Ratio And Proportion

Basic Ratio Definition

A way to compare the relative sizes of two or more quantities of the same kind, written with a colon (a:b).

Ratio Order Rule

The order of numbers in a ratio must match the order in the description (e.g., 3 red to 5 blue = 3:5).

Ratio to Fraction Conversion

Add the ratio parts together to get the denominator. For ratio 3:5, total parts = 8, so the fraction of red is 3/8.

Simplifying Ratios Method

Divide all numbers in the ratio by their Highest Common Factor (HCF) until no common factors remain except 1.

Simplifying Ratios Example

Simplify 12:18 -> HCF of 12 and 18 is 6 -> (12÷6):(18÷6) = 2:3.

Ratio Units Rule

Ratios must always be in the same units before simplifying (e.g., 50p : £2 becomes 50p : 200p -> 1:4).

Ratio Sharing (Add-Divide-Multiply)

Step 1: Add ratio parts for total parts. Step 2: Divide total amount by total parts to get 1 part. Step 3: Multiply 1 part by each ratio number.

Ratio Sharing Worked Example

Share £60 in ratio 2:3:5 -> Total parts = 2+3+5=10. Value of 1 part = 60/10 = £6. Parts: 2x6=£12, 3x6=£18, 5x6=£30.

Map Scales in Ratios

Expressed as a ratio without units (e.g., 1:25,000 means 1 cm on the map equals 25,000 cm or 250 m in real life).

Recipe Scaling with Ratios

Multiply or divide all ingredients by the same scale factor to change batch sizes.

Direct Proportion Definition

Two quantities are in direct proportion if one increases at the exact same rate as the other (if one doubles, the other doubles).

Direct Proportion Formula

y ∝ x implies y = kx (where k is the constant of proportionality).

Unitary Method Definition

Finding the cost or value of 1 single item first, then multiplying to scale up to the target amount.

Unitary Method Worked Example

4 pens cost £3.20. Find cost of 7 pens -> 1 pen = 3.20 / 4 = £0.80. 7 pens = 7 x 0.80 = £5.60.

Inverse Proportion Definition

Two quantities are in inverse proportion if as one increases, the other decreases at the exact same rate (if one doubles, the other halves).

Inverse Proportion Formula

y ∝ 1/x implies y = k/x or x * y = k (constant product).

Inverse Proportion Worked Example

3 workers take 8 hours to build a wall. How long for 4 workers? -> Total person-hours = 3 x 8 = 24. Time for 4 workers = 24 / 4 = 6 hours.

Direct vs Inverse Graph Shapes

Direct proportion is a straight line through the origin (0,0). Inverse proportion is a curved hyperbola that never touches the axes.

Direct vs Inverse Key Equations

Direct proportion has constant ratio (y/x = k). Inverse proportion has constant product (x * y = k).

Best Buy Methods

Option 1: Find cost per unit (lower cost is better). Option 2: Find quantity per penny/pound (higher quantity is better).

Best Buy Worked Example

Shop A: 500g for £2.50 (£0.50/100g). Shop B: 750g for £3.60 (£0.48/100g). Shop B is the best buy.

Speed Formula & Units

Speed = Distance / Time (Units: mph, km/h, or m/s).

Density Formula & Units

Density = Mass / Volume (Units: g/cm³ or kg/m³).

Pressure Formula & Units

Pressure = Force / Area (Units: N/m² or Pascals Pa).

Reverse Percentage Method

1. Represent original as 100%. 2. Find new percentage multiplier. 3. Divide current amount by multiplier to find original value.

Reverse Percentage Worked Example

Shirt costs £36 after a 10% discount. Original price = 36 / 0.90 = £40.

Compound Interest Concept

Interest earned on the original amount PLUS any interest already accumulated in previous periods.

Compound Interest Formula

Final Amount = P * (1 + r/100)^n (where P = Principal, r = percentage rate, n = number of periods).

Compound Interest Worked Example

Invest £1,000 at 4% for 3 years -> 1000 * (1.04)³ = £1,124.86.