Estimate before calculating exactly
Mental ability in data questions begins with a rough answer. An estimate tells you whether an option is plausible and catches place-value mistakes. If a chart shows about 780 units out of a total near 3,100, the share is close to one quarter, about 25%; an answer near 75% cannot be right. Round only for the first check. Return to the exact values when options are close. Distinguish a percent of a total from a percent change between two values: one uses the total as base, the other uses the initial value.
Choose a helpful form
Common fractions shorten calculation: one eighth is 12.5%, one sixth is about 16.67%, three quarters is 75%. For 15% of 240, take 10% (24) plus 5% (12) to get 36. For an average, combine totals before dividing; do not average subgroup averages unless their group sizes are equal. If one school has 20 pupils averaging 40 and another has 30 averaging 50, the combined mean is (20×40+30×50)/50=46, not 45.
Order of operations and units
Rewrite percentages as fractions or decimals only once. In a multi-step expression, do multiplication and division before addition and subtraction. When a chart is in thousands but an option is in lakhs, convert units before comparing options. Keep a small margin estimate alongside the exact working. After choosing an option, use the estimate as a final sanity check. The practice set emphasises speed without abandoning exact arithmetic; record whether a miss was due to the base, the unit or a calculation slip.
A two-minute number check
Suppose a table reports 48, 52, 61 and 39 calls over four shifts. Pair numbers that simplify: 48+52=100 and 61+39=100, so the total is 200 and the mean is 50. The pairing is faster than a running column sum and makes a mistaken total such as 190 easy to notice. If the question instead asks for the range, use the largest and smallest observations: 61−39=22. A total, mean and range are different summaries of the same data; identify the requested summary before calculating.
For 18% of 250, compute 20% (50) and subtract 2% (5), giving 45. For 250 as a percentage of 400, simplify the fraction 250/400 to 5/8, or 62.5%. These two questions reverse the role of the base. In a multiple-choice item, mark the base under the words “of”, “out of” or “compared with”. When that base changes, the percentage generally changes even if the two raw numbers do not.
Check answers without repeating the whole calculation
Use an inverse operation. If a missing addend was obtained as 740−285=455, check that 455+285=740. If a 25% share is reported as 90, its implied total is 90×4=360. If a five-observation average is 46, the implied total is 230. Inverse checks are especially valuable when two options differ by a single digit. They can also reveal an impossible answer: a subgroup count cannot exceed the stated total, and a percentage of a whole cannot exceed 100% unless the question asks for growth or a ratio against a smaller base.
Practice method
Attempt ten items without a calculator, then classify each error. If the method was right but arithmetic was wrong, repeat the same operation with new numbers. If the method was wrong, rewrite the question in words before choosing a formula. For instance, “one fifth of the total” becomes “divide the total by five”; “one fifth more than the original” becomes “original plus original divided by five.” This translation step is the core mental skill. Speed grows from recognising the relation, not from rushing the arithmetic.
Practice — 50 questions
Attempt every item before reading the answers. Use the chart or caselet printed with the question.
1. A's income is 25% more than B's income. By what percent is B's income less than A's income?
A. 20% B. 100% C. 200% D. 40%
2. What is 24% of 9250?
A. 2220 B. 22200 C. 7030 D. 222
3. A's income is 150% more than B's income. By what percent is B's income less than A's income?
A. 37.5% B. 33⅓% C. 60% D. 12.5%
4. A's income is 37.5% less than B's income. By what percent is B's income more than A's income?
A. 50% B. 75% C. 60% D. 16⅔%
5. In an examination Arjun scored 35% marks and failed by 100 marks. If the pass percentage is 45%, what is the maximum marks of the examination?
A. 1000 B. 1100 C. 500 D. 450
6. What percent of 4200 is 2100?
A. 20% B. 25% C. 100% D. 50%
7. A's income is 50% less than B's income. By what percent is B's income more than A's income?
A. 100% B. 16⅔% C. 12.5% D. 33⅓%
8. In an examination Amit scored 34% marks and failed by 12 marks. If the pass percentage is 40%, what is the maximum marks of the examination?
A. 80 B. 100 C. 200 D. 30
9. The population of a village is 8,000. If it decreases at the rate of 10% per annum, what will be its population after 2 years?
A. 9,680 B. 7,200 C. 6,480 D. 6,400
10. In an election between two candidates, the winner got 60% of the valid votes and won by 2,400 votes. What was the total number of votes polled?
A. 4,000 B. 14,400 C. 6,000 D. 12,000
11. In an election between two candidates, the winner got 60% of the valid votes and won by 400 votes. What was the total number of votes polled?
A. 2,400 B. 2,000 C. 800 D. 1,000
12. The length of a rectangle is first decreased by 60% and then increased by 40%. What is the net percentage change?
A. decrease of 19% B. decrease of 20% C. increase of 44% D. decrease of 44%
13. The price of wheat rises by 60%. By what percent must a family reduce its consumption of wheat so that its expenditure on wheat remains the same?
A. 37.5% B. 12.5% C. 25% D. 28⁴⁄₇%
14. In an examination Asha scored 25% marks and failed by 80 marks. If the pass percentage is 33%, what is the maximum marks of the examination?
A. 500 B. 330 C. 320 D. 1000
15. The population of a village is 48. If it increases at the rate of 25% per annum, what will be its population after 2 years?
A. 27 B. 60 C. 75 D. 72
16. The population of a village is 1,10,000. If it decreases at the rate of 10% per annum, what will be its population after 2 years?
A. 88,000 B. 99,000 C. 89,100 D. 1,33,100
17. A's income is 25% more than B's income. By what percent is B's income less than A's income?
A. 20% B. 75% C. 12.5% D. 16⅔%
18. A's income is 150% more than B's income. By what percent is B's income less than A's income?
A. 80% B. 200% C. 300% D. 60%
19. In an examination Rahul scored 45% marks and passed by 100 marks. If the pass percentage is 35%, what is the maximum marks of the examination?
A. 1000 B. 1100 C. 500 D. 350
20. In an election between two candidates, the winner got 62% of the valid votes and won by 2,880 votes. What was the number of votes the winner received?
A. 7,440 B. 4,560 C. 12,000 D. 2,880
21. A's income is 20% less than B's income. By what percent is B's income more than A's income?
A. 200% B. 75% C. 25% D. 12.5%
22. In an examination Priya scored 20% and failed by 20 marks, while Suresh scored 35% and passed by 10 marks. What is the maximum marks of the examination?
A. 200 B. 100 C. 300 D. 400
23. In an election between two candidates, 15% of the votes were declared invalid and the winner got 58% of the valid votes and won by 1,360 votes. What was the total number of votes polled?
A. 12,000 B. 10,000 C. 5,000 D. 8,500
24. In an election between two candidates, 10% of the votes were declared invalid and the winner got 56% of the valid votes and won by 270 votes. What was the number of votes the winner received?
A. 2,250 B. 990 C. 1,260 D. 270
25. In an election between two candidates, 5% of the votes were declared invalid and the winner got 60% of the valid votes and won by 3,800 votes. What was the number of votes the winner received?
A. 3,800 B. 11,400 C. 7,600 D. 19,000
26. The average age of 12 family members is 19 years. When the newborn's age is included, the average rises by 3. Find the newborn's age.
A. 19 years B. 58 years C. 22 years D. 55 years
27. Find the average of the first 65 even numbers.
A. 132 B. 65 C. 67 D. 66
28. The average age of 13 family members is 54 years. When the newborn's age is included, the average rises by 1. Find the newborn's age.
A. 55 years B. 68 years C. 67 years D. 41 years
29. The average of 3 observations is 58. If the average of 2 of them is 70, find the value of the remaining observation.
A. 70 B. 34 C. 174 D. 58
30. The average age of 10 girls is recorded. When a girl aged 58 years is replaced by a new one, the average age rises by 1 years. Find the age of the new girl.
A. 59 years B. 10 years C. 48 years D. 68 years
31. The average age of 17 students is 44 years. When the teacher's age is included, the average falls by 2. Find the teacher's age.
A. 8 years B. 78 years C. 42 years D. 10 years
32. The average of 16 numbers is 471. Find the sum of these numbers.
A. 7520 B. 7536 C. 7552 D. 471
33. The sum of 7 numbers is 2401. Find their average.
A. 343 B. 344 C. 342 D. 2401
34. The sum of 10 numbers is 774. The average of the first 7 of them is 75, and the next number is 73. Find the average of the last 2 numbers.
A. 93 B. 176 C. 88 D. 75
35. The average of 20 numbers is 94. If each number is multiplied by 4, find the new average.
A. 356 B. 376 C. 396 D. 94
36. The average of 4 observations is 39. If the average of 3 of them is 27, find the value of the remaining observation.
A. 27 B. 75 C. 156 D. 39
37. A batsman scores 116 runs in his 17th innings and thus raises his average by 6. Find his previous batting average.
A. 20 B. 14 C. 31 D. 8
38. The average age of 17 family members is 42 years. When the newborn's age is included, the average falls by 1. Find the newborn's age.
A. 59 years B. 41 years C. 25 years D. 24 years
39. The average age of 12 members is recorded. When a member aged 50 years is replaced by a new one, the average age rises by 1 years. Find the age of the new member.
A. 38 years B. 12 years C. 62 years D. 51 years
40. The average of 7 observations is 26. If the average of 6 of them is 23, find the value of the remaining observation.
A. 44 B. 26 C. 182 D. 23
41. The average age of 6 members is recorded. When a member aged 35 years is replaced by a new one, the average age rises by 6 years. Find the age of the new member.
A. 41 years B. 71 years C. 35 years D. 36 years
42. The average age of 10 workers is 51 years. When the new worker's age is included, the average falls by 4. Find the new worker's age.
A. 7 years B. 11 years C. 91 years D. 47 years
43. The average age of 9 members is recorded. When a member aged 60 years is replaced by a new one, the average age rises by 4 years. Find the age of the new member.
A. 96 years B. 36 years C. 64 years D. 24 years
44. The sum of 7 numbers is 658. The average of the first 4 of them is 39, and the next number is 66. Find the average of the last 2 numbers.
A. 94 B. 436 C. 218 D. 39
45. The average age of 23 workers is 56 years. When the new worker's age is included, the average rises by 1. Find the new worker's age.
A. 79 years B. 57 years C. 33 years D. 80 years
46. The average age of 11 students is 31 years. When the teacher's age is included, the average rises by 3. Find the teacher's age.
A. 34 years B. 64 years C. 31 years D. 67 years
47. The average of P, Q and R is 89; the average of Q, R and S is 39; and P + S = 200. Find P.
A. 89 B. 175 C. 161 D. 25
48. The average age of 6 players is recorded. When a player aged 45 years is replaced by a new one, the average age rises by 2 years. Find the age of the new player.
A. 57 years B. 33 years C. 12 years D. 47 years
49. A batsman scores 145 runs in his 11th innings and thus raises his average by 1. Find his previous batting average.
A. 133 B. 135 C. 134 D. 145
50. A batsman scores 115 runs in his 11th innings and thus raises his average by 3. Find his previous batting average.
A. 93 B. 85 C. 79 D. 82
Answers and explanations
Check the value and the unit before comparing your working with the calculation.
1. A. A is more than B by 1/4 of B. Take B = 4 units, then A = 5 units. B is less than A by 1 units out of 5, i.e. 1/5 = 20%. (Rule: more by a/b ⇒ less by a/(a+b).)
APDI26-01-01 | Percentage | Easy
2. A. Percent means 'per hundred', so 24% of 9250 = (24/100) × 9250 = 222000/100 = 2220.
APDI26-01-02 | Percentage | Easy
3. C. A is more than B by 3/2 of B. Take B = 2 units, then A = 5 units. B is less than A by 3 units out of 5, i.e. 3/5 = 60%. (Rule: more by a/b ⇒ less by a/(a+b).)
APDI26-01-03 | Percentage | Easy
4. C. A is less than B by 3/8 of B. Take B = 8 units, then A = 5 units. B is more than A by 3 units out of 5, i.e. 3/5 = 60%. (Rule: less by a/b ⇒ more by a/(b−a).)
APDI26-01-04 | Percentage | Easy
5. A. Difference between pass marks and marks obtained = (45 − 35)% of maximum = 10% of M = 100. So M = 100 × 100/10 = 1000.
APDI26-01-05 | Percentage | Easy
6. D. Required percentage = (part/whole) × 100 = (2100/4200) × 100 = 210000/4200 = 50%.
APDI26-01-06 | Percentage | Easy
7. A. A is less than B by 1/2 of B. Take B = 2 units, then A = 1 units. B is more than A by 1 units out of 1, i.e. 1/1 = 100%. (Rule: less by a/b ⇒ more by a/(b−a).)
APDI26-01-07 | Percentage | Easy
8. C. Difference between pass marks and marks obtained = (40 − 34)% of maximum = 6% of M = 12. So M = 12 × 100/6 = 200.
APDI26-01-08 | Percentage | Easy
9. C. Population after 2 years = P × (1 − 10/100)^2 = 8,000 × (9/10)^2 = 6,480.
APDI26-01-09 | Percentage | Medium
10. D. Winner 60%, loser 40% of valid votes; margin = 20% of valid votes = 2,400 ⇒ valid votes = 12,000. Winner's votes = 60% of 12,000 = 7,200.
APDI26-01-10 | Percentage | Medium
11. B. Winner 60%, loser 40% of valid votes; margin = 20% of valid votes = 400 ⇒ valid votes = 2,000. Winner's votes = 60% of 2,000 = 1,200.
APDI26-01-11 | Percentage | Medium
12. D. Net change = a + b + ab/100 with a = -60, b = 40: -60 + (40) + (-60 × 40)/100 = -44%. A 44% decrease. (Simply adding the two percentages, -20%, ignores the compounding term.)
APDI26-01-12 | Percentage | Medium
13. A. Price rises by 3/5: new price = 8/5 of old. To keep expenditure fixed, consumption must become 5/8 of old, a reduction of 3/8 = 3/8 = 37.5%.
APDI26-01-13 | Percentage | Medium
14. D. Difference between pass marks and marks obtained = (33 − 25)% of maximum = 8% of M = 80. So M = 80 × 100/8 = 1000.
APDI26-01-14 | Percentage | Medium
15. C. Population after 2 years = P × (1 + 25/100)^2 = 48 × (5/4)^2 = 75.
APDI26-01-15 | Percentage | Medium
16. C. Population after 2 years = P × (1 − 10/100)^2 = 1,10,000 × (9/10)^2 = 89,100.
APDI26-01-16 | Percentage | Medium
17. A. A is more than B by 1/4 of B. Take B = 4 units, then A = 5 units. B is less than A by 1 units out of 5, i.e. 1/5 = 20%. (Rule: more by a/b ⇒ less by a/(a+b).)
APDI26-01-17 | Percentage | Medium
18. D. A is more than B by 3/2 of B. Take B = 2 units, then A = 5 units. B is less than A by 3 units out of 5, i.e. 3/5 = 60%. (Rule: more by a/b ⇒ less by a/(a+b).)
APDI26-01-18 | Percentage | Medium
19. A. Difference between pass marks and marks obtained = (35 − 45)% of maximum = 10% of M = 100. So M = 100 × 100/10 = 1000.
APDI26-01-19 | Percentage | Medium
20. A. Winner 62%, loser 38% of valid votes; margin = 24% of valid votes = 2,880 ⇒ valid votes = 12,000. Winner's votes = 62% of 12,000 = 7,440.
APDI26-01-20 | Percentage | Medium
21. C. A is less than B by 1/5 of B. Take B = 5 units, then A = 4 units. B is more than A by 1 units out of 4, i.e. 1/4 = 25%. (Rule: less by a/b ⇒ more by a/(b−a).)
APDI26-01-21 | Percentage | Medium
22. A. Pass marks = 20% of M + 20 = 35% of M − 10. So (35 − 20)% of M = 20 + 10 = 30, giving M = 30 × 100/15 = 200.
APDI26-01-22 | Percentage | Difficult
23. B. Winner 58%, loser 42% of valid votes; margin = 16% of valid votes = 1,360 ⇒ valid votes = 8,500. Valid votes are 85% of the total ⇒ total = 10,000. Winner's votes = 58% of 8,500 = 4,930.
APDI26-01-23 | Percentage | Difficult
24. C. Winner 56%, loser 44% of valid votes; margin = 12% of valid votes = 270 ⇒ valid votes = 2,250. Valid votes are 90% of the total ⇒ total = 2,500. Winner's votes = 56% of 2,250 = 1,260.
APDI26-01-24 | Percentage | Difficult
25. B. Winner 60%, loser 40% of valid votes; margin = 20% of valid votes = 3,800 ⇒ valid votes = 19,000. Valid votes are 95% of the total ⇒ total = 20,000. Winner's votes = 60% of 19,000 = 11,400.
APDI26-01-25 | Percentage | Difficult
26. B. New average = 19 + 3 = 22. New member's value = newAvg × (n+1) − oldAvg × n = 22 × 13 − 19 × 12 = 58.
APDI26-01-26 | Average | Easy
27. D. Using the standard formula for the average of first 65 even numbers: average = n+1 = 65+1 = 66. Memorising these standard formulas saves time versus listing and adding every term.
APDI26-01-27 | Average | Easy
28. B. New average = 54 + 1 = 55. New member's value = newAvg × (n+1) − oldAvg × n = 55 × 14 − 54 × 13 = 68.
APDI26-01-28 | Average | Easy
29. B. Total of all 3 = 3 × 58 = 174. Total of known 2 = 2 × 70 = 140. Missing value = 174 − 140 = 34.
APDI26-01-29 | Average | Easy
30. D. Change in total = n × k = 10 × 1 = 10. New value = old value + 10 = 58 + 10 = 68.
APDI26-01-30 | Average | Easy
31. A. New average = 44 − 2 = 42. New member's value = newAvg × (n+1) − oldAvg × n = 42 × 18 − 44 × 17 = 8.
APDI26-01-31 | Average | Easy
32. B. Sum of a group of numbers = average value × number of terms = 471 × 16 = 7536.
APDI26-01-32 | Average | Easy
33. A. Average of a group of numbers = sum of the numbers/number of terms = 2401/7 = 343.
APDI26-01-33 | Average | Easy
34. C. Sum of first 7 = 75 × 7 = 525. Remaining sum after removing these and the next number 73: 774 − 525 − 73 = 176. Average of last 2 = 176/2 = 88.
APDI26-01-34 | Average | Medium
35. B. Multiplying every value by k multiplies the average by k too: new average = 94 × 4 = 376.
APDI26-01-35 | Average | Medium
36. B. Total of all 4 = 4 × 39 = 156. Total of known 3 = 3 × 27 = 81. Missing value = 156 − 81 = 75.
APDI26-01-36 | Average | Medium
37. B. New average = score − (n−1)×k = 116 − 16×6 = 20. Previous average = new average − k = 20 − 6 = 14.
APDI26-01-37 | Average | Medium
38. D. New average = 42 − 1 = 41. New member's value = newAvg × (n+1) − oldAvg × n = 41 × 18 − 42 × 17 = 24.
APDI26-01-38 | Average | Medium
39. C. Change in total = n × k = 12 × 1 = 12. New value = old value + 12 = 50 + 12 = 62.
APDI26-01-39 | Average | Medium
40. A. Total of all 7 = 7 × 26 = 182. Total of known 6 = 6 × 23 = 138. Missing value = 182 − 138 = 44.
APDI26-01-40 | Average | Medium
41. B. Change in total = n × k = 6 × 6 = 36. New value = old value + 36 = 35 + 36 = 71.
APDI26-01-41 | Average | Medium
42. A. New average = 51 − 4 = 47. New member's value = newAvg × (n+1) − oldAvg × n = 47 × 11 − 51 × 10 = 7.
APDI26-01-42 | Average | Medium
43. A. Change in total = n × k = 9 × 4 = 36. New value = old value + 36 = 60 + 36 = 96.
APDI26-01-43 | Average | Medium
44. C. Sum of first 4 = 39 × 4 = 156. Remaining sum after removing these and the next number 66: 658 − 156 − 66 = 436. Average of last 2 = 436/2 = 218.
APDI26-01-44 | Average | Medium
45. D. New average = 56 + 1 = 57. New member's value = newAvg × (n+1) − oldAvg × n = 57 × 24 − 56 × 23 = 80.
APDI26-01-45 | Average | Medium
46. D. New average = 31 + 3 = 34. New member's value = newAvg × (n+1) − oldAvg × n = 34 × 12 − 31 × 11 = 67.
APDI26-01-46 | Average | Medium
47. B. P+Q+R = 267, Q+R+S = 117, so P − S = 267 − 117 = 150. With P + S = 200, adding gives 2P = 350, so P = 175.
APDI26-01-47 | Average | Difficult
48. A. Change in total = n × k = 6 × 2 = 12. New value = old value + 12 = 45 + 12 = 57.
APDI26-01-48 | Average | Difficult
49. C. New average = score − (n−1)×k = 145 − 10×1 = 135. Previous average = new average − k = 135 − 1 = 134.
APDI26-01-49 | Average | Difficult
50. D. New average = score − (n−1)×k = 115 − 10×3 = 85. Previous average = new average − k = 85 − 3 = 82.
APDI26-01-50 | Average | Difficult