Section 8, arithmetic

Ratio, Proportion, Partnership & Variation

Ratio hides inside ages, incomes, coins, mixtures and fruit-seller stories, so it shows up in nearly every paper. The one move to master: write every quantity as “parts × k”, then find what stays constant.

How CAT asks it

Quantities in a ratio change (people join, items are sold, years pass) and a new ratio is given; you find the original or final numbers. Partnership shares, coin values, incomes against expenses and “partly fixed, partly variable” costs also recur. Answers usually must be whole numbers, which is itself a clue.

Concepts you must know

  • a : b = ka : kb. Write A = ak and B = bk.
  • Combine ratios through the common term: A : B = 3 : 4 and B : C = 6 : 7 → make B equal (12) → 9 : 12 : 14.
  • If a/b = c/d = e/f = k, then (a + c + e)/(b + d + f) = k as well.
  • Componendo-dividendo: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d).
  • Proportion a : b :: c : d means ad = bc. The mean proportional of a and b is √(ab).
  • Partnership: profit share is proportional to capital × time.
  • Variation: direct y = kx; inverse y = k/x; joint z = kxy; “partly fixed, partly variable” C = F + vn.
  • Ages: the difference between two people’s ages never changes.
  • 2A = 3B = 4C means A : B : C = 1/2 : 1/3 : 1/4 = 6 : 4 : 3 (multiply through by 12).

Thought process

  1. Write each quantity as parts × k. Use a different multiplier for each unrelated ratio.
  2. Find what stays constant: an age gap, a component nobody touched, a total.
  3. Equate before and after through that constant, then solve for k.
  4. Check that the answer gives whole numbers where it must (people, coins, fruits).

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Combining ratios through a common term.
  2. 2Before/after changes: students join, fruits are sold, water is added; a new ratio is given.
  3. 3Incomes and expenditures in ratios, with savings given.
  4. 4Coins and notes: the number ratio differs from the value ratio.
  5. 5Partnership: capital × months; partners joining late or leaving early; a working partner’s salary paid first.
  6. 6Variation: fixed + variable cost; inverse or inverse-square relations.
  7. 7Ages: ratios now, n years ago and n years hence.
  8. 8Integer clues: totals must be multiples of the ratio sum; count the possible values.

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1, combining

    A : B = 3 : 4 and B : C = 6 : 7. Find A : B : C.

    Show solution

    Make B = 12. A : B = 9 : 12 and B : C = 12 : 14. Answer: 9 : 12 : 14.

  2. Example 2, incomes and expenditures

    The incomes of X and Y are in the ratio 5 : 4 and their expenditures 3 : 2. Each saves ₹8,000. Find X’s income.

    Show solution

    5k − 3m = 8,000 and 4k − 2m = 8,000. The second gives m = 2k − 4,000; substituting, 5k − 6k + 12,000 = 8,000, so k = 4,000. Check: X spends 12,000; Y earns 16,000 and spends 8,000. Answer: ₹20,000.

  3. Example 3, before/after

    Two classes are in the ratio 7 : 5. After 12 students join the smaller class, the ratio becomes 7 : 6. How big was the larger class?

    Show solution

    The larger class did not change and is 7 parts in both ratios, so the parts are the same size. The smaller class went from 5 to 6 parts, so 1 part = 12. Answer: 84.

  4. Example 4, ages

    Ten years ago, a father’s and son’s ages were in the ratio 5 : 1. Ten years from now they will be 7 : 3. Find the father’s present age.

    Show solution

    The age gap is 4 parts in both ratios, so the parts are equal. The father’s 7 parts later = his 5 parts earlier + 20 years, so 2 parts = 20 and 1 part = 10. Father now = 50 + 10. Check: 50 : 10 and 70 : 30. Answer: 60.

  5. Example 5, partnership

    A starts a business with ₹60,000. B joins after 4 months with ₹80,000, and C joins after 7 months with ₹1,00,000. The year’s profit is ₹93,000. Find C’s share.

    Show solution

    Capital × months (in thousands): A = 60 × 12 = 720, B = 80 × 8 = 640, C = 100 × 5 = 500; total 1,860. C’s share = 500/1,860 × 93,000. Answer: ₹25,000.

  6. Example 6, coins

    A bag holds ₹1, 50p and 25p coins in the ratio 2 : 3 : 4 by number, worth ₹216 in total. How many 25p coins are there?

    Show solution

    One set of 2 + 3 + 4 coins is worth 2 + 1.5 + 1 = ₹4.50. Sets = 216 ÷ 4.5 = 48. Answer: 192.

  7. Example 7, variation

    A school bus costs ₹19,000 a month for 30 students and ₹25,000 for 50; the cost is partly fixed and partly per student. What is the cost for 70 students?

    Show solution

    20 more students cost 6,000, so ₹300 each; fixed cost = 19,000 − 9,000 = 10,000. For 70: 10,000 + 21,000. Answer: ₹31,000.

  8. Example 8, integer clue

    A basket has apples and oranges in the ratio 5 : 3. After 13 apples and 3 oranges are sold, the ratio becomes 3 : 2. How many fruits were there at first?

    Show solution

    (5k − 13)/(3k − 3) = 3/2, so 10k − 26 = 9k − 9 and k = 17. Initially 85 + 51. Check: 72 : 48 = 3 : 2. Answer: 136.

Traps

  • Age ratios change over time; age gaps do not.
  • In partnership, multiply capital by time.
  • In coin questions, the number ratio is not the value ratio.
  • Never reuse the same k for two different ratio statements.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    Divide ₹7,800 among A, B and C so that 2A = 3B = 4C. Find C’s share.

    Show answer

    ₹1,800: the ratio is 6 : 4 : 3, so C = 3/13 × 7,800.

  2. 2

    A : B = 2 : 3, B : C = 4 : 5 and C : D = 6 : 7. Find A : D.

    Show answer

    16 : 35: 2/3 × 4/5 × 6/7 = 48/105.

  3. 3

    Two numbers are in the ratio 3 : 5. Subtracting 9 from each gives the ratio 12 : 23. Find the smaller number.

    Show answer

    33: 69k − 207 = 60k − 108 gives k = 11.

  4. 4

    The incomes of P and Q are in the ratio 3 : 2 and their expenditures 5 : 3. Each saves ₹2,000. Find P’s income.

    Show answer

    ₹12,000: from 3k − 5m = 2,000 and 2k − 3m = 2,000, m = 2,000 and k = 4,000.

  5. 5

    A and B invest in the ratio 3 : 5 for 8 and 6 months respectively. Find their profit-sharing ratio.

    Show answer

    4 : 5: 24 : 30.

  6. 6

    y varies inversely as x², and y = 8 when x = 3. Find y when x = 6.

    Show answer

    2: k = 8 × 9 = 72, and 72/36 = 2.

  7. 7

    Find the mean proportional of 9 and 16.

    Show answer

    12: √144.

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