Q4 - Ratio, proportion and cost models

1. Why this topic matters here

Expected load: 3.00 questions, 2.3% of a ~130-question paper (quantitative section, tier 1, confidence Medium).

That figure is a proportional allocation of the section's budget by evidence weight, not a count of anything observed, and the 130-question structure itself is [LIKELY], not published. It ranks at tier 1 because of its estimated marks per study hour (3.00, against a paper-wide mean of 2.52). The only direct evidence is 1 item on IndiaBIX placement paper 3874. It is undated and anonymous, but the IDs of the papers on either side of it are all 2007-08 L&T ECC (construction-division) papers, and the next dated paper above that block is February 2012, so it is probably from 2007-08 [INFERRED] and it is not a PGET paper. See section 10.

2. Exactly what gets asked

The observed question from an old (probably 2007-08) IndiaBIX paper:

"For 15 people each pays Rs 20, for 20 people each pays Rs 18, for 40 people how much each?"

This is a fixed cost plus variable cost per head model. Convert each statement into a total, then solve two simultaneous equations:

15 people × Rs 20 = Rs 300 total
20 people × Rs 18 = Rs 360 total
Difference: 5 extra people cost Rs 60 → Rs 12 per head (variable)
Fixed: 300 − (15 × 12) = Rs 120
40 people: 120 + (40 × 12) = Rs 600 → Rs 15 per head

Sub-pattern	How it appears	Roughly how often
Fixed + variable cost model	Two (people, rate) pairs, cost per head asked for a third count	The observed question; expect ~1 of the 2-3
Direct proportion	"If 5 machines make 200 units in 4 hours..."	Common, usually combined with inverse
Inverse proportion	More workers, fewer days; more speed, less time	Common
Partnership profit sharing	Capital ratios, sometimes with different time periods	Occasional, 1 in 3 papers
Simple ratio split	Divide an amount in a given ratio	Occasional, easiest of the set
Ratio chain / combining	a:b and b:c given, find a:c	Occasional
Mean / third proportional	"Mean proportional between 4 and 25"	Rare

3. Concept notes

3.1 Ratio basics

A ratio a : b compares two quantities of the same unit. It is a fraction, a/b, and it can be scaled freely: 3 : 4 = 6 : 8 = 30 : 40.

To split a total T in the ratio a : b : c:

Total parts = a + b + c
One part = T / (a + b + c)
Share of A = a × T / (a + b + c)

Example: Rs 1,200 split 2 : 3 : 5 → 10 parts, one part = 120, shares = 240, 360, 600.

Ratio of fractions: multiply every term by the LCM of the denominators. 1/2 : 1/3 : 1/4 → multiply by 12 → 6 : 4 : 3.

Combining two ratios where the middle term is common: for a : b = 3 : 4 and b : c = 6 : 7, make b the same in both (LCM of 4 and 6 is 12): a : b = 9 : 12, b : c = 12 : 14, so a : b : c = 9 : 12 : 14 and a : c = 9 : 14.

3.2 Proportion

Four quantities are in proportion when a : b = c : d, written a : b :: c : d, and then a × d = b × c (product of extremes = product of means). This cross-multiplication is the single most-used tool in the topic.

Mean proportional between a and c is √(a × c). Mean proportional between 4 and 25 is 10.
Third proportional to a and b is b²/a. Third proportional to 9 and 12 is 144/9 = 16.

A quantity x added to each of four numbers keeps them proportional only when (a+x)(d+x) = (b+x)(c+x) — a quadratic in which the x² terms usually cancel, leaving a linear equation.

3.3 Direct and inverse variation

Direct: y = kx. Double x, double y.

Example: 4 kg costs Rs 200 → 7 kg costs 7 × 50 = Rs 350.

Inverse: y = k/x, equivalently x₁y₁ = x₂y₂. Double x, halve y.

Example: 8 workers take 12 days → 6 workers take 8 × 12 / 6 = 16 days.

Multi-variable (the chain rule): write the relation as a product and keep units consistent.

Work = Rate × Machines × Hours

Example: 5 machines, 200 units, 4 hours → per machine-hour = 200 / (5 × 4) = 10 units. Then 8 machines for 6 hours = 8 × 6 × 10 = 480 units.

3.4 The fixed + variable cost model

Total cost = Fixed cost F + (variable rate v × quantity n).

Total = F + v·n

Given two observations (n₁, Total₁) and (n₂, Total₂):

v = (Total₂ − Total₁) / (n₂ − n₁)
F = Total₁ − v·n₁

Two warnings that decide whether you get this right:

The question may give the per-unit rate, not the total. "Each pays Rs 20" with 15 people means total = 300. Convert first, always.
The question may ask for the per-unit rate at the new quantity, not the total. Compute the total, then divide.

Profit is shared in the ratio of capital × time.

Case	Profit ratio
Same time	`C₁ : C₂`
Different times	`C₁T₁ : C₂T₂`
Money added/withdrawn mid-term	Split each partner's period into segments and sum

Example: A invests Rs 5,000 for 12 months, B invests Rs 6,000 for 8 months. Ratio = 60,000 : 48,000 = 5 : 4. On a profit of Rs 9,000, B gets 4/9 × 9000 = Rs 4,000.

For a sleeping partner (invests money, no work), the ratio is still capital × time; the active partner's extra is taken out first as a salary if the question says so.

3.6 Speed shortcuts worth drilling

Situation	Shortcut
Ratio `a : b`, sum known	One part = Sum / (a + b)
Ratio `a : b`, difference known	One part = Difference / (a − b)
Two numbers, ratio and HCF known	Numbers = `aH` and `bH`; LCM = a·b·H
Two numbers, ratio and LCM known	Product = LCM × HCF
Adding x to make four terms proportional	Cross-multiply, cancel `x²`, solve linear

For ratio 5 : 7 with HCF 6: numbers are 30 and 42, LCM = 5 × 7 × 6 = 210.

4. Must-memorise block

Core

a : b = ka : kb for any k ≠ 0
Split in a : b : c → one part = Total / (a + b + c)
a × d = b × c when a : b :: c : d
Mean proportional of a, c = √(ac); third proportional to a, b = b²/a

Variation

Direct: x₁/y₁ = x₂/y₂
Inverse: x₁y₁ = x₂y₂
Machines/hours/work: M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂

Cost model

Total = F + v·n
v = ΔTotal / Δn; F = Total − v·n

Partnership

Profit ratio = Capital × Time
Same capital, different time → ratio of times
Same time, different capital → ratio of capitals

Numbers

Product of two numbers = LCM × HCF
Ratio a : b, HCF h → numbers ah, bh; LCM = abh
Speed and time: inverse; Speed and distance (fixed time): direct
Work and time: inverse; Work and workers (fixed time): direct

5. Worked examples

Example 1 - the observed pattern (old paper)

For 15 people each pays Rs 20; for 20 people each pays Rs 18. For 40 people, how much does each pay?

Fast method: Totals: 15 × 20 = 300; 20 × 18 = 360. ΔTotal = 60 over Δn = 5 → v = Rs 12 per head. F = 300 − (15 × 12) = Rs 120. For 40: Total = 120 + 480 = Rs 600 → per head = Rs 15.

Slow method: setting up F + 15v = 300 and F + 20v = 360 and subtracting algebraically takes 90 seconds instead of 30. Same arithmetic, more writing. Do the subtraction in your head.

Example 2 - fixed + variable with the rate given

A caterer charges Rs 60 per plate for 50 plates and Rs 51 per plate for 80 plates. Find the fixed charge.

Totals: 50 × 60 = 3,000; 80 × 51 = 4,080. Δ = 1,080 over 30 plates → v = Rs 36 per plate. F = 3,000 − (50 × 36) = 3,000 − 1,800 = Rs 1,200.

Note the trap: the per-plate rate falls as plates rise. That is the whole point of a fixed cost — it is the observation that gives it away, not a contradiction.

Example 3 - partnership with unequal time

A, B and C invest in the ratio 2 : 3 : 5 for 6, 4 and 3 months respectively. Total profit is Rs 13,000. Find C's share.

Profit ratio = 2×6 : 3×4 : 5×3 = 12 : 12 : 15 = 4 : 4 : 5. Total parts = 13 → one part = 13,000 / 13 = 1,000. C = 5 × 1,000 = Rs 5,000.

Slow method: computing each product as a rupee figure for every partner wastes 40 seconds. Reduce the ratio to lowest terms first, then divide.

Example 4 - chain rule

If 5 machines produce 200 units in 4 hours, how many units will 8 machines produce in 6 hours?

Rate per machine-hour = 200 / (5 × 4) = 10 units. Output = 8 × 6 × 10 = 480 units.

Notice the two-step structure: reduce everything to a unit rate, then scale up. Any "X machines, Y hours" question reduces to this.

Example 5 - making four terms proportional

What must be added to each of 3, 5, 12 and 18 so the results are proportional?

Set up: (3 + x) / (5 + x) = (12 + x) / (18 + x). Cross-multiply: (3+x)(18+x) = (5+x)(12+x) 54 + 21x + x² = 60 + 17x + x² 4x = 6 → x = 1.5.

The x² cancelling is not a coincidence — it always cancels in this pattern. Knowing that, you can cross-multiply and jump straight to 4x = 6.

6. Traps and gotchas

Trap	Why it catches people	How to avoid it
Using the per-head rate as if it were the total	"Each pays Rs 20" reads like a total	Multiply rate × count the instant you read it
Assuming a constant per-head rate	The observed question deliberately varies it	If the rate changes with the count, it is a fixed + variable model
Answering the total when the per-head rate was asked	Both numbers appear in your working	Underline the unit asked for: "how much each"
Forgetting to multiply by time in partnerships	Capital ratio alone gives the wrong split	Ask: "did they invest for the same number of months?"
Mixing direct and inverse variation	"More workers → fewer days" feels backwards	More of X giving less of Y is inverse; write `x₁y₁ = x₂y₂`
Cancelling the `x` in `(3+x)/(5+x)`	Looks like a fraction to simplify	You can only cancel factors, never added terms
Computing LCM from ratio + HCF by multiplying the ratio terms	`5 × 7 = 35` is not the LCM	LCM = ratio product × HCF = 5 × 7 × 6 = 210
Leaving a blank because the topic "looks" small	No negative marking	A 25% blind guess beats a guaranteed zero

7. Practice questions

1. Rs 1,200 is divided among A, B and C in the ratio 2 : 3 : 5. What is C's share? a) 240 b) 360 c) 600 d) 500

2. For 15 people each pays Rs 20. For 20 people each pays Rs 18. For 40 people, how much does each pay? a) 12 b) 15 c) 16 d) 18

3. If a : b = 3 : 4 and b : c = 6 : 7, find a : c. a) 7 : 9 b) 21 : 24 c) 9 : 14 d) 4 : 7

4. 8 workers build a wall in 12 days. How many days will 6 workers take? a) 9 b) 16 c) 14 d) 18

5. A invests Rs 5,000 for 12 months and B invests Rs 6,000 for 8 months. Total profit is Rs 9,000. What is B's share? a) 5,000 b) 4,000 c) 4,500 d) 3,600

6. A service charges a fixed fee plus an amount per head. For 10 people the total is Rs 1,000; for 25 people the total is Rs 1,750. What is the total for 40 people? a) 2,500 b) 2,250 c) 2,000 d) 3,000

7. Two numbers are in the ratio 5 : 7 and their HCF is 6. What is their LCM? a) 42 b) 105 c) 210 d) 126

8. In 60 litres of a mixture, milk and water are in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1? a) 20 litres b) 10 litres c) 30 litres d) 40 litres

9. If 5 machines produce 200 units in 4 hours, how many units will 8 machines produce in 6 hours? a) 384 b) 720 c) 600 d) 480

10. A, B and C invest in the ratio 2 : 3 : 5 for 6, 4 and 3 months respectively. Total profit is Rs 13,000. Find C's share. a) 5,000 b) 4,500 c) 4,000 d) 6,000

11. A caterer charges a fixed amount plus a rate per plate. For 50 plates the charge is Rs 60 per plate; for 80 plates it is Rs 51 per plate. What is the fixed charge? a) 800 b) 1,000 c) 1,500 d) 1,200

12. What must be added to each of 3, 5, 12 and 18 so that the results are in proportion? a) 2 b) 3 c) 6 d) 1.5

ANSWER KEY

1. c) 10 parts, one part = 120; C = 5 × 120 = 600. 2. b) Totals 300 and 360; v = 60/5 = 12; F = 300 − 180 = 120; for 40 → (120 + 480)/40 = 15. 3. c) Make b common: a : b : c = 9 : 12 : 14, so a : c = 9 : 14. 4. b) Inverse proportion: 8 × 12 / 6 = 16 days. 5. b) Ratio = 60,000 : 48,000 = 5 : 4; B = 4/9 × 9,000 = 4,000. 6. a) v = 750/15 = 50; F = 1,000 − 500 = 500; total = 500 + 40×50 = 2,500. 7. c) Numbers are 30 and 42; LCM = 5 × 7 × 6 = 210. 8. a) Milk 40, water 20; need 40 water, so add 20 litres. 9. d) Unit rate = 200/(5×4) = 10; output = 8 × 6 × 10 = 480. 10. a) Profit ratio = 12 : 12 : 15 = 4 : 4 : 5; one part = 1,000; C = 5,000. 11. d) Totals 3,000 and 4,080; v = 1,080/30 = 36; F = 3,000 − 1,800 = 1,200. 12. d) (3+x)(18+x) = (5+x)(12+x) → 54 + 21x = 60 + 17x → 4x = 6 → x = 1.5.

8. Resources

Ratio and Proportion - Aptitude Questions and Answers — https://www.indiabix.com/aptitude/ratio-and-proportion/ — MCQ set with options, answers and explanations (verified).
Partnership - Aptitude Questions and Answers — https://www.indiabix.com/aptitude/partnership/ — MCQ set with options, answers and explanations (verified).
Client Challenge — https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates — could not be checked by me (blocks automated access) - open it yourself before relying on it.
artofproblemsolving.com — https://artofproblemsolving.com/wiki/index.php/Ratio — could not be checked by me (blocks automated access) - open it yourself before relying on it.
Aptitude Questions and Answers — https://www.geeksforgeeks.org/aptitude-questions-and-answers/ — index of question sets - open a sub-topic for the actual questions.

9. Self-check

Answer yes or no to each. Six yes answers means you can stop studying this topic.

Can you convert "15 people each pays Rs 20" into a total in under five seconds?
Can you write the fixed-plus-variable rule Total = F + v·n from memory and compute v and F from two totals without setting up simultaneous equations on paper?
Do you instantly know that "more workers, fewer days" means inverse proportion, and can you apply x₁y₁ = x₂y₂?
Can you compute a partnership profit ratio using capital × time without being prompted?
Do you know that LCM = ratio product × HCF, and that product of two numbers = LCM × HCF?
Can you spot the difference between "find the total" and "find the amount each pays" in a word problem?

10. Sources

Evidence status: confidence Medium. High needs a recent observed question, and none exists for any topic.

Observed on an old paper [REPORTED, OLD]. IndiaBIX placement paper 3874. It is undated and anonymous, but the IDs of the papers on either side of it are all 2007-08 L&T ECC (construction-division) papers, and the next dated paper above that block is February 2012, so it is probably from 2007-08 [INFERRED] and it is not a PGET paper. Items bearing on this topic:

OBS-A04 - For 15 people each has to pay Rs.20. For 20 people each has to pay Rs.18. For 40 people how much has each to pay?

Named by a first-hand account [REPORTED]. A candidate from SVNIT Surat (2021 batch, L&T Limited, civil branch) listed among the common L&T topics: "ratio and proportions". Source: IndiaBIX placement paper 6792.

Not evidence of the current paper. The official notice names only "analytical, verbal, and numerical ability" and "domain-specific knowledge"; nothing in it lists topics. See research/05-source-audit.md.