Combining Groups: Add Totals, Not Averages
When two groups are put together, the new average is not the plain average of the two averages. Go back to totals. Each group's total is its average times its count; add the totals and divide by everyone.
Here and are the sizes of the groups and and are their averages.
Boys and girls. A class has 30 boys with average marks 52 and 20 girls with average marks 62.
- Boys' total = 30 × 52 = 1560 and girls' total = 20 × 62 = 1240.
- Class total = 2800 for 50 students, so the class average is 56.
The plain average of 52 and 62 is 57. That is wrong because there are more boys than girls, and the bigger group pulls the answer towards itself.
Three facts that hold every time:
- The combined average lies between the two group averages.
- It sits closer to the bigger group.
- It equals the plain average only when the two groups are the same size.
These three facts alone knock out two options in many questions.
Ratios and percentages work as counts. You do not need the actual numbers of boys and girls, only how they compare. If boys : girls = 3 : 2, treat it as 3 boys and 2 girls: (3 × 52 + 2 × 62) ÷ 5 = 280 ÷ 5 = 56. "60% of the class are boys" and "the girls are of the boys" give the same weights, 3 : 2.
The quick deviation way. Start from the smaller average and add a share of the gap:
Girls are of the class and score 10 more than the boys, so the class average is . With awkward numbers this saves a lot of multiplying.
Passed and failed students. In a test, 45 students passed with average marks 64 and 15 failed with average marks 28. The totals are 45 × 64 = 2880 and 15 × 28 = 420, so the class has 3300 marks for 60 students and the class average is 55. The labels change from paper to paper (sections A and B, officers and clerks, men and women staff, juniors and seniors), but the method stays the same.
Average speed over a journey is also a weighted average, with time as the weight (see Chapter 10).
Three or More Groups, and Averages from a Table
The rule does not change when there are more groups: add every group's total and divide by the total count.
Counts given. Sections A, B and C have 30, 45 and 25 students, with average marks 66, 72 and 60.
- Totals: 30 × 66 = 1980, 45 × 72 = 3240 and 25 × 60 = 1500.
- All together: 6720 marks for 100 students, so the average is 6720 ÷ 100 = 67.2.
Dividing 6720 by 3, the number of sections, is a slip that SSC options are built to catch.
Group sizes in a ratio. Three teams of workers earn average daily wages of ₹640, ₹720 and ₹600, and the numbers of workers are in the ratio 2 : 3 : 5. Use 2, 3 and 5 as the counts:
(2 × 640 + 3 × 720 + 5 × 600) ÷ 10 = (1280 + 2160 + 3000) ÷ 10 = 6440 ÷ 10 = ₹644.
The base-value shortcut. Big multiplications are where the time goes. Pick a round base (₹600 here), see how far each group is above it, and average only those gaps:
- The ₹640 team is 40 above, the ₹720 team is 120 above and the ₹600 team is 0 above.
- Weighted gaps: 2 × 40 + 3 × 120 + 5 × 0 = 440, spread over 10 workers, which is 44.
- Average = 600 + 44 = ₹644.
Gaps below the base count as minus. With three or four groups this is usually the fastest way.
Averages from a table of counts. CGL papers sometimes give a table of values and how many people have each value. Every column is a small group, so this is a weighted average too.
| Marks (out of 10) | 4 | 6 | 7 | 8 | 10 |
|---|---|---|---|---|---|
| Number of students | 3 | 5 | 8 | 6 | 3 |
Total marks = 4 × 3 + 6 × 5 + 7 × 8 + 8 × 6 + 10 × 3 = 12 + 30 + 56 + 48 + 30 = 176. The number of students is 3 + 5 + 8 + 6 + 3 = 25, so the average is 176 ÷ 25 = 7.04.
The traps are dividing by 5 (the number of columns) and taking the plain average of the five mark values, which is 7. If a table shows two sections side by side, find each section's total the same way and then combine them like two groups.
Finding a Missing Group Average or Total
Often the question gives the overall average and one group, and asks about the other group. Work with totals again: the missing total is the whole total minus the known total.
The rest of a team. An office has 80 employees with an average salary of ₹45,000. The 30 people in the sales team average ₹53,000. Find the average salary of the other 50.
- Whole total = 80 × 45,000 = ₹36,00,000.
- Sales team total = 30 × 53,000 = ₹15,90,000.
- The other 50 earn ₹20,10,000 in all, so their average is ₹40,200.
The balance way. Think of the overall average as a balance point. The sales team is ₹8,000 above it, and there are 30 of them, so they are ₹2,40,000 above in all. The other 50 must be ₹2,40,000 below in all, which is ₹4,800 each. Their average is 45,000 − 4,800 = ₹40,200. No lakh-sized totals needed.
Leaving out a top group and a bottom group. The average score of 60 students is 58. The top 6 average 88 and the bottom 4 average 23. Find the average of the remaining 50.
- Totals: the class has 60 × 58 = 3480, the top 6 have 6 × 88 = 528 and the bottom 4 have 4 × 23 = 92.
- The remaining 50 have 3480 − 528 − 92 = 2860, so their average is 57.2.
Balance check: the top 6 are 30 above the average each (+180 in all) and the bottom 4 are 35 below each (−140 in all). Removing them takes away a net +40, so the other 50 are 40 below in all, which is 0.8 each: 58 − 0.8 = 57.2.
One person counted inside the average. "The average monthly salary of 15 clerks and their section officer is ₹24,500. The officer earns ₹56,000. Find the clerks' average." The ₹24,500 covers 16 people, so they earn 16 × 24,500 = ₹3,92,000. Taking out the officer leaves ₹3,36,000 for 15 clerks, an average of ₹22,400.
A topper works the same way. If 30 students including the topper average 62 and the topper scored 91, the other 29 average (1860 − 91) ÷ 29 = 61. So does a class average that includes the teacher, or a team average that includes the coach.
One group's size is missing. "The average marks of a group are 48. Twenty of them average 54 and the rest average 44. How many are in the group?" With x people in the rest:
1080 + 44x = 960 + 48x, so 4x = 120 and x = 30. The group has 50 people. The distance rule below gets this faster.
[SSC Tip] Read whether the given average includes the special person (officer, teacher, topper) or not. "The average of 15 clerks and the officer" means 16 people; "the clerks average ₹22,400 and the officer earns ₹56,000" means the 15 clerks only.
Finding Group Sizes: The Distance Rule
When you know both group averages and the overall average, the sizes of the groups come out of distances alone.
Mark the three averages on a number line. The overall average A sits between the two group averages. Each group's size goes with the other group's distance from A:
So the group farther from the overall average is the smaller group.
Officers and workers. In a factory the average salary is ₹30,000. The 12 officers average ₹60,000 and the workers average ₹24,000. How many workers are there?
- Officers are 60,000 − 30,000 = 30,000 away from the overall average; workers are 30,000 − 24,000 = 6,000 away.
- Swap the distances: officers : workers = 6,000 : 30,000 = 1 : 5.
- 12 officers means 60 workers, and 72 people in all.
Check: 12 × 60,000 + 60 × 24,000 = 7,20,000 + 14,40,000 = 21,60,000, and 21,60,000 ÷ 72 = 30,000.
Ratio of boys to girls. Boys in a class average 58 marks, girls average 70, and the class averages 63. Boys : girls = (70 − 63) : (63 − 58) = 7 : 5. In a class of 48 that means 28 boys and 20 girls, and the girls make up of the class.
Why it works. The marks one group has above A must exactly balance what the other group is short below A. Here the 20 girls are 7 above each, which is 140 in all, and the 28 boys are 5 below each, which is also 140.
A group joins and the average moves. "The average marks of a class are 54. When 8 new students with average marks 72 join, the class average becomes 56. How many students were there at first?" Compare everyone with the new average 56. The old students are 2 below it and the newcomers are 16 above it. So old : new = 16 : 2 = 8 : 1, and with 8 newcomers there were 64 students at first. Check: 64 × 54 + 8 × 72 = 3456 + 576 = 4032, and 4032 ÷ 72 = 56.
The missing-size question above is quicker this way: 54 is 6 above 48 and 44 is 4 below, so the sizes are 4 : 6 = 2 : 3. Twenty people at 54 means 30 in the rest.
This distance rule is the same idea as alligation for mixtures (Chapter 6).
[SSC Tip] Swapping the distances is the classic mistake, and the swapped ratio is usually one of the options. Ask yourself: is the overall average closer to the bigger group? If not, flip your ratio.
Average Price per kg and Cost per Item
When different quantities are bought at different rates, the average rate is the total cost divided by the total quantity. It is a weighted average with quantity as the weight.
Two lots. A grocer buys 12 kg of rice at ₹40 per kg and 18 kg at ₹55 per kg. The total cost is 12 × 40 + 18 × 55 = 480 + 990 = ₹1,470 for 30 kg, so the average price is ₹49 per kg. The plain average of 40 and 55 is 47.5, which is wrong because more of the dearer rice was bought.
Quantities in a ratio. Three teas costing ₹180, ₹240 and ₹300 per kg are mixed in the ratio 3 : 2 : 1. The average is (3 × 180 + 2 × 240 + 1 × 300) ÷ 6 = 1320 ÷ 6 = ₹220 per kg.
Items work the same way. 40 pens at ₹12 and 60 pens at ₹17 cost 480 + 1020 = ₹1,500 for 100 pens, an average of ₹15 a pen.
Equal quantity or equal money? This is the main trap in price questions.
- The same quantity of each kind: the average price is the plain average.
- The same money on each kind: you get more of the cheaper kind, so the average price leans towards the cheaper price.
"Ravi spends ₹1,200 on apples at ₹80 per kg and another ₹1,200 on apples at ₹120 per kg. Find his average cost per kg." He gets 15 kg and 10 kg, which is 25 kg for ₹2,400. The average is ₹96 per kg, not ₹100. For two prices x and y with equal money spent on each,
Three prices, equal money. "A shop spends equal amounts on three lots of notebooks costing ₹20, ₹30 and ₹60 each. Find the average cost of a notebook." Pick a spend that all three prices divide, say ₹60 per lot. That buys 3, 2 and 1 notebooks, which is 6 notebooks for ₹180, so the average is ₹30. Picking such a number saves you from fractions.
Finding the quantity. "How many kg of rice at ₹40 per kg must be mixed with 18 kg at ₹55 per kg so that the mixture costs ₹49 per kg?" Use the distance rule: ₹40 is 9 away from ₹49 and ₹55 is 6 away, so cheap : dear = 6 : 9 = 2 : 3. With 18 kg of the dear rice, we need 12 kg of the cheap rice.
Average speed over equal distances works out the same way as equal money.
Weighted Percentages: Marks, Pass Rates and Returns
A percentage is a share of some base. When the bases differ, the average percentage is not the plain average of the percentages. Turn every percentage back into an actual amount, add the amounts, and divide by the total base.
Papers with different maximum marks. Neha scores 90% in a paper out of 50, 60% in a paper out of 100 and 72% in a paper out of 150.
- Marks: 45 + 60 + 108 = 213, out of 50 + 100 + 150 = 300.
- Overall percentage = 213 ÷ 300 × 100 = 71%.
The plain average of 90, 60 and 72 is 74%, which is the trap. The 150-mark paper carries the most weight, and she did only fairly well in it.
Two classes. In class A, 75% of 40 students passed; in class B, 60% of 60 students passed. That is 30 + 36 = 66 passes out of 100 students, so the overall pass rate is 66%, not 67.5%.
Boys and girls taking part. In a school, 45% of the 600 boys and 60% of the 400 girls play a sport. Players = 270 + 240 = 510 out of 1000, which is 51%.
A target percentage. "Sanjay needs 60% overall in three papers of 100, 100 and 200 marks. He scored 52% and 58% in the first two. What percentage does he need in the third?" He needs 60% of 400, which is 240 marks. He has 52 + 58 = 110, so he needs 130 out of 200, which is 65%. Averaging 52, 58 and x to 60 gives x = 70%, which forgets that the third paper is worth double.
Money at different rates. ₹40,000 earns 9% a year and ₹60,000 earns 14% a year. The total return is 3,600 + 8,400 = ₹12,000 on ₹1,00,000, so the average rate is 12%. The weights are the amounts invested, not the number of schemes.
Going the other way uses the distance rule. If ₹1,00,000 is split between 9% and 14% and the average rate is 12%, the split is (14 − 12) : (12 − 9) = 2 : 3, which is ₹40,000 and ₹60,000.
Returns on equal amounts. If the amounts are equal, the plain average works again: equal sums at 6%, 8% and 13% give an average rate of 9%. The question has to tell you the amounts are equal; do not assume it.
Salaries, Incomes and Increments
Salary questions are weighted averages with money as the value and headcount as the weight. What is new here is how a raise changes the average.
Average salary across groups. A firm has 20 workers averaging ₹15,000 a month and 5 supervisors averaging ₹40,000. The totals are ₹3,00,000 and ₹2,00,000, which is ₹5,00,000 for 25 people, so the average is ₹20,000.
Women and men from a percentage. A company has 80 employees, 35% of them women. The women average ₹48,000 and the men ₹56,000. There are 28 women and 52 men, so the average is (28 × 48,000 + 52 × 56,000) ÷ 80 = 42,56,000 ÷ 80 = ₹53,200. Shortcut: men are 65% of the staff and earn ₹8,000 more, so the average is 48,000 + 65% of 8,000 = ₹53,200.
How a raise changes the average. Keep the firm of 25 above, with average ₹20,000.
| Raise | Effect on the average | New average |
|---|---|---|
| Everyone gets ₹1,500 more | rises by ₹1,500 | ₹21,500 |
| Everyone gets 10% more | rises by 10% | ₹22,000 |
| Only the 20 workers get ₹1,250 each | rises by 20 × 1,250 ÷ 25 = ₹1,000 | ₹21,000 |
| Only the 5 supervisors get 10% | rises by 5 × 4,000 ÷ 25 = ₹800 | ₹20,800 |
Different percentage raises for different groups. Say the workers get 10% and the supervisors 5%. The workers' bill rises by ₹30,000 and the supervisors' by ₹10,000, so the total becomes ₹5,40,000 and the average becomes ₹21,600, a rise of 8%.
That 8% is a weighted average of 10% and 5%, but the weights are the salary totals (3 : 2), not the headcounts (4 : 1). Using headcounts gives 9%, a common wrong option.
Finding a raise. "The average salary of 40 staff is ₹25,000. After the 15 seniors get a raise, the average becomes ₹26,500. Find each senior's raise." The total went up by 40 × 1,500 = ₹60,000, shared by 15 seniors, so each senior got ₹4,000.
A cut works the same way. If the 20 workers each lose ₹500 for a month, the firm's average drops by 20 × 500 ÷ 25 = ₹400.
Average of Averages, and Two-Step Questions
When the plain average of averages works. Only when every group has the same size. Four sections of 30 students each average 60, 64, 70 and 66, so the whole school's average is (60 + 64 + 70 + 66) ÷ 4 = 65.
When it does not. "The average temperature for the first 10 days of a month was 30°C and for the next 20 days it was 36°C." The month's average is (300 + 720) ÷ 30 = 34°C, not 33°C. The days are the weights here.
When it cannot be found. "Class A averages 60 and class B averages 70. What is the average of both classes together?" Without the sizes, or at least their ratio, the answer could be anything between 60 and 70. SSC sometimes keeps "Cannot be determined" as the right option for exactly this.
[Tier 2] A change in one group's marks. If every boy's marks went up by 5 and the class average went up by 3, then the boys must be of the class. Each boy's 5 extra marks get shared over the whole class:
So if, instead, every girl (the other ) gained 10 marks, the class average would rise by .
[Tier 2] One student moves from one section to another. Section A has 21 students averaging 72 marks and section B has 19 students averaging 62. A student who scored 67 moves from A to B.
- A: (1512 − 67) ÷ 20 = 1445 ÷ 20 = 72.25, so A's average rises by 0.25.
- B: (1178 + 67) ÷ 20 = 1245 ÷ 20 = 62.25, so B's average also rises by 0.25.
Both averages go up, because 67 was below A's average and above B's. The average of all 40 students does not change: it is 2690 ÷ 40 = 67.25 before and after.
Going backwards:
- If a student leaves a group of n with average a and the average rises by x, the student's marks are a − (n − 1)x.
- If a student joins a group of m with average b and the average rises by y, the student's marks are b + (m + 1)y.
Here 72 − 20 × 0.25 = 67 and 62 + 20 × 0.25 = 67, so both give the same student.
Solved Examples
Question 1
In a class, the average weight of 27 boys is 48 kg and the average weight of 18 girls is 43 kg. What is the average weight of the whole class?
Answer. I add the totals, not the averages. Boys: 27 × 48 = 1296 kg. Girls: 18 × 43 = 774 kg. Together that is 2070 kg for 45 students, so the class average is 2070 ÷ 45 = 46 kg. Quick check: the boys are of the class and are 5 kg heavier, so the average is 43 + 3 = 46. It sits nearer the boys' 48 because there are more boys.
Question 2
The numbers of students in sections A and B of a school are in the ratio 5 : 3. The average marks of section A are 68 and those of section B are 76. Find the average marks of both sections together.
Answer. I treat the ratio as 5 students and 3 students. Total = 5 × 68 + 3 × 76 = 340 + 228 = 568 for 8 students, so the average is 568 ÷ 8 = 71. The plain average of 68 and 76 is 72, but section A is bigger, so the answer has to be nearer 68.
Question 3
The average marks of 180 students in a test were 52. Those who passed averaged 60 and those who failed averaged 36. How many students passed?
Answer. The passed group is 60 − 52 = 8 above the class average and the failed group is 52 − 36 = 16 below it. The sizes go the other way round, so passed : failed = 16 : 8 = 2 : 1. Out of 180, that is 120 passed and 60 failed. Check: 120 × 60 + 60 × 36 = 7200 + 2160 = 9360, and 180 × 52 = 9360.
Question 4
Three sections of Class 10 have 32, 40 and 28 students. Their average marks are 75, 62 and 70. What is the average mark of all the students?
Answer. Totals: 32 × 75 = 2400, 40 × 62 = 2480 and 28 × 70 = 1960. That makes 6840 marks for 100 students, so the average is 68.4. Dividing 6840 by 3, the number of sections, is the mistake to avoid.
Question 5
Three groups of athletes have average weights of 62 kg, 70 kg and 58 kg. The numbers of athletes in the groups are in the ratio 2 : 3 : 5. Find the average weight of all the athletes.
Answer. I use 2, 3 and 5 as the counts and take 58 kg as the base. The first group is 4 kg above it, the second 12 kg above and the third 0. The weighted extra is 2 × 4 + 3 × 12 + 5 × 0 = 44 kg over 10 athletes, which is 4.4 kg each. So the average is 58 + 4.4 = 62.4 kg. The long way gives the same: (124 + 210 + 290) ÷ 10 = 624 ÷ 10 = 62.4.
Question 6
The table shows the daily wages paid in a small factory.
| Daily wage (₹) | 500 | 600 | 700 | 800 | 1,000 |
|---|---|---|---|---|---|
| Number of workers | 6 | 10 | 12 | 8 | 4 |
What is the average daily wage?
Answer. Each column is a small group. I take ₹700 as the base and add up the gaps: 6 × (−200) + 10 × (−100) + 12 × 0 + 8 × 100 + 4 × 300 = −1200 − 1000 + 0 + 800 + 1200 = −200. There are 6 + 10 + 12 + 8 + 4 = 40 workers, so the average is 200 ÷ 40 = 5 below the base, which is ₹695. (The total wage bill is ₹27,800, and 27,800 ÷ 40 = 695.)
Question 7
The average monthly salary of 60 employees in an office is ₹38,000. The average salary of 24 of them is ₹47,000. What is the average salary of the remaining employees?
Answer. The 24 are ₹9,000 above the overall average, which is 24 × 9,000 = ₹2,16,000 extra. The other 36 must be ₹2,16,000 below in all, which is ₹6,000 each. So their average is 38,000 − 6,000 = ₹32,000. Check with totals: 60 × 38,000 = 22,80,000 and 24 × 47,000 = 11,28,000, which leaves 11,52,000 for 36 people, and 11,52,000 ÷ 36 = 32,000.
Question 8
The average score of 45 students in a test is 66. The top 5 students have an average of 96 and the bottom 4 have an average of 33. What is the average score of the remaining students?
Answer. The remaining students number 45 − 5 − 4 = 36. The whole class has 45 × 66 = 2970 marks, the top 5 have 5 × 96 = 480 and the bottom 4 have 4 × 33 = 132. The rest have 2970 − 480 − 132 = 2358, so their average is 2358 ÷ 36 = 65.5.
Question 9
The average monthly salary of 11 nurses and their head nurse is ₹31,500. The head nurse earns ₹48,000. What is the average salary of the 11 nurses?
Answer. The ₹31,500 average is for 12 people, so the total is 12 × 31,500 = ₹3,78,000. Taking out the head nurse leaves ₹3,30,000 for 11 nurses, so they average ₹30,000. Using 11 × 31,500 for the total is the usual slip.
Question 10
The average salary of all the staff in an office is ₹35,000. The 15 officers have an average salary of ₹63,000 and the rest of the staff have an average of ₹28,000. How many staff members are there in all?
Answer. Officers are 63,000 − 35,000 = 28,000 above the overall average, and the others are 35,000 − 28,000 = 7,000 below. Swapping the distances, officers : others = 7,000 : 28,000 = 1 : 4. With 15 officers there are 60 others, so the office has 75 staff. Check: 15 × 63,000 + 60 × 28,000 = 9,45,000 + 16,80,000 = 26,25,000, and 26,25,000 ÷ 75 = 35,000.
Question 11
In a class of 54 students, the boys average 62 marks and the girls average 71 marks. The class average is 66. How many girls are in the class?
Answer. The boys are 4 below the class average and the girls are 5 above it, so boys : girls = 5 : 4. That is 9 parts for 54 students, so 1 part is 6 students, and there are 24 girls (and 30 boys). Check: 30 × 62 + 24 × 71 = 1860 + 1704 = 3564 = 54 × 66.
Question 12
The average weight of the students in a hostel is 52 kg. When 6 new students with an average weight of 61 kg join, the average rises to 53.5 kg. How many students were there before?
Answer. I compare everyone with the new average, 53.5. The old students are 1.5 kg below it and the new ones are 7.5 kg above it. So old : new = 7.5 : 1.5 = 5 : 1, and with 6 new students there were 30 students before. Check: 30 × 52 + 6 × 61 = 1560 + 366 = 1926, and 1926 ÷ 36 = 53.5.
Question 13
A shopkeeper buys 30 kg of sugar at ₹46 per kg and 10 kg at ₹62 per kg. What is his average cost per kg?
Answer. Total cost = 30 × 46 + 10 × 62 = 1380 + 620 = ₹2,000 for 40 kg. Average = 2000 ÷ 40 = ₹50 per kg.
Question 14
A family spends the same amount of money on cooking oil in each of three years. The price was ₹75, ₹100 and ₹150 per litre in those years. What was the average price per litre over the three years?
Answer. Since the money is the same each year, I pick a yearly spend that all three prices divide: ₹300. That buys 4 litres, 3 litres and 2 litres, so 9 litres for ₹900 in all. Average price = 900 ÷ 9 = ₹100 per litre. The plain average, about ₹108.33, is wrong here because more oil was bought in the cheap year.
Question 15
How many kg of dal costing ₹96 per kg must be mixed with 30 kg of dal costing ₹120 per kg so that the mixture costs ₹105 per kg?
Answer. The cheap dal is 105 − 96 = 9 below the target price and the costly dal is 120 − 105 = 15 above it. So cheap : costly = 15 : 9 = 5 : 3. With 30 kg of costly dal, the cheap dal is 30 × = 50 kg. Check: 50 × 96 + 30 × 120 = 4800 + 3600 = 8400, and 8400 ÷ 80 = 105.
Question 16
Meera scored 76% in English (out of 50), 84% in Maths (out of 150) and 70% in Science (out of 100). What is her overall percentage?
Answer. I change each percentage to marks. English: 76% of 50 is 38. Maths: 84% of 150 is 126. Science: 70% of 100 is 70. That is 234 out of 300, and 234 ÷ 300 × 100 = 78%.
Question 17
In class A, 80% of the 30 students passed an exam. Classes A and B together have 90 students, and 60% of them passed. What percentage of class B passed?
Answer. Passed in all: 60% of 90 is 54. Passed in A: 80% of 30 is 24. So 54 − 24 = 30 of the 60 students in B passed, which is 50%. Averaging 80% and x% to get 60% would give 40%, but B is twice as big as A.
Question 18
Mr Rao puts ₹1,20,000 into two schemes, one paying 8% a year and the other 13% a year. His total yearly income from the two is ₹11,100. How much did he put into the 13% scheme?
Answer. His average rate is 11,100 ÷ 1,20,000 × 100 = 9.25%. The 8% scheme is 1.25 below that and the 13% scheme is 3.75 above it. So money at 8% : money at 13% = 3.75 : 1.25 = 3 : 1. The 13% scheme gets one quarter of ₹1,20,000, which is ₹30,000. Check: 8% of 90,000 is 7,200 and 13% of 30,000 is 3,900, and together they make ₹11,100.
Question 19
A company has 36 workers with an average salary of ₹16,000 and 9 managers with an average salary of ₹52,000. (i) Find the average salary of all 45 employees. (ii) If every worker gets a raise of ₹2,500, what is the new average salary?
Answer. (i) Total = 36 × 16,000 + 9 × 52,000 = 5,76,000 + 4,68,000 = ₹10,44,000. Average = 10,44,000 ÷ 45 = ₹23,200. (ii) The raise adds 36 × 2,500 = ₹90,000 to the total, which is 90,000 ÷ 45 = ₹2,000 per head. The new average is ₹25,200. Adding the full ₹2,500 to the average is the trap, since the managers got nothing.
Question 20
In an office, 30 clerks have an average salary of ₹20,000 and 10 officers have an average salary of ₹40,000. The clerks get a 15% raise and the officers a 10% raise. By what percentage does the average salary of the office go up?
Answer. The clerks' total is ₹6,00,000 and the officers' total is ₹4,00,000, so the office pays ₹10,00,000. Raises: 15% of 6,00,000 is 90,000 and 10% of 4,00,000 is 40,000, so the bill goes up by ₹1,30,000. That is 13% of ₹10,00,000, so the average goes up by 13% (from ₹25,000 to ₹28,250). Weighting 15% and 10% by headcount (3 : 1) gives 13.75%, which is wrong: the weights must be the salary totals (3 : 2).
Question 21
[Tier 2] In a class, if every boy's marks were raised by 6, the class average would go up by 4.5. By how much would the class average go up if, instead, every girl's marks were raised by 8?
Answer. Raising each boy by 6 lifts the class average by (share of boys) × 6. So 4.5 = share × 6, which makes the boys of the class. The girls are the other , so raising each girl by 8 lifts the class average by . The answer is 2 marks.
Question 22
[Tier 2] Section P has 23 students with average marks 58, and section Q has 17 students with average marks 66. One student moves from Q to P, and the average of P goes up by 0.25. What is the new average of Q?
Answer. P now has 24 students, so the student who joined brought P's old average plus 0.25 for each of the 24: 58 + 24 × 0.25 = 64 marks. Q had 17 × 66 = 1122 marks. Without this student it has 1122 − 64 = 1058 marks for 16 students, so its new average is 1058 ÷ 16 = 66.125. Both averages went up, because 64 is above P's average and below Q's.