Ratios and Proportions — Complete Guide with Examples

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

By the end of this guide you will understand what ratios and proportions are, how to simplify ratios, how to solve proportions using cross-multiplication, and how to find unit rates. You will be able to apply these skills to word problems involving recipes, maps, speed, and everyday comparisons.

Theory

What is a ratio?

A ratio compares two quantities by showing how much of one thing there is relative to another. You can write a ratio in three equivalent ways:

$a : b \qquad \frac{a}{b} \qquad a \text{ to } b$

For example, if a classroom has 12 boys and 18 girls, the ratio of boys to girls is:

$12 : 18 = \frac{12}{18} = \frac{2}{3}$

The simplified ratio is $2 : 3$, meaning for every 2 boys there are 3 girls.

Important: The order of a ratio matters. The ratio of boys to girls ($2 : 3$) is different from the ratio of girls to boys ($3 : 2$).

Simplifying ratios

To simplify a ratio, divide both parts by their Greatest Common Divisor (GCD) — exactly like simplifying a fraction:

$a : b = \frac{a}{\gcd(a,b)} : \frac{b}{\gcd(a,b)}$

For example, to simplify $45 : 60$: the GCD of 45 and 60 is 15, so $45 : 60 = 3 : 4$.

What is a proportion?

A proportion is an equation that says two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

For instance, $\frac{2}{3} = \frac{8}{12}$ is a proportion because both ratios simplify to the same value.

Cross-multiplication

Cross-multiplication is the most common method for solving proportions. If:

$\frac{a}{b} = \frac{c}{d}$

then:

$a \times d = b \times c$

This works because multiplying both sides of the equation by $b \times d$ eliminates the denominators. When one of the four values is unknown, cross-multiplication lets you solve for it in one step.

Unit rates

A unit rate expresses a ratio as a quantity per one unit. To find the unit rate, divide the first quantity by the second:

$\text{unit rate} = \frac{\text{total quantity}}{\text{number of units}}$

For example, if you travel 150 kilometers in 3 hours, the unit rate (speed) is:

$\frac{150 \text{ km}}{3 \text{ h}} = 50 \text{ km/h}$

Unit rates are especially useful for comparing prices, speeds, and efficiency.

Worked Examples

Example 1: Simplifying a ratio (easy)

Problem: A bag contains 20 red marbles and 35 blue marbles. Write the ratio of red to blue marbles in simplest form.

Step 1: Write the ratio. $20 : 35$

Step 2: Find the GCD of 20 and 35. Factors of 20: 1, 2, 4, 5, 10, 20. Factors of 35: 1, 5, 7, 35. The GCD is 5.

Step 3: Divide both parts by 5. $20 : 35 = \frac{20}{5} : \frac{35}{5} = 4 : 7$

Answer: $4 : 7$

Example 2: Solving a proportion with cross-multiplication (medium)

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

Step 1: Cross-multiply. $3 \times 20 = 5 \times x$ $60 = 5x$

Step 2: Divide both sides by 5. $x = \frac{60}{5} = 12$

Step 3: Verify by checking the proportion. $\frac{3}{5} = \frac{12}{20} = \frac{3}{5} \checkmark$

Answer: $x = 12$

Example 3: Finding the unit rate (medium)

Problem: A pack of 8 notebooks costs $14.40. What is the cost per notebook?

Step 1: Set up the unit rate. $\text{cost per notebook} = \frac{\$14.40}{8}$

Step 2: Divide. $\frac{14.40}{8} = 1.80$

Answer: Each notebook costs $1.80.

Example 4: Recipe scaling with proportions (medium)

Problem: A recipe uses 3 cups of flour for every 2 cups of sugar. If you want to use 9 cups of flour, how much sugar do you need?

Step 1: Set up the proportion. $\frac{3 \text{ flour}}{2 \text{ sugar}} = \frac{9 \text{ flour}}{x \text{ sugar}}$

Step 2: Cross-multiply. $3 \times x = 2 \times 9$ $3x = 18$

Step 3: Solve for $x$. $x = \frac{18}{3} = 6$

Answer: You need 6 cups of sugar.

Example 5: Map scale problem (challenging)

Problem: On a map, 2 cm represents 15 km. Two cities are 9 cm apart on the map. What is the actual distance between them?

Step 1: Set up the proportion using the map scale. $\frac{2 \text{ cm}}{15 \text{ km}} = \frac{9 \text{ cm}}{d \text{ km}}$

Step 2: Cross-multiply. $2 \times d = 15 \times 9$ $2d = 135$

Step 3: Solve for $d$. $d = \frac{135}{2} = 67.5$

Answer: The actual distance is 67.5 km.

Common Mistakes

Mistake 1: Mixing up the order of a ratio

❌ "There are 5 cats and 3 dogs, so the ratio of dogs to cats is $5 : 3$."

✅ "The ratio of dogs to cats is $3 : 5$."

Why this matters: The order of quantities in a ratio must match the order stated in the question. Swapping them gives the inverse relationship, which is a completely different answer.

Mistake 2: Cross-multiplying incorrectly

❌ $\frac{4}{x} = \frac{6}{9} \Rightarrow 4 \times 6 = 9 \times x$

✅ $\frac{4}{x} = \frac{6}{9} \Rightarrow 4 \times 9 = 6 \times x \Rightarrow 36 = 6x \Rightarrow x = 6$

Why this matters: Cross-multiplication means you multiply diagonally — the numerator of one fraction times the denominator of the other. Multiplying numerator-by-numerator or denominator-by-denominator produces the wrong equation.

Mistake 3: Forgetting to simplify the ratio

❌ The ratio of 12 to 8 is $12 : 8$.

✅ $12 : 8 = 3 : 2$ (divide both by GCD of 4).

Why this matters: Unsimplified ratios are technically correct but most problems expect the simplest form. Always check whether both parts share a common factor.

Practice Problems

Try these on your own before checking the answers:

1. Simplify the ratio $36 : 48$.
2. Solve for $x$: $\frac{7}{4} = \frac{x}{12}$.
3. A car travels 240 km in 4 hours. What is the speed in km/h?
4. If 5 oranges cost $3.75, how much do 12 oranges cost?
5. On a blueprint, 3 cm represents 1.5 m. A wall measures 11 cm on the blueprint. What is the actual length of the wall?

Click to see answers

1. GCD of 36 and 48 is 12. $36 : 48 = 3 : 4$
2. Cross-multiply: $7 \times 12 = 4 \times x \Rightarrow 84 = 4x \Rightarrow x = 21$
3. Unit rate: $\frac{240}{4} = 60$ km/h
4. Set up proportion: $\frac{5}{3.75} = \frac{12}{x}$. Cross-multiply: $5x = 3.75 \times 12 = 45$. So $x = 9$. The cost is $9.00.
5. Set up proportion: $\frac{3}{1.5} = \frac{11}{x}$. Cross-multiply: $3x = 1.5 \times 11 = 16.5$. So $x = 5.5$. The actual length is 5.5 m.

Summary

• A ratio compares two quantities; order matters and you should always simplify by dividing both parts by their GCD.
• A proportion is an equation of two equal ratios; solve it by cross-multiplying and isolating the unknown.
• A unit rate divides a total by the number of units to find the value per one unit — useful for comparing prices, speeds, and more.
• Ratios and proportions appear everywhere: recipes, maps, scale models, speed, and shopping.

Related Topics

• How to Simplify Ratios — Methods and Examples
• How to Solve Proportions Using Cross Multiplication
• Ratio Word Problems with Step-by-Step Solutions
• Unit Rate — How to Find and Compare Unit Rates
• Fractions — Complete Guide
• Percentages — Calculate, Convert, and Solve

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