Ratio is the quiet engine of CDS arithmetic. Partnerships, mixtures, time and work, speed and even similar triangles all lean on it. Variation, which the syllabus names separately, is ratio written as an equation. A candidate who is fluent here saves time across half the arithmetic section.
The one habit to build is the parts method. A ratio of 5 : 7 does not mean 5 and 7. It means 5 parts and 7 parts of some unknown size. Find the size of one part and everything else follows.
Ratio basics
- A ratio a : b compares two quantities of the same kind, in the same units.
- Multiplying or dividing both terms by the same non-zero number does not change the ratio.
- To divide a quantity Q in the ratio a : b, one part = Q ÷ (a + b). The shares are a parts and b parts.
Combining ratios
If A : B = 2 : 3 and B : C = 4 : 5, B has two different values. Make B the same in both by scaling to the LCM of 3 and 4, which is 12.
- A : B = 8 : 12 and B : C = 12 : 15
- So A : B : C = 8 : 12 : 15
Special ratios
| Name | For a : b | Example with 2 : 3 |
|---|---|---|
| Duplicate ratio | a² : b² | 4 : 9 |
| Sub-duplicate ratio | √a : √b | √2 : √3 |
| Triplicate ratio | a³ : b³ | 8 : 27 |
| Inverse ratio | b : a | 3 : 2 |
| Compound ratio of a : b and c : d | ac : bd | With 9 : 4, it is 18 : 12 = 3 : 2 |
Proportion
Four numbers a, b, c, d are in proportion when a : b = c : d, that is, when ad = bc (product of extremes equals product of means).
- Fourth proportional to a, b, c is d = bc ÷ a.
- Third proportional to a and b is c where a : b = b : c, so c = b² ÷ a.
- Mean proportional of a and b is x where a : x = x : b, so x = √(ab).
Componendo and dividendo
If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d). Why: add 1 to both sides to get (a + b)/b = (c + d)/d, subtract 1 to get (a − b)/b = (c − d)/d, then divide one by the other. The rule also works backwards, which is where it saves most time.
Variation
| Type | Statement | Equation | What stays fixed |
|---|---|---|---|
| Direct | x varies directly as y | x = ky | x ÷ y |
| Inverse | x varies inversely as y | xy = k | x × y |
| Joint | x varies directly as y and inversely as z | x = ky ÷ z | xz ÷ y |
| With powers | x varies as y² | x = ky² | x ÷ y² |
The method is always the same: write the equation with k, use the given pair of values to find k, then substitute the new values.
Worked questions
Question 1: A : B = 2 : 3 and B : C = 4 : 5. Divide ₹1,400 among A, B and C.
- A : B : C = 8 : 12 : 15, a total of 35 parts.
- One part = 1,400 ÷ 35 = 40.
- A = ₹320, B = ₹480, C = ₹600.
Question 2: What number must be subtracted from each of 15, 28, 20 and 38 so that the results are in proportion?
- Let the number be x. Then (15 − x)(38 − x) = (28 − x)(20 − x).
- 570 − 53x + x² = 560 − 48x + x², so 10 = 5x and x = 2.
- Check: 13 : 26 = 1 : 2 and 18 : 36 = 1 : 2.
Question 3: The incomes of two people are in the ratio 5 : 4 and their expenditures in the ratio 3 : 2. Each saves ₹2,000. Find their incomes.
- Let the incomes be 5x and 4x, and the expenditures 3y and 2y.
- 5x − 3y = 2,000 and 4x − 2y = 2,000. The second gives y = 2x − 1,000.
- Substituting: 5x − 6x + 3,000 = 2,000, so x = 1,000.
- Incomes = ₹5,000 and ₹4,000. Expenditures = 3,000 and 2,000, and both save 2,000.
Question 4: If (x + y)/(x − y) = 5/3, find x : y.
- Componendo and dividendo: x/y = (5 + 3)/(5 − 3) = 8/2 = 4.
- So x : y = 4 : 1. Check: (4 + 1)/(4 − 1) = 5/3.
Question 5: x varies directly as y and inversely as z. x = 6 when y = 4 and z = 2. Find x when y = 9 and z = 3.
- x = ky ÷ z. So 6 = k × 4 ÷ 2, giving k = 3.
- x = 3 × 9 ÷ 3 = 9.
Question 6: A bag has ₹1, 50-paise and 25-paise coins in the ratio 5 : 6 : 8. The total value is ₹210. How many coins of each kind are there?
- Let the numbers be 5x, 6x and 8x. Their values in rupees are 5x, 3x and 2x.
- 10x = 210, so x = 21.
- Coins: 105, 126 and 168.
Practice set
- Find the third proportional to 9 and 12.
- Find the fourth proportional to 3, 5 and 12.
- x varies as y². x = 18 when y = 3. Find x when y = 5.
- Find the mean proportional of 9 and 16.
- Divide ₹1,560 in the ratio 5 : 7.
- A : B = 3 : 4 and B : C = 6 : 7. Find A : C.
- Two numbers are in the ratio 3 : 5. If 9 is subtracted from each, the ratio becomes 12 : 23. Find the numbers.
- Find the compound ratio of 2 : 3, 6 : 5 and 5 : 8.
Answers
- 16. 12² ÷ 9 = 144 ÷ 9.
- 20. 5 × 12 ÷ 3.
- 50. 18 = k × 9 gives k = 2, and 2 × 25 = 50.
- 12. √(9 × 16) = √144.
- ₹650 and ₹910. One part = 1,560 ÷ 12 = 130.
- 9 : 14. Make B equal to 12: A : B : C = 9 : 12 : 14.
- 33 and 55. (3x − 9)/(5x − 9) = 12/23 gives 69x − 207 = 60x − 108, so x = 11.
- 1 : 2. (2 × 6 × 5) : (3 × 5 × 8) = 60 : 120.
What to do next
- Practise combining three ratios until you can do it without writing the LCM step
- Solve five variation questions using only the "find k first" method
- Apply the parts method in averages and mixtures and time and work
- Time yourself on the ratio and variation questions from recent CDS papers
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Union Public Service Commission website .
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