Ratio, Rate & Percentage
Ratio divides a total. Rate compares different units. Percentage change always uses the original value.
A. Ratio
Ratio
A ratio compares two or more quantities of the same unit. Written as a:b. Always simplify by dividing both sides by their HCF.
Example
Ratio of 15 to 25 → HCF is 5 → divide both → simplified ratio = 3:5
This means: for every 3 of one thing, there are 5 of the other.
This means: for every 3 of one thing, there are 5 of the other.
Dividing a Total in a Given Ratio
Add the ratio numbers to get total parts. Then each share = (its part ÷ total parts) × whole amount.
Example
Divide Rs.800 in ratio 3:5.
Total parts = 3+5 = 8.
First share = 3/8 × 800 = Rs.300
Second share = 5/8 × 800 = Rs.500
Total parts = 3+5 = 8.
First share = 3/8 × 800 = Rs.300
Second share = 5/8 × 800 = Rs.500
Dividing in a Ratio — Visual
Example 1 — Divide Rs.2400 in ratio 1:3
1Total parts = 1 + 3 = 4
2First share = 1/4 × 2400 = Rs.600
3Second share = 3/4 × 2400 = Rs.1800
✓Check: 600 + 1800 = 2400 ✓
Rs.600 and Rs.1800
Example 2 — Divide Rs.4500 in ratio 2:3:4
1Total parts = 2+3+4 = 9
21st = 2/9 × 4500 = Rs.1000
32nd = 3/9 × 4500 = Rs.1500
43rd = 4/9 × 4500 = Rs.2000
✓1000+1500+2000 = 4500 ✓
Rs.1000 : Rs.1500 : Rs.2000
B. Rate
Rate
A rate compares two quantities of different units. The most common example is speed — kilometres per hour.
Example
60 km/h means the car travels 60 kilometres for every 1 hour.
Rs.130 per kg means 1 kg costs Rs.130.
Rs.130 per kg means 1 kg costs Rs.130.
Unit Rate
The rate for exactly 1 unit of the second quantity. Find it by dividing.
Example
5 kg costs Rs.650 → unit rate = 650 ÷ 5 = Rs.130 per kg
Now cost of 8 kg = 8 × 130 = Rs.1040
Now cost of 8 kg = 8 × 130 = Rs.1040
Example 1 — A car travels 240 km in 4 hours. Find speed.
1Speed = Distance ÷ Time
2Speed = 240 ÷ 4 = 60 km/h
Speed = 60 km/h
Example 2 — 3 notebooks cost Rs.135. How much for 7?
1Unit rate = 135 ÷ 3 = Rs.45 per notebook
27 notebooks = 7 × 45 = Rs.315
Rs.315
Ratio vs Rate: Ratio = same unit (money:money). Rate = different units (km:hours, Rs.:kg).
C. Percentage
Percentage
Percentage means "out of 100". It tells you how large a part is compared to the whole.
Formula
Percentage = (Part ÷ Whole) × 100
Part = (Percentage ÷ 100) × Whole
Part = (Percentage ÷ 100) × Whole
Converting: Fraction ↔ Percentage
Fraction to percentage: multiply by 100. Percentage to fraction: divide by 100 and simplify.
Example
3/4 → 3/4 × 100 = 75%
40% → 40/100 = 4/10 = 2/5
40% → 40/100 = 4/10 = 2/5
Example 1 — A student scored 54 out of 75. What percentage?
1% = (Part ÷ Whole) × 100
2= (54 ÷ 75) × 100 = 0.72 × 100 = 72%
72%
Example 2 — Find 35% of Rs.2400
1Part = (35 ÷ 100) × 2400
2= 0.35 × 2400 = Rs.840
Rs.840
Example 3 — What percent is 45 of 180?
1% = (45 ÷ 180) × 100
2= 0.25 × 100 = 25%
25%
D. Percentage Change
Percentage Increase
When a value goes up — divide the increase by the original value, then multiply by 100.
Formula
% Increase = (New − Old) ÷ Old × 100
Percentage Decrease
When a value goes down — divide the decrease by the original value, then multiply by 100.
Formula
% Decrease = (Old − New) ÷ Old × 100
% Change — Always Divide by the ORIGINAL
Example 1 — Price rises from Rs.500 to Rs.600
1Increase = 600 − 500 = 100
2% increase = (100 ÷ 500) × 100 = 20%
20% increase
Example 2 — Salary drops from Rs.12000 to Rs.9000
1Decrease = 12000 − 9000 = 3000
2% decrease = (3000 ÷ 12000) × 100 = 25%
25% decrease
Example 3 — Find new price: Rs.800 increased by 15%
1Increase amount = 15% of 800 = 0.15 × 800 = Rs.120
2New price = 800 + 120 = Rs.920
Rs.920 (shortcut: 800 × 1.15 = 920)
⚠ Biggest mistake: Always divide by the ORIGINAL value — not the new value. If it was Rs.500 before, divide by 500.
Quick MCQ Revision
| Formula | Remember |
|---|---|
% | (Part ÷ Whole) × 100 |
% Change | (Change ÷ Original) × 100 — original always below |
Divide in ratio | (part ÷ total parts) × whole |
Unit rate | Total ÷ quantity → then × what you need |
| Ratio vs Rate | Ratio = same unit · Rate = different units |