Problems on Numbers: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 3)

Let the number be x. Multiplying it by 13 increases it by 372. So 13x = x + 372. That simplifies to (13 – 1)x = 372. Solving gives x = 372 / 12 = 31. Check: 13 × 31 = 403, and 403 – 31 = 372, which matches.

Let the number be x. The first condition says (x – 9) / 2 = 3. So x – 9 = 2 × 3 = 6. That gives x = 6 + 9 = 15. Now apply the second operation: subtract 8 from x and divide by 7. (15 – 8) / 7 = 7 / 7 = 1. Check: (15 – 9) / 2 = 6 / 2 = 3, which matches the first condition.

Let the number be x. The first condition says (x – 10) / 10 = 16. So x – 10 = 10 × 16 = 160. That gives x = 160 + 10 = 170. Now apply the second operation: subtract 10 from x and divide by 5. (170 – 10) / 5 = 160 / 5 = 32. Check: (170 – 10) / 10 = 160 / 10 = 16, which matches the first condition.

Let the number be x. The first condition says (x – 10) / 2 = 8. So x – 10 = 2 × 8 = 16. That gives x = 16 + 10 = 26. Now apply the second operation: subtract 10 from x and divide by 8. (26 – 10) / 8 = 16 / 8 = 2. Check: (26 – 10) / 2 = 16 / 2 = 8, which matches the first condition.

Let the number be x. The first condition says (x – 6) / 2 = 15. So x – 6 = 2 × 15 = 30. That gives x = 30 + 6 = 36. Now apply the second operation: subtract 4 from x and divide by 8. (36 – 4) / 8 = 32 / 8 = 4. Check: (36 – 6) / 2 = 30 / 2 = 15, which matches the first condition.

Let the number be x. The first condition says (x – 10) / 2 = 4. So x – 10 = 2 × 4 = 8. That gives x = 8 + 10 = 18. Now apply the second operation: subtract 8 from x and divide by 2. (18 – 8) / 2 = 10 / 2 = 5. Check: (18 – 10) / 2 = 8 / 2 = 4, which matches the first condition.

Let the number be x. The first condition says (x – 4) / 8 = 3. So x – 4 = 8 × 3 = 24. That gives x = 24 + 4 = 28. Now apply the second operation: subtract 8 from x and divide by 2. (28 – 8) / 2 = 20 / 2 = 10. Check: (28 – 4) / 8 = 24 / 8 = 3, which matches the first condition.

Let the number be x. The first condition says (x – 8) / 3 = 19. So x – 8 = 3 × 19 = 57. That gives x = 57 + 8 = 65. Now apply the second operation: subtract 5 from x and divide by 2. (65 – 5) / 2 = 60 / 2 = 30. Check: (65 – 8) / 3 = 57 / 3 = 19, which matches the first condition.

Let the number be x. The first condition says (x – 15) / 7 = 9. So x – 15 = 7 × 9 = 63. That gives x = 63 + 15 = 78. Now apply the second operation: subtract 6 from x and divide by 3. (78 – 6) / 3 = 72 / 3 = 24. Check: (78 – 15) / 7 = 63 / 7 = 9, which matches the first condition.

Let the number be x. The first condition says (x – 14) / 2 = 6. So x – 14 = 2 × 6 = 12. That gives x = 12 + 14 = 26. Now apply the second operation: subtract 2 from x and divide by 8. (26 – 2) / 8 = 24 / 8 = 3. Check: (26 – 14) / 2 = 12 / 2 = 6, which matches the first condition.

Let the number be x. The first condition says (x – 5) / 4 = 9. So x – 5 = 4 × 9 = 36. That gives x = 36 + 5 = 41. Now apply the second operation: subtract 1 from x and divide by 4. (41 – 1) / 4 = 40 / 4 = 10. Check: (41 – 5) / 4 = 36 / 4 = 9, which matches the first condition.

Let the number be x. The number multiplied by 1/3 of itself equals 108. So 1/3x² = 108. Dividing both sides by 1/3 gives x² = 108 / (1/3) = 324. Taking the positive square root gives x = 18. Check: 1/3 × 18 × 18 = 1/3 × 324 = 108, which matches.

Let the number be x. The number multiplied by 1/3 of itself equals 432. So 1/3x² = 432. Dividing both sides by 1/3 gives x² = 432 / (1/3) = 1296. Taking the positive square root gives x = 36. Check: 1/3 × 36 × 36 = 1/3 × 1296 = 432, which matches.

Let the number be x. The number multiplied by 1/4 of itself equals 576. So 1/4x² = 576. Dividing both sides by 1/4 gives x² = 576 / (1/4) = 2304. Taking the positive square root gives x = 48. Check: 1/4 × 48 × 48 = 1/4 × 2304 = 576, which matches.

Let the number be x. The number multiplied by 1/3 of itself equals 27. So 1/3x² = 27. Dividing both sides by 1/3 gives x² = 27 / (1/3) = 81. Taking the positive square root gives x = 9. Check: 1/3 × 9 × 9 = 1/3 × 81 = 27, which matches.

Let the number be x. The number multiplied by 3/4 of itself equals 972. So 3/4x² = 972. Dividing both sides by 3/4 gives x² = 972 / (3/4) = 1296. Taking the positive square root gives x = 36. Check: 3/4 × 36 × 36 = 3/4 × 1296 = 972, which matches.

Let the number be x. The number multiplied by 1/3 of itself equals 12. So 1/3x² = 12. Dividing both sides by 1/3 gives x² = 12 / (1/3) = 36. Taking the positive square root gives x = 6. Check: 1/3 × 6 × 6 = 1/3 × 36 = 12, which matches.

Let the number be x. The number multiplied by 3/4 of itself equals 1200. So 3/4x² = 1200. Dividing both sides by 3/4 gives x² = 1200 / (3/4) = 1600. Taking the positive square root gives x = 40. Check: 3/4 × 40 × 40 = 3/4 × 1600 = 1200, which matches.

Let the number be x. The number multiplied by 2/5 of itself equals 90. So 2/5x² = 90. Dividing both sides by 2/5 gives x² = 90 / (2/5) = 225. Taking the positive square root gives x = 15. Check: 2/5 × 15 × 15 = 2/5 × 225 = 90, which matches.

Let the number be x. The number plus 1/10 of itself equals 110. So x + 1/10x = 110. That simplifies to (11/10)x = 110. Solving gives x = 110 / (11/10) = 100. Check: 1/10 of 100 is 10, and 100 + 10 = 110, which matches.

Let the number be x. The number plus 1/9 of itself equals 410. So x + 1/9x = 410. That simplifies to (10/9)x = 410. Solving gives x = 410 / (10/9) = 369. Check: 1/9 of 369 is 41, and 369 + 41 = 410, which matches.

Let the number be x. The number plus 1/5 of itself equals 522. So x + 1/5x = 522. That simplifies to (6/5)x = 522. Solving gives x = 522 / (6/5) = 435. Check: 1/5 of 435 is 87, and 435 + 87 = 522, which matches.

Let the number be x. The number plus 1/10 of itself equals 891. So x + 1/10x = 891. That simplifies to (11/10)x = 891. Solving gives x = 891 / (11/10) = 810. Check: 1/10 of 810 is 81, and 810 + 81 = 891, which matches.

Let the number be x. The number plus 2/7 of itself equals 1296. So x + 2/7x = 1296. That simplifies to (9/7)x = 1296. Solving gives x = 1296 / (9/7) = 1008. Check: 2/7 of 1008 is 288, and 1008 + 288 = 1296, which matches.

Let the number be x. The number plus 2/7 of itself equals 1548. So x + 2/7x = 1548. That simplifies to (9/7)x = 1548. Solving gives x = 1548 / (9/7) = 1204. Check: 2/7 of 1204 is 344, and 1204 + 344 = 1548, which matches.