Paper I · 9 Logical Reasoning & Analytical Ability · 9.1 Number system and simplification

9.1.12 Problems on numbers: turning word statements into equations

Name the unknown, translate each phrase into symbols, and solve one or two simple equations; in MCQs, checking the options against the conditions is often faster still.

  • APPSC Paper I · Paper I
  • Standard level
  • 6 min read
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Core facts

  • Translation table:
WordsSymbols
a number increased by 7 / exceeds y by 7x + 7 / x = y + 7
3/5 of a number(3/5)x
two numbers differ by 6x − y = 6
the reciprocal of a number1/x
three consecutive numbersx − 1, x, x + 1
  • Sum S and difference D: larger = (S + D)/2, smaller = (S − D)/2.
  • Sum S and product P: (x − y)² = S² − 4P; x² + y² = S² − 2P.
  • For three consecutive numbers, use the middle as x: the sum is 3x and the sum of squares is 3x² + 2.
  • Fractions: call it n/d; each condition gives one equation.
  • Dividend = divisor × quotient + remainder.

Example 1. A number added to 3/5 of itself gives 64. Find the number. Solution. (8/5)x = 64. Answer: 40

Example 2. Two numbers add to 50 and differ by 14. Find them. Solution. (50 + 14)/2 = 32 and (50 − 14)/2 = 18. Answer: 32 and 18

Example 3. The difference between a number and two-fifths of it is 36. Find the number. Solution. (3/5)x = 36. Answer: 60

Explained

Type 1: sum and product given. Use the identities instead of solving for each number.

Example 4. Two numbers have sum 17 and product 72. Find (i) their difference and (ii) the sum of their squares. (a) 1, 145 (b) 1, 217 (c) 3, 145 (d) 7, 217 Solution. (x − y)² = 289 − 288 = 1, so the difference is 1. x² + y² = 289 − 144 = 145. (The numbers are 8 and 9.) Answer: (a) 1, 145

Type 2: number and its reciprocal.

Example 5. A number plus its reciprocal is 26/5. Find the number. Solution. x + 1/x = 5 + 1/5, so x = 5 or 1/5. Answer: 5 or 1/5

Type 3: fraction problems.

Example 6. If 1 is added to both the numerator and denominator of a fraction, it becomes 2/3. If 1 is subtracted from both, it becomes 1/2. Find the fraction. Solution. 3(n + 1) = 2(d + 1) and 2(n − 1) = d − 1. The second gives d = 2n − 1; substitute: 3n + 3 = 4n, so n = 3, d = 5. Check: 4/6 = 2/3 and 2/4 = 1/2. Answer: 3/5

Type 4: wrong operation. Set (correct result) − (wrong result) = the stated gap.

Example 7. A student multiplied a number by 3/5 instead of 5/3 and got an answer 32 less than the correct one. Find the number. Solution. (5/3 − 3/5)x = (16/15)x = 32. Answer: 30

Type 5: consecutive numbers.

Example 8. The squares of three consecutive natural numbers add to 434. Find the numbers. Solution. With middle x: 3x² + 2 = 434, so x² = 144, x = 12. Answer: 11, 12, 13

Type 6: dividing a number into parts.

Example 9. Divide 75 into two parts so that one-third of the first plus one-fifth of the second is 19. Solution. x/3 + (75 − x)/5 = 19. Multiply by 15: 5x + 225 − 3x = 285, so x = 30. Answer: 30 and 45

Type 7: quotient and remainder statements.

Example 10. Three times the larger of two numbers, divided by the smaller, gives quotient 5 and remainder 4. The numbers differ by 26. Find them. Solution. 3L = 5S + 4 and L = S + 26. So 3S + 78 = 5S + 4, S = 37, L = 63. Check: 189 = 5 × 37 + 4. Answer: 63 and 37

Speed tips.

  • In MCQs, test the options in the condition that is easiest to check (often the sum or difference) before writing any equation.
  • Clear fractions at once by multiplying through by the LCM of the denominators.

Common traps

  • ✗ "Three consecutive numbers: x, x + 1, x + 2, so the sum of squares is messy." ✓ Use the middle as x: sum of squares = 3x² + 2.
  • ✗ "x + 1/x = 26/5 gives only x = 5." ✓ The reciprocal 1/5 also works; check the options for both.
  • ✗ "Multiplied by 3/5 instead of 5/3: difference = (5/3 − 3/5) × 32." ✓ The gap equals (16/15)x, so x = 32 × 15/16 = 30.
  • ✗ "Sum 17, product 72: sum of squares = 17² − 72." ✓ Subtract 2P: 289 − 144 = 145.
  • ✗ "B says 41 − 15 = 26." ✓ The pair adds to 41 + 1: B says 27.

Quick revision

  • Name the unknown first; one condition, one equation.
  • Larger = (S + D)/2; smaller = (S − D)/2.
  • (x − y)² = S² − 4P; x² + y² = S² − 2P.
  • Middle of three consecutive numbers as x; squares sum to 3x² + 2.
  • Fraction n/d: two conditions, two equations.
  • Wrong operation: correct − wrong = gap.
  • Ratio of sum : difference gives the numbers as multiples of k.
  • Counting both ways: the pair adds to (last + first).
  • Check the answer in every condition of the question.
Go deeper: Deeper (beyond what APPSC Paper I usually asks)

Deeper Advanced

Example 11. The product of two numbers is 120 and the sum of their squares is 289. Find the numbers. (a) 8, 15 (b) 10, 12 (c) 6, 20 (d) 5, 24 Solution. (x + y)² = 289 + 240 = 529, so x + y = 23. (x − y)² = 289 − 240 = 49, so x − y = 7. Answer: (a) 8, 15

Example 12. The sum, difference and product of two numbers are in the ratio 7 : 1 : 60. Find the numbers. Solution. Let x + y = 7k and x − y = k: x = 4k, y = 3k. Product 12k² = 60k, so k = 5. Answer: 20 and 15

Example 13. Caselet. Of three numbers, the second is twice the first and the third is three times the second. Their sum is 108. (i) Find the largest. (ii) Find the average of the first two. Solution. x + 2x + 6x = 9x = 108, so x = 12; the numbers are 12, 24, 72. (i) 72. (ii) (12 + 24)/2 = 18. Answer: (i) 72 (ii) 18

Example 14. The sum of a number and its square is 182. Find the number (positive). Solution. x² + x − 182 = 0 = (x + 14)(x − 13). Answer: 13

  • Shortcut: x(x + 1) = 182 is a product of consecutive numbers: 13 × 14.

Example 15. A counts from 1 to 41 while B counts from 41 down to 1 at the same speed, starting together. What does B say when A says 15? Solution. At each moment the two numbers add to 41 + 1 = 42. 42 − 15. Answer: 27

  • "Think of a number" puzzles: write the steps as algebra. Double, add 15, multiply by 3, subtract 45, divide by 6: ((2x + 15) × 3 − 45)/6 = x. The trick always returns the starting number.

Sources: NCERT Mathematics textbooks, Classes 8–10 (linear equations in one variable, pairs of linear equations, quadratic equations)

Last checked 2026-10-17.

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