Average Basics and Averages of Numbers — SSC CGL 2027 Quantitative Aptitude (Maths) Chapter 5 Notes

What an Average Is

An average is one number that stands in for a whole list. Add up all the values and divide by how many there are:

$\text{Average} = \frac{\text{Sum of the values}}{\text{Number of values}}$

Tanvi scored 62, 75, 58, 81 and 69 in five class tests. The sum is 345 and there are 5 tests, so her average is 345 ÷ 5 = 69. If she had scored 69 in every test, her total would have been exactly the same. That is all an average means: the equal share each value would get if the total were spread out evenly.

The one formula that does everything. Rearranging the definition gives

$\text{Sum} = \text{Average} \times \text{Number of values}$

Almost every question in this chapter is solved by turning averages into totals, working with the totals, and turning back at the end. The same line answers three kinds of question:

Find the sum. A family's average monthly electricity bill over a year is ₹1,840, so the bill for the year is 1840 × 12 = ₹22,080.
Find the count. A shop sells 1,575 kg of rice at an average of 45 kg a day, so it took 1575 ÷ 45 = 35 days.
Find the average. 8 friends split a bill of ₹3,720 equally, so each pays 3720 ÷ 8 = ₹465.

Where the average sits. The average always lies between the smallest and the largest value. It can equal one of the values, but it can never be bigger than the largest or smaller than the smallest. If an option breaks this rule, strike it out at once. If all the values are equal, the average is that value.

The balance property. Measure each value from the average. Some are above it and some below, and the gaps always cancel. For Tanvi's marks the gaps from 69 are −7, +6, −11, +12 and 0, which add up to 0. This is why the average is called the balancing point of a list, and it is the idea behind the assumed-average method later in this section.

Units. The average carries the same unit as the values: kg for weights, ₹ for money, °C for temperatures, runs for scores. An average of marks out of 50 is still out of 50.

Reverse questions. Many questions give the average and ask about the total, or give the total of n numbers and then one more number. If the average of 8 numbers is 24.5, their sum is 8 × 24.5 = 196. If a ninth number, 38, is written down, the total becomes 234 and the average of all nine is 234 ÷ 9 = 26.

Key Point: Turn every average into a total (average × count). Totals can be added and subtracted; averages usually cannot.

Consecutive Numbers and Arithmetic Progressions

When numbers go up by the same step every time, you never need to add them. Lists like 17, 21, 25, 29, 33, 37, 41 (step 4) or 8, 9, 10, 11 (step 1) are arithmetic progressions, and for them

$\text{Average} = \frac{\text{first} + \text{last}}{2} = \text{the middle term}$

For 17, 21, …, 41 the average is (17 + 41) ÷ 2 = 29, which is also the 4th (middle) term. With an even number of terms there are two middle terms, and the average is halfway between them: for 8, 9, 10, 11 it is 9.5.

This works because the terms pair up from the outside in (first with last, second with second-last), and every pair has the same sum.

Consecutive numbers. For n consecutive numbers with average A:

List	Smallest	Largest	Largest minus smallest
Consecutive numbers (step 1)	$A - \frac{n - 1}{2}$	$A + \frac{n - 1}{2}$	n − 1
Consecutive odd or even numbers (step 2)	$A - (n - 1)$	$A + (n - 1)$	2(n − 1)

So if 7 consecutive numbers average 46, they run from 43 to 49. If 5 consecutive odd numbers average 57, they are 53, 55, 57, 59 and 61, and the product of the smallest and largest is 53 × 61 = 3233.

An even count of consecutive even numbers has an odd average. 6 consecutive even numbers with average 37 are 32, 34, 36, 38, 40 and 42. Students who expect the average to be one of the numbers get stuck here; with an even count, it falls between the two middle terms.

Smallest + largest. In any arithmetic progression the first and last terms add up to twice the average. If four consecutive even numbers average 33, the smallest and largest add up to 66, and you do not even need to find them (they are 30 and 36).

The next set. If n consecutive numbers average A, the next n consecutive numbers average A + n, because every number moves up by n. The average of 9 consecutive numbers is 52 (they are 48 to 56), so the next 9 (57 to 65) average 61. Put all 18 together and the average is the midpoint, 56.5. For consecutive odd or even numbers the next n average A + 2n.

Removing the middle term. In a progression with an odd number of terms, taking out the middle term leaves the average unchanged, since the middle term is the average. Taking out the first and last terms together does not change it either.

[SSC Tip] "Consecutive" on its own means step 1. Consecutive odd, consecutive even and consecutive multiples of 5 have steps 2, 2 and 5. In every case the largest term is the average plus half of (n − 1) steps: $A + \frac{(n - 1) \times \text{step}}{2}$ .

Averages of Standard Number Sets

A few standard lists come up as one-line questions, so learn their averages directly.

Set	Sum	Average	Quick check
First n natural numbers	$\frac{n(n + 1)}{2}$	$\frac{n + 1}{2}$	first 50: 25.5
First n odd numbers	$n^2$	n	first 25 odd numbers: 25
First n even numbers	$n(n + 1)$	$n + 1$	first 30 even numbers: 31
First n multiples of k	$\frac{kn(n + 1)}{2}$	$\frac{k(n + 1)}{2}$	first 15 multiples of 8: 64
Squares of the first n natural numbers	$\frac{n(n + 1)(2n + 1)}{6}$	$\frac{(n + 1)(2n + 1)}{6}$	n = 19: 130
Cubes of the first n natural numbers	$\left(\frac{n(n + 1)}{2}\right)^2$	$\frac{n(n + 1)^2}{4}$	n = 9: 225

For the squares with n = 19: the sum is 19 × 20 × 39 ÷ 6 = 2470, and 2470 ÷ 19 = 130. For the cubes with n = 9: the sum is 45² = 2025, and 2025 ÷ 9 = 225. Using the average column directly saves the division.

"First n" versus "up to n". These are different lists, and SSC sets options for both.

The first 25 odd numbers are 1, 3, …, 49. Their average is 25.
The odd numbers up to 25 are 1, 3, …, 25. There are only 13 of them, and their average is 13.
The first 45 even numbers end at 90 and average 46; the even numbers up to 45 are 2 to 44, which is 22 numbers averaging 23.

To count the odd numbers from 1 to N (N odd), use (N + 1) ÷ 2; for the even numbers from 2 to N (N even), use N ÷ 2.

The average of the squares is not the square of the average. For 1 to 19 the average is 10, and 10² = 100, but the average of the squares is 130. The same goes for cubes. One option is nearly always the "square of the average", so watch for it.

[Tier 2] More sums.

Set	Average	Check
Squares of the first n even numbers	$\frac{2(n + 1)(2n + 1)}{3}$	n = 7: 4 + 16 + … + 196 = 560, average 80
Squares of the first n odd numbers	$\frac{(2n - 1)(2n + 1)}{3}$	n = 8: 1 + 9 + … + 225 = 680, average 85
Cubes of the first n even numbers	$2n(n + 1)^2$	n = 4: 8 + 64 + 216 + 512 = 800, average 200
Cubes of the first n odd numbers	$n(2n^2 - 1)$	n = 5: 1 + 27 + 125 + 343 + 729 = 1225, average 245

The squares of the first n even numbers are 4 times the squares of 1 to n, so their average is 4 times $\frac{(n + 1)(2n + 1)}{6}$ . If you forget a formula in the hall, build it this way from the basic ones.

[SSC Tip] Test any "first n" formula with n = 3 in your head. The first 3 odd numbers 1, 3, 5 average 3; the squares 1, 4, 9 average $\frac{14}{3}$ , and the formula gives $\frac{4 \times 7}{6} = \frac{14}{3}$ too. If your formula fails for n = 3, it is the wrong one.

Numbers in a Range: Multiples, Remainders and Primes

"Find the average of all multiples of 7 between 100 and 300" uses the progression rule: find the first and last numbers that qualify, and take their average.

Steps.

First qualifying number: the smallest multiple of 7 above 100 is 105 (7 × 15).
Last qualifying number: the largest multiple below 300 is 294 (7 × 42).
Average = (105 + 294) ÷ 2 = 199.5.
If the count is asked: 42 − 15 + 1 = 28 multiples, or (294 − 105) ÷ 7 + 1 = 28.

The trap answer is (100 + 300) ÷ 2 = 200, which uses the limits instead of the actual first and last multiples.

"Between" versus "from". "Between 25 and 100" leaves out 25 and 100; "from 25 to 100" includes them. For multiples of 8 between 25 and 100, the list is 32 to 96 and the average is 64, not the midpoint 62.5 of the limits.

Two-digit and three-digit numbers.

Set	First	Last	Count	Average
All two-digit numbers	10	99	90	54.5
Two-digit odd numbers	11	99	45	55
Two-digit even numbers	10	98	45	54
All three-digit numbers	100	999	900	549.5
Three-digit multiples of 5	100	995	180	547.5

Numbers that leave a remainder. Three-digit numbers that leave remainder 3 when divided by 7 are 101, 108, …, 997, still a progression with step 7. The first is 7 × 14 + 3 = 101 and the last is 7 × 142 + 3 = 997. Average = (101 + 997) ÷ 2 = 549, and the count is 142 − 14 + 1 = 129.

Primes. Primes do not form a progression, so you must list them and add. Know the primes up to 100 cold:

Range	Primes	Count	Sum
1 to 10	2, 3, 5, 7	4	17
11 to 20	11, 13, 17, 19	4	60
21 to 30	23, 29	2	52
31 to 40	31, 37	2	68
41 to 50	41, 43, 47	3	131
51 to 60	53, 59	2	112
61 to 70	61, 67	2	128
71 to 80	71, 73, 79	3	223
81 to 90	83, 89	2	172
91 to 100	97	1	97

The 25 primes below 100 add up to 1060. The first 10 primes (2 to 29) add up to 129, so their average is 12.9. The primes between 50 and 70 are 53, 59, 61 and 67; they add up to 240, so their average is 240 ÷ 4 = 60.

Points that cost marks: 1 is not a prime; 2 is the only even prime, so the first six primes are 2, 3, 5, 7, 11 and 13 (average $\frac{41}{6}$ , about 6.83), not 3, 5, 7, 11, 13 and 17; and 51 (3 × 17), 57 (3 × 19), 87 (3 × 29) and 91 (7 × 13) look prime but are not.

[Tier 2] Multiples of one number but not another. The multiples of 4 up to 60 that are not multiples of 8 are 4, 12, 20, …, 60, a progression with step 8. Their average is (4 + 60) ÷ 2 = 32.

The Assumed-Average Method and the Missing Number

Adding six four-digit numbers by hand is slow and error-prone. The balance property gives a shortcut.

Assumed average. Guess a round number near the middle of the list, write down how far each value is from it, and correct the guess:

$\text{Average} = \text{Assumed average} + \frac{\text{sum of deviations}}{n}$

A factory's output over six days was 1,248, 1,263, 1,239, 1,257, 1,271 and 1,240 units. Take 1,250 as the guess.

Output	1,248	1,263	1,239	1,257	1,271	1,240
Deviation from 1,250	−2	+13	−11	+7	+21	−10

The deviations add up to +18, so the average is 1250 + 18 ÷ 6 = 1,253 units.

Any guess works; a good guess just keeps the deviations small. For 4,537, 4,542, 4,529 and 4,548, take 4,540: the deviations −3, +2, −11 and +8 add up to −4, so the average is 4540 − 1 = 4,539.

The missing number. If the average of n numbers is known and all but one are given, the missing number is the total minus the sum of the rest.

The average of 6 numbers is 47 and five of them are 38, 52, 41, 60 and 45. The total must be 6 × 47 = 282; the five add up to 236, so the sixth is 46.

The deviation route is faster when the numbers sit near the average. From 47 the five deviations are −9, +5, −6, +13 and −2, which total +1. The sixth number has to cancel this, so it is 1 below 47, which is 46.

Two missing numbers. The average of 7 numbers is 54 and five of them add up to 262. The other two total 7 × 54 − 262 = 116. If one is 8 more than the other, they are 54 and 62.

Numbers given in terms of x. The average of x, x + 3, x + 7, x + 10 and x + 15 is 29. Adding: 5x + 35 = 145, so x = 22, and the largest number is 37. A quicker route: the extra parts 0, 3, 7, 10 and 15 average 7, so x + 7 = 29 and x = 22.

When Every Value Changes by the Same Amount

If the same operation is done to every value in a list, the average goes through the same operation.

Every value is …	The new average is	An average of 24 becomes
increased by k	$A + k$	increase by 6: 30
decreased by k	$A - k$	decrease by 6: 18
multiplied by k	$kA$	multiply by 6: 144
divided by k	$\frac{A}{k}$	divide by 6: 4
multiplied by k, then m added	$kA + m$	multiply by 2, add 5: 53

The reason is simple. If each of n values goes up by k, the total goes up by nk, and dividing by n gives a rise of k. The usual mistake is to add nk to the average.

Order matters for mixed operations. "Multiply by 3 and then subtract 8" turns an average of 26 into 3 × 26 − 8 = 70. "Subtract 8 and then multiply by 3" gives 3 × (26 − 8) = 54. Read the order carefully.

Working backwards. Each number in a list is multiplied by 4 and then 7 is added, and the new average is 91. Undo the steps in reverse order: subtract 7 to get 84, then divide by 4 to get 21. The original average was 21.

Percentage changes. If every value rises by 15%, the average rises by 15% too: an average salary of ₹24,000 becomes ₹27,600.

Only some values change. Then work with the total. Of 12 numbers averaging 40, five are each increased by 6. The total goes up by 30, so the average goes up by 30 ÷ 12 = 2.5, to 42.5.

Changes that follow a pattern. The average of 10 numbers is 36. The 1st is increased by 1, the 2nd by 2, and so on, up to the 10th, which is increased by 10. The total goes up by 1 + 2 + … + 10 = 55, so the average rises by 5.5, to 41.5. Put simply, the average rises by the average of the increases, and the increases 1 to 10 average 5.5.

Variants use the standard sums from earlier in this section: increases of 2, 4, 6, … (the first n even numbers, average n + 1) or decreases of 1, 3, 5, … (the first n odd numbers, average n). If 16 numbers average 30 and they are increased by 2, 4, …, 32, the average rises by 17, to 47.

When a question gives only the average and then says "each value is …", you never need the actual values.

Averages Over Days: Temperatures, Sales and Visitors

Day-wise questions look different but run on the same totals.

Monday to Wednesday versus Tuesday to Thursday. Two groups of three days share Tuesday and Wednesday. Subtracting the totals cancels the shared days:

$(M + T + W) - (T + W + Th) = M - Th$

So Monday minus Thursday is 3 × (difference of the two averages).

The average temperature for Monday to Wednesday was 36.5°C and for Tuesday to Thursday 37.2°C. Thursday was 38.4°C. Then Monday − Thursday = 3 × (36.5 − 37.2) = −2.1, so Monday was 38.4 − 2.1 = 36.3°C. A quick check: the later group has the higher average, so Thursday must be warmer than Monday.

The same works for four days: Monday to Thursday against Tuesday to Friday gives Monday − Friday = 4 × (difference of the averages).

When the first and last day are linked. The average for Monday to Thursday was 33°C and for Tuesday to Friday 34.5°C. Friday was 20% hotter than Monday. Here Friday − Monday = 4 × 1.5 = 6, and Friday = 1.2 × Monday, so 0.2 × Monday = 6. Monday was 30°C and Friday 36°C.

A week split into overlapping halves. The average temperature of a week was 31°C, the average of the first four days 30°C and of the last four days 32.5°C. Thursday belongs to both halves:

first four days: 4 × 30 = 120
last four days: 4 × 32.5 = 130
whole week: 7 × 31 = 217

Thursday = 120 + 130 − 217 = 33°C. The part on overlapping groups below does the same for any list of numbers.

Sundays in a month. A library gets 510 visitors on average on Sundays and 240 on other days. In a 30-day month that begins on a Sunday, the Sundays fall on the 1st, 8th, 15th, 22nd and 29th, so there are 5 of them.

$\text{Average per day} = \frac{5 \times 510 + 25 \times 240}{30} = \frac{8550}{30} = 285$

Assuming 4 Sundays gives 276, the usual wrong option. To count: a 30-day month has two weekdays that come 5 times (the day it starts on and the next one); a 31-day month has three.

Average daily sales. A shop's sales for six days were ₹18,400, ₹21,250, ₹19,800, ₹22,600, ₹20,150 and ₹24,400. Take ₹21,000 as the assumed average. The deviations are −2,600, +250, −1,200, +1,600, −850 and +3,400, which add up to +600, so the average is 21,000 + 100 = ₹21,100.

Overlapping Groups: First k and Last k

This type comes up often in recent papers. A list of n numbers is split into a first group and a last group, and one or more numbers belong to both groups (or to neither). The rule never changes:

$\text{shared part} = \text{first-group total} + \text{last-group total} - \text{whole total}$

One shared number (the middle one). The average of 11 numbers is 62. The first 6 average 59 and the last 6 average 66. The 6th number is in both groups:

$6\text{th} = 6 \times 59 + 6 \times 66 - 11 \times 62 = 354 + 396 - 682 = 68$

One number left out (a gap). The average of 13 numbers is 42; the first 6 average 39 and the last 6 average 44. Now the 7th number is in neither group, so the subtraction goes the other way:

7th = 13 × 42 − 6 × 39 − 6 × 44 = 546 − 234 − 264 = 48.

How to tell the two apart: add the group sizes and compare with n. Here 6 + 6 = 12 is more than 11 in the first case, so one number is counted twice; 6 + 6 = 12 is less than 13 in the second, so one number is left out.

Two shared numbers. The average of 12 numbers is 45; the first 7 average 43 and the last 7 average 49. 7 + 7 = 14 is two more than 12, so the 6th and 7th are both counted twice: 6th + 7th = 301 + 343 − 540 = 104. To split them you need one more fact; if the 7th is 6 more than the 6th, they are 49 and 55.

First five, next four and the rest. The average of 15 numbers is 28. The first five average 25 and the next four average 31. The 10th number is 26. Find the average of the last five.

whole: 15 × 28 = 420
first five: 125; next four: 124; the 10th: 26
last five: 420 − 125 − 124 − 26 = 145, so their average is 29

First and last of a short list. The average of the first four of five numbers is 26 and of the last four is 29. The middle three are common to both, so last − first = 4 × (29 − 26) = 12. If the first is 20, the last is 32.

A shared value counted twice in a story. "The average of a batch of 9 parcels is 36 kg; the first five average 33 kg and the last five 41 kg" is the same question in words: the 5th parcel is in both, and it weighs 165 + 205 − 324 = 46 kg.

Three or Four Numbers from Pair Conditions

Some questions describe a few numbers only through averages of pairs, or through "the average of the others".

Averages of every pair. The average of A and B is 41, of B and C is 47, and of C and A is 52. Turn each into a sum: A + B = 82, B + C = 94 and C + A = 104. Adding all three counts each number twice:

$2(A + B + C) = 82 + 94 + 104 = 280 \Rightarrow A + B + C = 140$

Then A = 140 − 94 = 46, B = 140 − 104 = 36 and C = 140 − 82 = 58. The average of all three is $\frac{140}{3}$ , about 46.67.

Only two pairs. If the average of P and Q is 30 and of Q and R is 34, the sums are 60 and 68, and subtracting gives R − P = 8. This is the same cancelling idea as the Monday-Wednesday questions.

Three and three. The average of A, B and C is 44 and of B, C and D is 47. If D is 59, then A − D = 3 × (44 − 47) = −9, so A = 50.

"The other three". Four numbers average 42, so they total 168. The first is 12 more than the average of the other three. If the first is A, the other three total 168 − A, so

$A = \frac{168 - A}{3} + 12 \Rightarrow 3A = 168 - A + 36 \Rightarrow A = 51$

A short form: if one of n numbers is d more than the average of the rest, it is $\frac{n - 1}{n} \times d$ more than the average of all n. Here 42 + $\frac{3}{4}$ × 12 = 51. If it is d less than the average of the rest, subtract instead.

Numbers linked to each other. Of three numbers, the first is twice the second and the second is 4 more than the third; their average is 28. Take the third as t: the second is t + 4 and the first is 2t + 8. Adding, 4t + 12 = 84, so t = 18 and the numbers are 44, 22 and 18.

[Tier 2] Average and product. The average of two numbers is 17 and their product is 280. Their sum is 34, and

$(a - b)^2 = (a + b)^2 - 4ab = 1156 - 1120 = 36$

so the difference is 6 and the numbers are 20 and 14.

Solved Examples

Question 1

The average of 13 consecutive natural numbers is 47. What is the largest of these numbers? (a) 60 (b) 53 (c) 59 (d) 54

Answer. For an odd count of consecutive numbers, the middle one is the average. With 13 numbers the 7th is 47, and there are 6 numbers after it. So the largest is 47 + 6 = 53, and the answer is (b) 53. If you take 47 as the smallest number, you get 47 + 12 = 59.

Question 2

The average of six consecutive even numbers is 49. What is the product of the smallest and the largest of these numbers? (a) 2392 (b) 2365 (c) 2385 (d) 2376

Answer. With an even count, 49 sits between the two middle numbers, 48 and 50. So the six numbers are 44, 46, 48, 50, 52 and 54. Smallest × largest = 44 × 54 = 2376, so the answer is (d) 2376. Quick check: 44 + 54 = 98, which is 2 × 49.

Question 3

The average of 11 consecutive numbers is 67. If the next 5 consecutive numbers are also included, what is the average of all 16 numbers? (a) 69.5 (b) 72 (c) 75 (d) 70

Answer. The 11 numbers run from 67 − 5 = 62 to 67 + 5 = 72. The next five are 73 to 77. Now all 16 numbers form one list from 62 to 77, so the average is (62 + 77) ÷ 2 = 69.5. The answer is (a) 69.5.

Question 4

The average of the squares of the first 17 natural numbers is: (a) 81 (b) 1785 (c) 105 (d) 153

Answer. I use the average formula $\frac{(n + 1)(2n + 1)}{6}$ with n = 17: $\frac{18 \times 35}{6} = 105$ . The answer is (c) 105. Squaring the average 9 gives 81, which is not the same as averaging the squares.

Question 5

The average of the cubes of the first 7 natural numbers is: (a) 784 (b) 112 (c) 98 (d) 64

Answer. The sum of the first 7 cubes is $\left(\frac{7 \times 8}{2}\right)^2 = 28^2 = 784$ , and 784 ÷ 7 = 112. Or straight from $\frac{n(n + 1)^2}{4} = \frac{7 \times 64}{4} = 112$ . The answer is (b) 112.

Question 6

What is the difference between the average of the first 45 odd numbers and the average of all odd numbers from 1 to 45? (a) 22 (b) 0 (c) 23 (d) 21

Answer. The first 45 odd numbers are 1, 3, …, 89, and their average is 45. The odd numbers from 1 to 45 are 1, 3, …, 45; there are 23 of them, and their average is (1 + 45) ÷ 2 = 23. The difference is 45 − 23 = 22, so the answer is (a) 22.

Question 7

What is the average of all multiples of 11 between 200 and 500? (a) 350 (b) 357.5 (c) 346.5 (d) 352

Answer. The first multiple of 11 above 200 is 209 (11 × 19) and the last one below 500 is 495 (11 × 45). They form a progression, so the average is (209 + 495) ÷ 2 = 352. The answer is (d) 352. 350 uses the limits 200 and 500, which are not multiples of 11.

Question 8

What is the average of all prime numbers between 10 and 40? (a) 25 (b) 20 (c) 22.5 (d) 24

Answer. The primes are 11, 13, 17, 19, 23, 29, 31 and 37. That is 8 primes, and they add up to 180. The average is 180 ÷ 8 = 22.5, so the answer is (c) 22.5. Primes do not form a progression, so the midpoint rule does not work here.

Question 9

The average of 3,725, 3,739, 3,718, 3,746, 3,731 and 3,733 is: (a) 3,732 (b) 3,730 (c) 3,734 (d) 3,728

Answer. I take 3,730 as the assumed average. The deviations are −5, +9, −12, +16, +1 and +3, which add up to +12. So the average is 3730 + 12 ÷ 6 = 3,732. The answer is (a) 3,732.

Question 10

The average of 8 numbers is 61. Six of them are 57, 71, 48, 66, 73 and 56, and the remaining two are equal. What is each of the remaining two numbers? (a) 117 (b) 61 (c) 59 (d) 58.5

Answer. The total of all 8 is 8 × 61 = 488. The six given numbers add up to 371. So the two equal numbers together make 488 − 371 = 117, and each is 58.5. The answer is (d) 58.5. 117 is the total of the two, not each one.

Question 11

The average of x, x + 2, x + 5, x + 9 and x + 14 is 30. What is the average of the smallest and the largest of these numbers? (a) 30 (b) 31 (c) 38 (d) 33

Answer. The extra parts 0, 2, 5, 9 and 14 add up to 30, so the five numbers add up to 5x + 30 = 150. That gives x = 24, and the numbers are 24, 26, 29, 33 and 38. The average of the smallest and largest is (24 + 38) ÷ 2 = 31, so the answer is (b) 31. These numbers are not evenly spaced, so the average of the two ends is not the overall average 30.

Question 12

The average of 15 numbers is 26. Each number is multiplied by 3 and then 8 is subtracted from it. What is the new average? (a) 54 (b) 78 (c) 70 (d) 62

Answer. Whatever is done to every number is done to the average too. So the new average is 3 × 26 − 8 = 78 − 8 = 70. The answer is (c) 70. Subtracting 8 first and then multiplying gives 3 × 18 = 54, which reverses the order.

Question 13

The average of 20 numbers is 42. The first number is increased by 2, the second by 4, the third by 6, and so on, so that the 20th number is increased by 40. What is the new average? (a) 63 (b) 62 (c) 82 (d) 52.5

Answer. The increases are the first 20 even numbers, and their total is 20 × 21 = 420. Spread over 20 numbers, that raises the average by 21. So the new average is 42 + 21 = 63. The answer is (a) 63.

Question 14

The average temperature of Monday, Tuesday and Wednesday was 31.4°C, and that of Tuesday, Wednesday and Thursday was 32.6°C. If the temperature on Thursday was 36°C, what was the temperature on Monday? (a) 34.8°C (b) 39.6°C (c) 33.6°C (d) 32.4°C

Answer. Tuesday and Wednesday are in both groups, so subtracting the totals leaves Monday − Thursday = 3 × (31.4 − 32.6) = −3.6. So Monday = 36 − 3.6 = 32.4°C. The answer is (d) 32.4°C. Since the second group is warmer, Monday must be cooler than Thursday.

Question 15

The average daily sales of a shop for a week (Monday to Sunday) were ₹14,600. The average for Monday to Thursday was ₹13,900 and for Thursday to Sunday ₹15,500. What were the sales on Thursday? (a) ₹14,700 (b) ₹15,400 (c) ₹14,600 (d) ₹16,200

Answer. Thursday is counted in both groups of four days. Monday to Thursday total 4 × 13,900 = ₹55,600, Thursday to Sunday total 4 × 15,500 = ₹62,000, and the whole week totals 7 × 14,600 = ₹1,02,200. So Thursday = 55,600 + 62,000 − 1,02,200 = ₹15,400. The answer is (b) ₹15,400.

Question 16

A month of 31 days begins on a Saturday. A museum has an average of 528 visitors on Sundays and 280 on other days. What is the average number of visitors per day in that month? (a) 312 (b) 404 (c) 320 (d) 316

Answer. Day 1 is a Saturday, so the Sundays are the 2nd, 9th, 16th, 23rd and 30th. That is 5 Sundays and 26 other days. Total visitors = 5 × 528 + 26 × 280 = 2640 + 7280 = 9920, and 9920 ÷ 31 = 320. The answer is (c) 320. Taking 4 Sundays gives 9672 ÷ 31 = 312.

Question 17

The average of 11 numbers is 37. The average of the first 6 numbers is 34.5 and that of the last 6 numbers is 41. What is the 6th number? (a) 46 (b) 37 (c) 41 (d) 52

Answer. The 6th number is in both groups. First six total 6 × 34.5 = 207, last six total 6 × 41 = 246, and all eleven total 11 × 37 = 407. So the 6th number = 207 + 246 − 407 = 46. The answer is (a) 46.

Question 18

The average of 25 numbers is 55. The average of the first 12 is 52 and that of the last 12 is 57. What is the 13th number? (a) 55 (b) 54.5 (c) 61 (d) 67

Answer. 12 + 12 = 24, one less than 25, so the 13th number is in neither group. All 25 total 25 × 55 = 1375, the first 12 total 624 and the last 12 total 684. So the 13th = 1375 − 624 − 684 = 67. The answer is (d) 67.

Question 19

The average of 14 numbers is 64. The average of the first 5 is 60 and that of the next 4 is 66.5. If the 10th number is 70, what is the average of the last 4 numbers? (a) 66 (b) 65 (c) 82.5 (d) 52

Answer. All 14 total 14 × 64 = 896. The first 5 total 300, the next 4 total 266, and the 10th is 70. That leaves 896 − 300 − 266 − 70 = 260 for the last 4, so their average is 65. The answer is (b) 65.

Question 20

The average of A and B is 47, the average of B and C is 53, and the average of A and C is 56. What is the value of C? (a) 50 (b) 44 (c) 62 (d) 52

Answer. As sums: A + B = 94, B + C = 106 and A + C = 112. Adding all three gives 2(A + B + C) = 312, so A + B + C = 156. Then C = 156 − 94 = 62. The answer is (c) 62.

Question 21

The average of five numbers is 32. The first number is 15 more than the average of the other four. What is the first number? (a) 44 (b) 47 (c) 29 (d) 45

Answer. All five total 160. If the first is x, the other four total 160 − x, so $x = \frac{160 - x}{4} + 15$ . Multiplying by 4: 4x = 160 − x + 60, so 5x = 220 and x = 44. The answer is (a) 44. Check: the other four total 116 and average 29, and 44 − 29 = 15.