Ratio Word Problems: Income, Coins, Groups, Salaries, Mixtures and Scale
Ratio Word Problems: Income, Coins, Groups, Salaries, Mixtures and Scale
Ratio word problems are the same algebra in different clothes: write every quantity as a multiple of an unknown (5x, 4x), translate the condition into one equation, and solve. The table below gives the set-up for each story type NTPC uses, so you never start from a blank page.
1. Set-up Table for Every Story Type
| Story | What to write | Key equation |
|---|---|---|
| Income, expenditure, savings | Incomes ax, bx; expenditures cy, dy | Income − Expenditure = Savings |
| Equal savings | ax − cy = bx − dy | Subtracting gives a direct x : y relation |
| Coins by number in ratio a : b : c | Numbers ak, bk, ck | Total value = Σ (number × face value) |
| Coins by value in ratio a : b : c | Values ak, bk, ck | Number = value ÷ face value of the coin |
| Group joins or leaves | Boys 7x, girls 5x | (7x ± change) : (5x ± change) = new ratio |
| Salaries with fixed increment | 2x + I and 3x + I | Set equal to the new ratio |
| Salaries with percentage rise | 2x × (1 + p/100) and 3x × (1 + q/100) | New ratio = 2(100 + p) : 3(100 + q) |
| Mixture, one component added | Keep the other component fixed | Milk/(water + w) = new ratio |
| Mixture, part replaced | Remove the same fraction of each component | Each component × (1 − removed/total) |
| Numbers with sum of squares | Numbers ak, bk, ck | (a² + b² + c²)k² = given sum |
| Map scale 1 : n | Real length = map length × n | Real area = map area × n² |
✗ Coins with values in ratio 3 : 4 : 10 means 3 : 4 : 10 coins | ✓ Value ratio and number ratio differ; divide each value by its face value
2. Mixture Rule Worth Memorising
When a fraction f of a mixture is taken out, every component falls by the same fraction f. If the removed quantity is replaced by water, only water is added back. After n such operations, milk left = original milk × (1 − f)ⁿ.
3. Scale and Area
A map scale 1 : 2,00,000 means 1 cm on the map = 2,00,000 cm = 2 km on the ground. Areas scale by the SQUARE: 1 cm² on this map = 4 km² = 400 hectares (1 km² = 100 hectares).
हिंदी नोट: आय-व्यय के प्रश्नों में आय = व्यय + बचत लिखिए। सिक्कों के प्रश्नों में पहले देखिए कि अनुपात सिक्कों की संख्या का है या उनके मूल्य का; मूल्य को सिक्के के अंकित मूल्य से भाग देने पर संख्या मिलती है।
Exam Pointer: Verified NTPC patterns: income-expenditure-savings (January 2021, June 2025 Graduate CBT-1), coins in a purse (April 2016), group joining or leaving (January 2021, January 2017 CBT-2), salaries with percentage rises (February 2021), numbers in a ratio with a given sum (April 2016) and scale applied to area (2020-21 cycle). Mixture change has no verified NTPC shift yet, so it is tagged syllabus-based; its method is reused in the Average and Mixture chapter.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Income, expenditure and savings
[PYQ: NTPC CBT-1 4-Jan-2021 Shift-2 | NTPC Graduate CBT-1 10-Jun-2025 Shift-3]
EXAM LEVEL
Q. The incomes of A and B are in the ratio 5 : 4 and their expenditures in the ratio 3 : 2. If each saves ₹8,000, find their incomes.
Incomes 5x and 4x, expenditures 3y and 2y. Equal savings: 5x − 3y = 4x − 2y, so x = y. Then savings = 5x − 3x = 2x = 8,000, x = 4,000. Incomes are ₹20,000 and ₹16,000 (expenditures ₹12,000 and ₹8,000).
Answer: ₹20,000 and ₹16,000
EXAMATLAS LEVEL
Q. The incomes of A and B are in the ratio 7 : 5 and their expenditures in the ratio 3 : 2. A saves ₹6,000 and B saves ₹5,000. Find their total expenditure.
7x − 3y = 6,000 and 5x − 2y = 5,000. Multiply the second by 1.5: 7.5x − 3y = 7,500. Subtract the first: 0.5x = 1,500, so x = 3,000. Then 3y = 21,000 − 6,000 = 15,000 and y = 5,000. Expenditures are ₹15,000 and ₹10,000, total ₹25,000. Check: incomes ₹21,000 and ₹15,000 leave savings of ₹6,000 and ₹5,000.
Answer: ₹25,000
Pattern 2: Coins in a bag or purse
[PYQ: NTPC CBT-1 28-Apr-2016 Shift-2]
EXAM LEVEL
Q. A bag contains ₹1, 50 paise and 25 paise coins in the ratio 2 : 3 : 4 by number. The total amount is ₹90. How many 25 paise coins are there?
Numbers 2k, 3k, 4k. Value = 2k × 1 + 3k × 0.5 + 4k × 0.25 = 2k + 1.5k + k = 4.5k = 90, so k = 20. Number of 25 paise coins = 4 × 20 = 80.
Answer: 80
EXAMATLAS LEVEL
Q. A box contains ₹1, ₹2 and ₹5 coins whose total VALUES are in the ratio 3 : 4 : 10. The total amount is ₹544. Find the total number of coins.
Values are 3k, 4k, 10k with 17k = 544, so k = 32. Values: ₹96, ₹128, ₹320. Numbers: 96 ÷ 1 = 96, 128 ÷ 2 = 64, 320 ÷ 5 = 64. Total = 224 coins. Reading 3 : 4 : 10 as a count ratio gives a wrong answer.
Answer: 224
Pattern 3: Two groups: the ratio changes when some join or leave
[PYQ: NTPC CBT-1 7-Jan-2021 Shift-1 | NTPC CBT-2 18-Jan-2017 Shift-3]
EXAM LEVEL
Q. The ratio of boys to girls in a class is 7 : 5. If 8 more girls join, the ratio becomes 1 : 1. How many students were there originally?
Boys 7x, girls 5x. After joining: 7x = 5x + 8, so x = 4. Originally 12x = 48 students (28 boys, 20 girls).
Answer: 48
EXAMATLAS LEVEL
Q. In a school the ratio of boys to girls is 4 : 3. If 20 boys leave and 10 girls join, the ratio becomes 6 : 7. How many girls were there originally?
(4x − 20)/(3x + 10) = 6/7, so 28x − 140 = 18x + 60, giving 10x = 200 and x = 20. Girls originally = 3 × 20 = 60. Check: boys 80 → 60, girls 60 → 70, ratio 6 : 7.
Answer: 60
Pattern 4: Salaries in a ratio with increments or percentage rises
[PYQ: NTPC CBT-1 8-Feb-2021 Shift-2]
EXAM LEVEL
Q. The salaries of A and B are in the ratio 2 : 3. If each gets an increment of ₹4,000, the ratio becomes 3 : 4. Find B's original salary.
(2x + 4,000)/(3x + 4,000) = 3/4, so 8x + 16,000 = 9x + 12,000 and x = 4,000. B's salary = 3x = ₹12,000. Check: 12,000 : 16,000 = 3 : 4.
Answer: ₹12,000
EXAMATLAS LEVEL
Q. The salaries of P and Q are in the ratio 3 : 5. P gets a 40% rise and Q a 20% rise. If Q's new salary is ₹48,000, find the new ratio and P's original salary.
New ratio = 3 × 140 : 5 × 120 = 420 : 600 = 7 : 10. Q's original = 48,000 ÷ 1.2 = ₹40,000, so P's original = 40,000 × 3/5 = ₹24,000. Check: P's new salary 24,000 × 1.4 = 33,600, and 33,600 : 48,000 = 7 : 10.
Answer: 7 : 10; ₹24,000
Pattern 5: Mixture ratio change: adding or replacing
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. 60 litres of a mixture contains milk and water in the ratio 7 : 3. How much water must be added to make the ratio 3 : 2?
Milk = 42 L and water = 18 L. Milk does not change: 42/(18 + w) = 3/2, so 84 = 54 + 3w and w = 10 litres.
Answer: 10 litres
EXAMATLAS LEVEL
Q. A vessel holds 80 litres of milk and water in the ratio 5 : 3. 16 litres of the mixture are taken out and replaced with water, and the process is repeated once more. Find the final ratio of milk to water.
Removing 16 of 80 litres removes 1/5 of each component. Start: milk 50, water 30. After the first operation: milk 40, water 24 + 16 = 40. After the second: milk 32, water 32 + 16 = 48. Final ratio 32 : 48 = 2 : 3. Formula check: milk = 50 × (4/5)² = 32.
Answer: 2 : 3
Pattern 6: Numbers in a ratio with a given sum, difference or sum of squares
[PYQ: NTPC CBT-1 3-Apr-2016 Shift-3]
EXAM LEVEL
Q. Three numbers are in the ratio 2 : 3 : 5 and their sum is 250. Find the largest number.
10 parts = 250, one part = 25, largest = 5 × 25 = 125.
Answer: 125
EXAMATLAS LEVEL
Q. Three numbers are in the ratio 3 : 4 : 5 and the sum of their squares is 1,250. Find the sum of the numbers.
(9 + 16 + 25)k² = 50k² = 1,250, so k² = 25 and k = 5. Numbers 15, 20, 25 with sum 60. Dividing 1,250 by 12 (the sum of ratio terms) is the trap.
Answer: 60
Pattern 7: Map scale and real area
[PYQ: NTPC 2020-21 cycle]
EXAM LEVEL
Q. On a map 1 cm represents 25 km. Two towns are 7.2 cm apart on the map. Find the actual distance.
7.2 × 25 = 180 km.
Answer: 180 km
EXAMATLAS LEVEL
Q. A map is drawn to the scale 1 : 2,00,000. A rectangular farm measures 3 cm by 2 cm on the map. Find its actual area in hectares.
1 cm on the map = 2,00,000 cm = 2 km. Real sides: 6 km and 4 km, so the area is 24 km². Since 1 km² = 100 hectares, the area is 2,400 hectares. Multiplying 6 cm² by 2 km gives the classic wrong answer; areas scale by the square.
Answer: 2,400 hectares
60-Second Revision
- Income − Expenditure = Savings; equal savings link x and y directly.
- Coins: check whether the ratio is of numbers or of values; number = value ÷ face value.
- Groups and salaries: write ak, bk, apply the change, equate to the new ratio.
- Percentage rises: new ratio = a(100 + p) : b(100 + q).
- Replacement: each component falls by the same fraction; milk left = milk × (1 − f)ⁿ.
- Scale 1 : n multiplies lengths by n and areas by n².