Concept 1
A ratio compares like quantities
The ratio a:b means a/b, with b≠0. Convert quantities to the same unit first. Thus 2 m:50 cm = 200:50 = 4:1, not 2:50.
Order matters: if boys:girls=3:5, then girls:boys=5:3.
Concept 2
Equivalent ratios preserve comparison
Multiply or divide both terms by the same non-zero number: 12:18 = 2:3 = 20:30. Reduce using the HCF.
| Ratio | Common factor | Lowest form |
|---|---|---|
| 24:36 | 12 | 2:3 |
| 1.5:2.5 | Clear decimals, then ÷5 | 3:5 |
| 2/3:5/6 | Multiply by LCM 6 | 4:5 |
Concept 3
A proportion equates two ratios
a:b=c:d exactly when ad=bc. In 4:7=20:35, both cross-products equal 140. The fourth proportional to a, b and c is bc/a.
Concept 4
Divide a total in a given ratio
To divide ₹840 in 3:4, total parts=7. One part=840/7=120, so the shares are ₹360 and ₹480.
General form: shares of total T in a:b are T×a/(a+b) and T×b/(a+b).
Concept 5
Join and compound ratios carefully
If A:B=2:3 and B:C=4:5, equalise the common term B. The LCM of 3 and 4 is 12, giving A:B=8:12 and B:C=12:15; therefore A:B:C=8:12:15.
The compound ratio of a:b and c:d is ac:bd. Thus the compound ratio of 2:3 and 5:7 is 10:21.
Concept 6
Direct and inverse proportion
Direct
Both quantities change by the same factor: more items at a fixed rate cost more. y/x stays constant.
Inverse
One rises as the other falls: more workers need fewer days for fixed work. xy stays constant.
Test the relationship: doubling one quantity doubles the other in direct proportion, but halves the other in inverse proportion.
Applications
How exams use ratios
Partnership
Profit is divided in the ratio of capital×time. ₹20,000 for 12 months and ₹30,000 for 8 months contribute equally: 240,000:240,000=1:1.
Mixtures
If milk:water=5:2 in 21 L, milk=21×5/7=15 L and water=6 L.
Ages
If ages are 3:5 and total 48, one part is 6, so the ages are 18 and 30.
Speed and time
For the same distance, speed and time are inverse: speed ratio 3:4 gives time ratio 4:3.
Worked methods
Examples with reasoning
Find x if 6:15=x:35
6×35=15x, so x=14.
Divide ₹2,700 among A, B and C in 2:3:4
Total parts=9 and one part=₹300. Shares are ₹600, ₹900 and ₹1,200.
If 8 workers finish in 15 days, how long for 12 workers?
Workers and days are inverse. 8×15=12×d, so d=10 days.
A:B=3:4 and B:C=2:5
Make B equal: 3:4 becomes 3:4 and 2:5 becomes 4:10 after multiplying by 2. Therefore A:B:C=3:4:10.
Partners invest ₹40,000 for 9 months and ₹60,000 for 6 months
The capital-time ratio is 360,000:360,000=1:1, so profit is shared equally.
Exam traps
Errors that break the comparison
Different units
Convert both quantities to the same unit before forming the ratio.
Reversing order
a:b and b:a are reciprocals, not interchangeable labels.
Adding ratio terms
2:3 plus 4:5 is not 6:8; ratios are comparisons, not ordinary pairs.
Unequal middle term
When joining A:B and B:C, first make both B values equal.
Direct versus inverse
Ask whether doubling one quantity doubles or halves the other.
Ignoring time in partnership
Use capital×duration, not capital alone, when investment periods differ.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Make units identical, preserve order, then reduce by the HCF.
- a:b=c:d exactly when ad=bc.
- To divide total T in a:b, use T×a/(a+b) and T×b/(a+b).
- Equalise the shared term before joining two ratios.
- Direct proportion keeps y/x constant; inverse proportion keeps xy constant.
- Partnership share is proportional to capital×time.
Source ledger
Official foundations
- SSC CGL official syllabus: Ratio and Proportion
- NCERT official mathematics exemplar directory
- NCERT Direct and Inverse Proportion exemplar
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Reviewed 24 September 2026.
Square roots and surds: exact forms, estimation and simplification.