Ratio and proportion Revision Notes

A ratio compares the sizes of two or more quantities, showing how many times one quantity contains another. For example, a ratio of 20: 30: 40 compares three quantities.

To simplify a ratio, divide all parts by their greatest common divisor (GCD). This reduces the ratio to its simplest form without changing the relative sizes.

For example, consider the ratio 20: 30: 40. The GCD of 20, 30, and 40 is 10. Dividing each term by 10 gives 2: 3: 4, which is the simplest form.

• Identify the greatest common divisor of all terms.
• Divide each term by the GCD.
• Write the simplified ratio maintaining the order of terms.

When a quantity is divided in a given ratio, the total quantity is split into parts proportional to the ratio numbers.

To divide a quantity, first find the sum of the ratio parts. Then, find the value of one part by dividing the total quantity by this sum. Finally, multiply the value of one part by each ratio number to find each share.

For example, divide 90 litres of juice into the ratio 2: 3: 4. The sum of parts is 2 + 3 + 4 = 9. One part equals 90 ÷ 9 = 10 litres. The shares are: 2 × 10 = 20 litres, 3 × 10 = 30 litres, and 4 × 10 = 40 litres.

• Add the ratio numbers to find total parts.
• Divide the total quantity by total parts to find one part.
• Multiply one part by each ratio number for each share.

Recipes often need to be adjusted for different numbers of servings. Proportional reasoning helps scale ingredient quantities up or down while maintaining the correct ratios.

If a recipe for 4 people requires 200 g of flour, to make it for 6 people, multiply the flour amount by 6 ÷ 4 = 1.5. So, 200 g × 1.5 = 300 g of flour is needed.

This method ensures the proportions between ingredients remain consistent, preserving the recipe's balance and taste.

• Identify the original quantity and the new quantity or servings.
• Calculate the scale factor as new quantity ÷ original quantity.
• Multiply each ingredient by the scale factor.

Maps use scales to represent real distances proportionally. A scale like 1: 50,000 means 1 cm on the map equals 50,000 cm (or 500 m) in reality.

To find the real distance, multiply the map measurement by the scale factor.

For example, if two towns are 3 cm apart on a map with scale 1: 50,000, the real distance is 3 cm × 50,000 = 150,000 cm = 1.5 km.

• Read the scale carefully and understand the units.
• Multiply the map distance by the scale factor to find real distance.
• Convert units as necessary for clarity.

Ratios help compare quantities to find the best value, such as price per unit weight or volume.

For example, if a 500 g packet of rice costs $2.50 and a 750 g packet costs $3.60, calculate the cost per gram for each:

500 g packet: $2.50 ÷ 500 g = $0.005 per gram; 750 g packet: $3.60 ÷ 750 g = $0.0048 per gram. The 750 g packet offers better value as the cost per gram is lower.

• Calculate unit cost or value by dividing total cost by quantity.
• Compare unit costs to determine the best value.
• Use ratios to express and compare these unit values clearly.