
$\frac{2}{75} = \frac{5}{x}$. Cross-multiply: $2x = 375$, so $x = 187.5$
4.5 cups
$\frac{3}{8} = \frac{x}{12}$. Cross-multiply: $3 \cdot 12 = 8x$, so $36 = 8x$, thus $x = 4.5$
60 mph
$\frac{240}{4} = 60$ miles per hour
24 oz for $4.32
First deal: $\frac{3.20}{16} = $0.20$ per oz
Second deal: $\frac{4.32}{24} = $0.18$ per oz (better deal)

Key Takeaways

The order matters in ratios. The ratio of boys to girls (3:5) is different from the ratio of girls to boys (5:3). Always keep track of what you're comparing.

Cross-multiplication is your friend. It's fast, reliable, and works every time for solving proportions.

Check your work. After solving, substitute your answer back into the proportion to make sure both sides equal the same value.

Real-world problems require units. Always include units in your setup and answer. "12" is meaningless—"12 miles" or "12 dollars" gives context.

Proportions assume constant rates. When a recipe doubles from 4 to 8 servings, we assume every ingredient doubles. In reality, some things (like spices) might not scale perfectly, but mathematically we treat them as proportional.

Why This Matters

Proportions are one of the most practical topics in all of mathematics. You'll use them for:

Cooking and baking (scaling recipes)
Shopping (comparing prices)
Travel (maps, distances, speeds)
Construction and design (scale drawings)
Science (concentrations, dilutions)
Finance (currency conversion, interest rates)

Once you master proportions, you have a powerful tool for solving countless everyday problems. The key is recognizing when two quantities are in proportion and setting up the equation correctly.