A ratio is in its simplest form when both numbers cannot be divided by any common number (other than 1). This makes the ratio easier to understand.

Steps to Simplify a Ratio

1. Write the two quantities as a fraction
2. Find the Greatest Common Divisor (GCD) of both numbers
3. Divide both numbers by the GCD
4. Write the simplified ratio using a colon

Worked Example

A Tanzanian shopkeeper has 150 kg of rice and 250 kg of beans. Find the ratio of rice to beans in simplest form.

Step 1: Write as a fraction: $\frac{150}{250}$

Step 2: Find the GCD of 150 and 250, which is 50

Step 3: Divide both numbers by 50: $150 \div 50 = 3$ $250 \div 50 = 5$

Step 4: Write the simplified ratio: 3:5

So the ratio of rice to beans is 3:5.

Another Example

Express the ratio 8:12 in simplest form.

Step 1: Write as a fraction: $\frac{8}{12}$

Step 2: The GCD of 8 and 12 is 4

Step 3: Divide both by 4: $8 \div 4 = 2$ $12 \div 4 = 3$

Answer: 2:3

When comparing quantities with different units, first convert them to the same unit.

Worked Example

Find the ratio of 2 hours to 90 minutes.

Step 1: Convert 2 hours to minutes: $2 \times 60 = 120 \text{ minutes}$

Step 2: Write the ratio as a fraction: $\frac{120}{90}$

Step 3: Simplify (divide both by 30): $\frac{120 \div 30}{90 \div 30} = \frac{4}{3}$

Answer: 4:3

A proportion is an equation that shows two ratios are equal. It tells us that two comparisons have the same relationship.

For example, if 3 shirts cost TSh 15,000, then 6 shirts would cost TSh 30,000. We can write this as a proportion:

$3:15,000 = 6:30,000$

Or using fractions: $\frac{3}{15000} = \frac{6}{30000}$

Checking if Two Ratios Form a Proportion

Two ratios form a proportion if their simplified forms are equal.

Example: Do the ratios 2:4 and 3:6 form a proportion?

• Simplify 2:4 = 1:2
• Simplify 3:6 = 1:2

Since both simplify to 1:2, they form a proportion.

In everyday life in Tanzania, ratios and proportions are used when mixing concrete for building, cooking food for a large group, or calculating exchange rates when changing money. For example, if a farmer wants to mix fertilizer with water in the ratio 1:4 (1 part fertilizer to 4 parts water), understanding ratios helps the farmer prepare the correct mixture for better crop yield.