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Ratio and proportion for RRB NTPC: methods, patterns and practice

Sharing money, combining two ratios, incomes and savings, coins in a bag, and "if 6 is added to each number" problems. Ratio questions in RRB NTPC follow a few clear patterns once you think in parts. Each method with the reason it works, six worked questions and practice.

29 Sept 2026 7 min read

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In this guide
  1. Think in parts
  2. Combining two ratios
  3. Ratios given as fractions
  4. Proportionals
  5. Direct and inverse proportion
  6. Worked questions at NTPC level
  7. Common mistakes
  8. Practice set
  9. What to do next

A ratio compares two quantities of the same kind. A proportion says that two ratios are equal. That is all the theory there is. What makes the topic fast or slow is one habit: seeing a ratio as a number of equal parts.

Ratio also sits inside other topics. Partnership is profit shared in a ratio. Mixtures, ages and speed questions are often ratio questions with a story around them. If you can handle parts cleanly, those chapters get easier.

Think in parts

If money is shared in the ratio 2 : 3 : 4, the total is cut into 2 + 3 + 4 = 9 equal parts. The first person gets 2 of them, the second 3, the third 4.

So the method is always: add the ratio terms, find one part, then multiply.

Why it works: a ratio tells you how many equal parts each share contains, never the size of a part. The total fixes the size of one part, and everything else follows.

A ratio does not change if you multiply or divide every term by the same number. 8 : 12 is the same as 2 : 3. That is why you can scale ratios freely when combining them.

Combining two ratios

If you know A : B and B : C, make the B terms equal, then read off A : B : C.

A : B = 2 : 3 and B : C = 4 : 5. B is 3 in one and 4 in the other; the LCM is 12. Multiply the first ratio by 4 and the second by 3: A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15.

Why it works: B is the same person in both ratios. Until B is written as the same number of parts, the parts in the two ratios are different sizes and cannot be compared.

Ratios given as fractions

If shares are in the ratio 1/2 : 1/3 : 1/4, multiply every term by the LCM of the denominators (12) to get 6 : 4 : 3. The ratio is unchanged because each term was multiplied by the same number.

Proportionals

TermMeaningFormulaExample
Fourth proportional to a, b, cd in a : b = c : dd = bc ÷ a4, 6, 10 → 15
Third proportional to a, bc in a : b = b : cc = b² ÷ a4, 8 → 16
Mean proportional of a and bx in a : x = x : bx = √(ab)9, 16 → 12

All three come from one rule: in a : b = c : d, the product of the outer terms equals the product of the inner terms, so ad = bc.

Direct and inverse proportion

  • Direct: if one quantity doubles, so does the other. Cost of cloth and its length; distance and time at a fixed speed.
  • Inverse: if one doubles, the other halves. Workers and days for a fixed job; speed and time for a fixed distance.

Decide which kind you have before writing an equation. Treating an inverse relation as direct is a common source of wrong answers in time and work.

Worked questions at NTPC level

Q1. ₹5,200 is divided among A, B and C in the ratio 1/2 : 1/3 : 1/4. Find each share.

Multiply by 12: the ratio becomes 6 : 4 : 3, which is 13 parts. One part = 5,200 ÷ 13 = 400.
Shares: A ₹2,400, B ₹1,600, C ₹1,200.

Q2. A : B = 2 : 3 and B : C = 4 : 5. If the three together have ₹10,500, how much does C have?

From the section above, A : B : C = 8 : 12 : 15, which is 35 parts. One part = 10,500 ÷ 35 = 300.
C has 15 × 300 = ₹4,500. (A has ₹2,400 and B has ₹3,600.)

Q3. Two numbers are in the ratio 3 : 4. If 6 is added to each, the ratio becomes 4 : 5. Find the numbers.

Let the numbers be 3x and 4x. Then (3x + 6)/(4x + 6) = 4/5.
Cross-multiply: 5(3x + 6) = 4(4x + 6), so 15x + 30 = 16x + 24, and x = 6.
The numbers are 18 and 24. Check: 24 : 30 = 4 : 5.

Q4. The incomes of two people are in the ratio 5 : 3 and their expenses in the ratio 9 : 5. Each saves ₹2,000 a month. Find their incomes.

Let incomes be 5x and 3x, and expenses 9y and 5y. Savings are equal, so 5x − 9y = 3x − 5y, which gives 2x = 4y, so x = 2y.
Put this into 3x − 5y = 2,000: 6y − 5y = 2,000, so y = 2,000 and x = 4,000.
Incomes: ₹20,000 and ₹12,000. Check: expenses are ₹18,000 and ₹10,000, and both save ₹2,000.

Q5. A bag holds ₹1, 50-paise and 25-paise coins in the ratio 5 : 6 : 8 by number. The total value is ₹210. How many coins of each kind are there?

Convert the ratio of numbers into a ratio of values: 5 × 1 : 6 × 0.5 : 8 × 0.25 = 5 : 3 : 2, which is 10 parts.
One part = 210 ÷ 10 = ₹21. Values: ₹105, ₹63 and ₹42.
Coins: 105 one-rupee, 63 ÷ 0.5 = 126 fifty-paise, and 42 ÷ 0.25 = 168 twenty-five-paise. So 105, 126 and 168.
Check: 105 : 126 : 168, divided by 21, is 5 : 6 : 8.

Q6. What number must be added to each of 6, 14, 18 and 38 so that the results are in proportion?

Let the number be x. Then (6 + x)(38 + x) = (14 + x)(18 + x).
Expand: 228 + 44x + x² = 252 + 32x + x². So 12x = 24 and x = 2.
Check: 8 : 16 = 20 : 40, both equal to 1 : 2.

Common mistakes

  • Dividing by the wrong number of parts. In 2 : 3 : 4, one part is the total ÷ 9, not ÷ 3.
  • Combining ratios without making the common term equal. A : B = 2 : 3 and B : C = 4 : 5 do not give 2 : 3 : 5.
  • Mixing the ratio of numbers with the ratio of values in coin questions.
  • Mixing up third and fourth proportionals. The third proportional uses only two numbers.
  • Adding to the ratio instead of to the numbers. Adding 6 to each of 3 : 4 gives 9 : 10, which is meaningless. Add 6 to the actual numbers, 3x and 4x.

Practice set

  1. Divide ₹900 in the ratio 4 : 5.
  2. A : B = 3 : 4 and B : C = 6 : 7. Find A : B : C.
  3. Find the third proportional to 4 and 8.
  4. Find the mean proportional of 4 and 25.
  5. Two numbers are in the ratio 5 : 7. If 4 is subtracted from each, the ratio becomes 3 : 5. Find the numbers.
  6. Divide ₹9,400 in the ratio 1/3 : 1/4 : 1/5.
  7. The salaries of A and B are in the ratio 4 : 5. After each gets a raise of ₹2,000, the ratio becomes 9 : 11. Find their salaries.
  8. Find the fourth proportional to 4, 6 and 10.

Answers:

  1. ₹400 and ₹500. 9 parts of ₹100 each.
  2. 9 : 12 : 14. Make B equal to 12: 9 : 12 and 12 : 14.
  3. 16. 8² ÷ 4.
  4. 10. √(4 × 25) = √100.
  5. 10 and 14. (5x − 4)/(7x − 4) = 3/5 gives 25x − 20 = 21x − 12, so x = 2. Check: 6 : 10 = 3 : 5.
  6. ₹4,000, ₹3,000 and ₹2,400. Multiply by 60: 20 : 15 : 12, which is 47 parts of ₹200.
  7. ₹16,000 and ₹20,000. (4x + 2,000)/(5x + 2,000) = 9/11 gives 44x + 22,000 = 45x + 18,000, so x = 4,000.
  8. 15. 6 × 10 ÷ 4.

What to do next

  • Solve 20 ratio questions this week, and for each one write the number of parts before you calculate.
  • Practise ten "combine two ratios" questions until making the common term equal is automatic.
  • Move on to averages and then mixture and alligation, which uses ratios throughout.

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 Railway Recruitment Boards website .

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