What is a ratio?
A ratio compares two (or more) amounts of the same kind. 2 red and 3 blue is written 2 : 3 (say "2 to 3"). Order matters: red : blue = 2 : 3 but blue : red = 3 : 2.
Both parts must be in the same unit. 50 cm : 2 m becomes 50 cm : 200 cm = 1 : 4.
Simplifying and equivalent ratios
Divide both parts by their highest common factor: 12 : 18 → ÷6 → 2 : 3. Multiplying both parts by the same number gives an equivalent ratio: 2 : 3 = 4 : 6 = 10 : 15.
A ratio can also be written as a fraction of the whole: in 2 : 3, red is 2/5 of all blocks and blue is 3/5.
Sharing an amount in a given ratio
Share 30 in the ratio 2 : 3:
- Add the parts: 2 + 3 = 5.
- One part = 30 ÷ 5 = 6.
- Multiply: 2 × 6 = 12 and 3 × 6 = 18. Check: 12 + 18 = 30.
If you know one share instead of the total, find one part from it. Example: in 2 : 3, red is 14, so one part is 7 and blue is 21.
Proportion and direct proportion
A proportion says two ratios are equal: a : b = c : d, or a/b = c/d. Cross-multiply to find a missing value: 3/4 = x/20 → 4x = 60 → x = 15.
Two quantities are in direct proportion if their ratio stays the same: y = k·x. The graph is a straight line through the origin. k is the unit rate (cost of 1 kg, km per litre).
Unitary method
5 pens cost ₹60 → 1 pen costs ₹12 → 8 pens cost ₹96.
Scale drawings and measurement
A scale 1 : 200 means real length = 200 × drawing length. 3.5 cm on the plan → 700 cm = 7 m. Areas scale by the square: 1 : 200 in length is 1 : 40 000 in area.
Inverse proportion
Two quantities are in inverse proportion if their product stays the same: x·y = k, so y = k/x. Double x and y halves. The graph is a curve (a hyperbola), not a straight line.
Example: 6 taps fill a tank in 8 hours (k = 48 tap-hours). 4 taps take 48 ÷ 4 = 12 hours.
Test: if y/x is constant → direct. If x·y is constant → inverse. If neither → not proportional.
Key formulas and definitions
- a : b = (a ÷ h) : (b ÷ h), h = HCF
- Share of A = total × a ÷ (a + b)
- a/b = c/d ⇔ a·d = b·c
- Direct: y = k·x (y/x constant)
- Inverse: x·y = k (y = k/x)
Worked examples
1. Simplify 45 : 60.
HCF = 15. 45 ÷ 15 : 60 ÷ 15 = 3 : 4.
2. Write 1.5 kg : 600 g in simplest form.
Same unit: 1500 g : 600 g. ÷300 → 5 : 2.
3. Share ₹840 between Asha and Ben in the ratio 3 : 4.
3 + 4 = 7 parts. One part = 840 ÷ 7 = ₹120. Asha 3 × 120 = ₹360, Ben 4 × 120 = ₹480.
4. Solve x : 12 = 5 : 8.
x/12 = 5/8 → 8x = 60 → x = 7.5.
5. A car uses 6 L of fuel for 90 km. How far on 10 L?
Direct proportion. Unit rate 90 ÷ 6 = 15 km per litre. 10 × 15 = 150 km.
6. 4 machines finish a job in 15 hours. How long for 6 machines?
Inverse: 4 × 15 = 60 machine-hours. 60 ÷ 6 = 10 hours.
7. A plan has scale 1 : 50. A room is 9 cm by 7 cm on the plan. Find its real area in m².
Real sides: 9 × 50 = 450 cm = 4.5 m and 7 × 50 = 350 cm = 3.5 m. Area = 4.5 × 3.5 = 15.75 m².
Common mistakes
- Mixing units: 30 cm : 1 m is not 30 : 1; it is 30 : 100 = 3 : 10.
- Dividing the total by one part instead of the sum of the parts.
- Swapping the order: boys : girls is not girls : boys.
- Treating every increasing pair as direct proportion: check that y/x stays constant.