Solve ratio and proportion problems

To simplify a ratio, divide both terms by their greatest common divisor (GCD).

Example 1: A classroom has 45 girls and 40 boys. Find the ratio of boys to girls in simplest form.

• Write the ratio: 40:45
• Write as a fraction: $\frac{40}{45}$
• Find the GCD of 40 and 45, which is 5
• Divide both terms by 5: $\frac{40 \div 5}{45 \div 5} = \frac{8}{9}$

Answer: 8:9

Example 2: Express the ratio 120:90 in simplest form.

• Write as a fraction: $\frac{120}{90}$
• Find the GCD of 120 and 90, which is 30
• Divide both terms by 30: $\frac{120 \div 30}{90 \div 30} = \frac{4}{3}$

Answer: 4:3

Before comparing quantities, convert them to the same unit.

Example: Find the ratio of 2 hours to 90 minutes.

• Convert 2 hours to minutes: $2 \times 60 = 120$ minutes
• Write the ratio: 120:90
• Simplify: divide both by 30 → $\frac{120}{90} = \frac{4}{3}$

Answer: 4:3

A proportion is an equation stating that two ratios are equal. We use cross-multiplication to solve for an unknown value.

Example: If 3 pens cost TZS 1,500, how much will 7 pens cost?

Let $x$ be the cost of 7 pens.

Set up the proportion:

$\frac{3}{1500} = \frac{7}{x}$

Cross-multiply:

$3x = 7 \times 1500$ $3x = 10,500$ $x = \frac{10,500}{3} = 3,500$

Answer: TZS 3,500

Example: A recipe uses 4 cups of flour to make 10 biscuits. How much flour is needed for 25 biscuits?

Let $x$ be the amount of flour needed.

$\frac{4}{10} = \frac{x}{25}$

Cross-multiply:

$4 \times 25 = 10x$ $100 = 10x$ $x = 10$

Answer: 10 cups of flour

• Check that both quantities in a ratio have the same unit
• Find the GCD to simplify ratios
• For proportions, set up the equation with the same order on both sides
• Cross-multiply to find the unknown value

In Tanzania, ratios and proportions are used when sharing money among friends, mixing ingredients for cooking, or calculating ingredient quantities for selling local foods like chipsi mayai or vitumbua. For example, if 5 kilograms of rice cost TZS 25,000 at a market in Dar es Salaam, a student can use proportions to find the cost of 8 kilograms — a practical skill for family shopping and budgeting.