Rates,Ratios,Proportion and Percentages Notes, Quizzes & Revision

Key Definitions

• Ratio — a comparison of two quantities using ":" or "/". Example 2:3 or 2/3. Read "2 to 3".
• Rate — a comparison of two quantities with different units (e.g., km/h, Ksh per kg, items per minute). Example: 60 km in 2 h → rate = 30 km/h.
• Proportion — an equation that shows two ratios are equal. E.g., 2/3 = 4/6.
• Percentage — a number out of 100. 1% = 1/100. Write as 25% = 25/100 = 0.25.

Ratios — how to use them

Write a ratio in simplest form by dividing both parts by their highest common factor (HCF).

Example — simplify: 8:12 → divide both by 4 → 2:3.

Sharing a quantity in a ratio (unitary method):

1. Add the parts of the ratio to get total parts.
2. Divide the total quantity by total parts to find one part.
3. Multiply by each part to get each share.

Worked example: Share Ksh 300 in the ratio 2:3.

Total parts = 2 + 3 = 5. One part = 300 ÷ 5 = Ksh 60. So first share = 2 × 60 = Ksh 120. Second share = 3 × 60 = Ksh 180.

Rates

A rate compares quantities with different units. Often find a unit rate (value of one unit).

Example: A car travels 180 km in 3 hours. Unit rate = 180 ÷ 3 = 60 km/hour.

Another example: Farmer sells 15 kg of maize for Ksh 1,200. Price per kg = 1200 ÷ 15 = Ksh 80 per kg.

Proportion

Two ratios form a proportion when they are equal. Use cross-multiplication to check or find an unknown.

Rule (cross-multiplication): a/b = c/d ⇔ a × d = b × c.

Example: Is 3/5 = 6/10? Cross-multiply: 3×10 = 30 and 5×6 = 30 → yes, they are equal.

Find the missing value: 4/9 = x/27. Cross-multiply: 4×27 = 9x → 108 = 9x → x = 12.

Percentages

Percent means "per hundred". To convert:

• Fraction → percent: multiply by 100. Example: 1/4 = 0.25 → 0.25×100 = 25%.
• Decimal → percent: ×100. Example: 0.07 = 7%.
• Percent → decimal: divide by 100. Example: 12% = 12/100 = 0.12.

Find percentage of a number: percent × number (as decimal).

Worked example: What is 25% of 240? 25% = 0.25 → 0.25 × 240 = 60.

Increase and decrease by a percentage:

• Increase by p%: Multiply by (1 + p/100). Example: Increase Ksh 200 by 15% → 200 × 1.15 = Ksh 230.
• Decrease by p%: Multiply by (1 − p/100). Example: Decrease Ksh 500 by 10% → 500 × 0.90 = Ksh 450.

Connections: Ratio, Fraction & Percentage

If ratio a:b is part of a whole, you can convert:

1. Fraction = a / (a + b).
2. Percentage = [a / (a + b)] × 100%.

Example: From earlier ratio 2:3, fraction for first part = 2/(2+3) = 2/5 = 0.4 → 40%.

Practice Questions

1. Write 45:30 in simplest form.
2. Share Ksh 420 in the ratio 3:4. Find each share.
3. A motorcycle covers 150 km in 2.5 hours. What is its speed in km/h?
4. Find x if 7/12 = x/36.
5. Convert 3/8 to a percentage.
6. What is 18% of Ksh 850?
7. A shirt costs Ksh 1200. After a 25% discount what is the selling price?
8. If 5 pens cost Ksh 225, what is the price per pen (unit rate)?

Answers

1. 45:30 = divide by 15 → 3:2.
2. Total parts = 3 + 4 = 7. One part = 420 ÷ 7 = 60. Shares: 3×60 = Ksh 180 and 4×60 = Ksh 240.
3. Speed = 150 ÷ 2.5 = 60 km/h.
4. 7/12 = x/36 → 7×36 = 12x → 252 = 12x → x = 21.
5. 3/8 = 0.375 → 37.5%.
6. 18% of 850 = 0.18 × 850 = Ksh 153.
7. 25% discount = 0.25 × 1200 = 300 off. Selling price = 1200 − 300 = Ksh 900.
8. Price per pen = 225 ÷ 5 = Ksh 45 per pen.