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Numerical Aptitude Syllabus / Module 6 · Ratio & Proportion / Week 6

Week 6: Ratio, Proportion & Variation

A ratio isn't a fraction you calculate — it's a single unknown multiplier you assign once and reuse. That reframe is what makes ratio problems fast, and it's the same idea that underlies partnership, mixture and time-and-work problems throughout the rest of this course.

Module 6 of 20 Week 6 of 20 ~2–3 Hours Practice Problems Included

By the end of this week, you'll be able to

  • Set up any ratio as a single unknown multiplier and solve directly
  • Tell direct and inverse variation apart from a problem's wording
  • Split a partnership's profit correctly when investments differ in both amount and time
1 Ratios as a Single Multiplier 2 Combining & Comparing Ratios 3 Direct vs. Inverse Variation 4 Partnership & Profit-Sharing 5 Practice Problems 6 Knowledge Check

1. Ratios as a Single Multiplier

"Two numbers are in the ratio 3:5, and their sum is 64. Find the numbers." The fast approach: call the numbers 3x and 5x, not two separate unknowns.

the single-multiplier setup
Numbers are 3x and 5x (NOT two independent variables --
one shared unknown x)

3x + 5x = 64
8x = 64
x = 8

Numbers: 3x = 24, 5x = 40
Check: 24 + 40 = 64 -- correct, and 24:40 simplifies to 3:5

This single-multiplier trick generalizes immediately to three or more terms (a:b:c becomes ax, bx, cx) and is the fastest path through almost every ratio word problem — one equation, one unknown, instead of setting up two separate variables and a system to solve.

2. Combining & Comparing Ratios

"A:B = 2:3 and B:C = 4:5. Find A:B:C." The two ratios share B, but with different values (3 and 4) — they need to be scaled to a common B before combining.

combining two ratios via a common term
A:B = 2:3
B:C = 4:5

B's value differs (3 vs 4) -- scale both ratios so B matches.
LCM(3,4) = 12

A:B = 2:3  -> multiply by 4  ->  8:12
B:C = 4:5  -> multiply by 3  ->  12:15

Now B matches (12 in both):
A:B:C = 8:12:15

This is exactly Week 2's LCM idea, reused: find the LCM of the shared term's two values, scale each ratio up to match it, then read off the combined ratio directly. The same technique handles chains of three or more linked ratios by combining two at a time.

3. Direct vs. Inverse Variation

Two quantities vary directly if one increasing makes the other increase proportionally; they vary inversely if one increasing makes the other decrease proportionally. Telling them apart from the wording is the actual skill.

direct variation — "more of one means more of the other"
"Cost varies DIRECTLY with quantity bought"
  -> Cost/Quantity = constant  ->  Cost = k x Quantity

If 5 books cost 250, find the cost of 8 books.
  250/5 = 50 (cost per book, the constant k)
  8 x 50 = 400
inverse variation — "more of one means LESS of the other"
"Time taken varies INVERSELY with number of workers"
  -> Time x Workers = constant

If 6 workers finish a job in 10 days, how long would 15
workers take?
  6 x 10 = 60 (the constant)
  60 / 15 = 4 days

The reliable test: hold everything else fixed and ask "if this quantity doubles, does the other one also roughly double (direct), or roughly halve (inverse)?" — more workers finishing a fixed job faster is inverse; more items costing proportionally more at a fixed price is direct. This same inverse relationship reappears explicitly in Week 8's time-and-work chapter.

4. Partnership & Profit-Sharing

When partners invest different amounts for different lengths of time, profit isn't split by investment amount alone — it's split by investment × time, since a rupee invested for twice as long contributes twice as much to the business.

partnership — investment x time
A invests 4,000 for 12 months.
B invests 6,000 for 8 months.
Total profit: 3,700. Find each partner's share.

A's "capital-months" = 4,000 x 12 = 48,000
B's "capital-months" = 6,000 x 8  = 48,000

Ratio A:B = 48,000 : 48,000 = 1:1

Despite investing LESS money, A gets an EQUAL share --
because A's money was invested for longer, exactly
compensating for the smaller amount.

A's share = 3,700 x 1/2 = 1,850
B's share = 3,700 x 1/2 = 1,850

This is Section 1's single-multiplier ratio technique again, just applied to a two-factor product (money × time) instead of a raw quantity — set up the capital-months for each partner, form the ratio, then split the total profit by that ratio directly.

5. Practice Problems

Practice

30 problems, no calculator — Easy, Medium & Tough

Set up every ratio problem with a single multiplier before solving. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.

Easy (1–10)

  1. Simplify the ratio 24:36.
  2. If a:b = 3:5, find a/b as a decimal.
  3. Divide 60 into two parts in the ratio 2:3.
  4. Find the ratio of 750g to 2kg.
  5. If 5 pens cost Rs. 60, find the cost of 8 pens.
  6. If a:b=2:3 and b:c=3:4, find a:c.
  7. Find the value of x if 3:x = 9:15.
  8. A sum of Rs. 900 is divided between A and B in the ratio 4:5. Find A's share.
  9. If x varies directly with y, and x=10 when y=2, find x when y=5.
  10. Find the ratio of 1 hour to 45 minutes.
Easy — Answers

1) 24:36=2:3. 2) 3/5=0.6. 3) Parts: 24 and 36. 4) 750:2000=3:8. 5) 60/5=12/pen; 8×12=96. 6) a:b=2:3, b:c=3:4 (b matches) → a:c=2:4=1:2. 7) 3/x=9/15 → x=5. 8) 900×4/9=400. 9) k=5; x=5×5=25. 10) 60:45=4:3.

Medium (11–20)

  1. Divide Rs. 3,600 among A, B and C in the ratio 2:3:4.
  2. If A:B=2:3 and B:C=4:5, find A:B:C.
  3. Two numbers are in the ratio 5:7. If their sum is 144, find the numbers.
  4. If 8 men can build a wall in 12 days, how many days will 6 men take?
  5. A sum of money is divided among A, B, C in the ratio 1/2 : 1/3 : 1/4. Find A's share if the total is Rs. 2,600.
  6. The ratio of two numbers is 3:4 and their HCF is 4. Find the numbers.
  7. If a:b = 5:6 and b:c = 4:5, find a:b:c.
  8. Two numbers are in ratio 4:5. If 10 is added to each, the new ratio becomes 6:7. Find the numbers.
  9. A, B and C invest in a business in the ratio 2:3:5. If the total profit is Rs. 5,000, find C's share.
  10. If x varies inversely with y, and x=6 when y=8, find x when y=12.
Medium — Answers

11) Parts=9; each=400; shares 800,1200,1600. 12) LCM(3,4)=12: A:B=8:12, B:C=12:15 → A:B:C=8:12:15. 13) Parts=12; each=12; numbers 60 and 84. 14) 8×12=6×d → d=16 days. 15) Ratio 6:4:3 (×12); parts=13; A=2600×6/13=1,200. 16) Coprime ratio 3:4 → numbers=4×3=12 and 4×4=16. 17) LCM(6,4)=12: a:b=10:12, b:c=12:15 → a:b:c=10:12:15. 18) (4x+10)/(5x+10)=6/7 → x=5 → numbers 20 and 25. 19) Parts=10; C=5000×5/10=2,500. 20) 6×8=x×12 → x=4.

Tough (21–30)

  1. A sum of money is divided among A, B, C such that A:B = 2:3 and B:C = 4:5. If C gets Rs. 350 more than A, find the total sum.
  2. Two numbers are in the ratio 3:5. If 9 is added to each, the new ratio becomes 3:4. Find the numbers.
  3. The ratio of the ages of A and B is 3:4. Five years hence, the ratio will be 4:5. Find their present ages.
  4. A, B and C share a sum of money such that A gets twice as much as B, and B gets thrice as much as C. If C gets Rs. 800, find the total sum.
  5. If (a+b):(a-b) = 5:1, find a:b.
  6. The salaries of A, B and C are in the ratio 3:4:5. If C's salary is Rs. 4,000 more than A's, find B's salary.
  7. Three friends invest Rs. 4,000, Rs. 6,000 and Rs. 10,000 respectively in a business for 1 year. Find the ratio in which they should share a profit of Rs. 6,000.
  8. If x:y = 3:4, find (3x+2y):(3x-2y).
  9. A bag contains coins of Rs. 1, 50p and 25p in the ratio 5:6:8. If the total value is Rs. 210, find the number of 50p coins.
  10. Two vessels contain milk and water in the ratios 3:1 and 5:3 respectively. In what ratio should quantities from the two vessels be mixed to get a mixture with milk and water in ratio 2:1?
Tough — Answers

21) A:B:C=8:12:15 (scaled to match B); C-A=7 parts=350 → part=50; total=35×50=1,750. 22) (3x+9)/(5x+9)=3/4 → x=3 → numbers 9 and 15. 23) (3x+5)/(4x+5)=4/5 → x=5 → ages 15 and 20. 24) C=800, B=2400, A=4800; total=8,000. 25) a+b=5a-5b → 6b=4a → a:b=3:2. 26) C-A=2 parts=4000 → part=2,000; B=4×2000=8,000. 27) Investment ratio=2:3:5 (same time); shares=1200,1800,3000. 28) x=3,y=4 → (9+8):(9-8)=17:1. 29) Value: 5k×1+6k×0.5+8k×0.25=10k=210 → k=21; 50p coins=6×21=126. 30) Alligation gives vessel1:vessel2=1:2.

6. Knowledge Check

Four quick questions. Expand each to check your answer.

Q1

Why does writing two numbers in a 3:5 ratio as "3x and 5x" (one shared unknown) work, rather than needing two separate unknowns?

A ratio only fixes the relative proportion between two numbers, not their absolute values — any pair of numbers with the same 3:5 relationship can be written as some common multiplier times 3 and times 5. Using one shared unknown x captures exactly that one degree of freedom, and any additional given fact (like a sum) pins down its specific value.

Q2

When combining A:B = 2:3 and B:C = 4:5, why can't you just write A:B:C = 2:3:5?

B is represented by different values in the two ratios (3 in the first, 4 in the second), so they can't be directly stitched together without first making B consistent between them. Both ratios need to be scaled so B has the same value in each (using their LCM) before A, B and C can be combined into one valid three-term ratio.

Q3

Why does "time taken to finish a job" vary inversely with "number of workers," rather than directly?

Adding more workers to a fixed amount of total work means the work gets divided among more people, so each worker's share shrinks and the job finishes sooner — more workers means LESS time, the defining behavior of an inverse relationship, not a direct one where both quantities would move in the same direction.

Q4

Why is a partnership's profit split by (investment × time), not by investment amount alone?

Money invested for a longer period contributes more to the business's overall operation than the same money invested briefly, so a fair split has to account for both how much was invested and for how long. "Capital-months" (or capital-days) captures both factors in a single number, which is why two partners can end up with equal shares despite investing different amounts, if their investment durations compensate.

← Back to Week 5: Simple & Compound Interest Up next Week 7: Averages, Mixtures & Alligation →