There is only one formula in this whole chapter: total = average × how many. Every one of the 34 questions the Punjab Police SI has asked on averages and mixtures since 2016 is that line rearranged, and the fastest candidates never think in averages at all — they convert to totals, add or subtract, and convert back at the end. The chapter splits cleanly into four moves. Totals: find a missing value by subtracting two totals. Replacement: when one member is swapped, the total moves by n × the change in average. Weighting: groups of different sizes can never have their averages averaged. Alligation: the cross that turns two concentrations and a target into a mixing ratio.
34
SI questions
2016–2023
27
on averages
simple and weighted
7
on mixtures
alligation and replacement
1
formula to learn
total = average × n
Work in totals, not averages
The one habit that halves your time.
KEY TERM
total = average × how many
Read any average question twice and write down the two totals it describes. The answer is almost always their difference. “The average of 12 observations is 18; adding a 13th makes the average 20” is really 12 × 18 = 216 and 13 × 20 = 260, so the new observation is 44 — no equation, two multiplications and a subtraction.
average × n→total→difference of totals
QUICK CHECK
The mean of 12 observations is 18. If one more observation is included, the new mean becomes 20. Find out the 13th observation. (SI 16 Oct 2022)
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Answer: 44
Old total = 12 × 18 = 216. New total = 13 × 20 = 260. The 13th observation is 260 − 216 = 44.
A second way to see it: the new value must supply 18 for itself plus 2 extra for each of the 13 slots, so 18 + 26 = 44.
QUICK CHECK
The average weight of a class of 25 students is 42 kg. When the teacher's weight is also included, the average weight increases by 2 kg. What is the teacher's weight? (SI 15 Jul 2023)
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Answer: 94
Students' total = 25 × 42 = 1050. With the teacher there are 26 people and the average is 42 + 2 = 44, so the new total is 26 × 44 = 1144. The teacher weighs 1144 − 1050 = 94 kg.
Faster still: the teacher must carry the new average 44 for himself plus the 2 kg rise for each of the 25 students — 44 + 50 = 94.
QUICK CHECK
If the arithmetic mean of the following data is 39, then what will be the value of p? 16, 7, 31, 25, 29, 11, 51, p, 63 (SI 2 Jul 2023)
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Answer: 118
Nine values with mean 39 means the total must be 9 × 39 = 351. The eight known values add to 16 + 7 + 31 + 25 + 29 + 11 + 51 + 63 = 233. So p = 351 − 233 = 118.
QUICK CHECK
The weights (in kg) of 9 members of a school cricket team are 50, 63, x, 62, 64, 48, 51, 55, 48. If the mean weight is 56 kg, then the value of x is _________. (SI 29 Jul 2023)
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Answer: 63
Total needed = 9 × 56 = 504. The eight known weights add to 50 + 63 + 62 + 64 + 48 + 51 + 55 + 48 = 441. So x = 504 − 441 = 63.
QUICK CHECK
The average of the monthly rainfalls for January, February, March and April was 22 cm and for February, March, April and May it was 24 cm. If the monthly rainfall in May was 20 cm, find the monthly rainfall in January. (SI 16 Oct 2022)
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Answer: 12 cm
Jan + Feb + Mar + Apr = 4 × 22 = 88 and Feb + Mar + Apr + May = 4 × 24 = 96. Since May = 20, the middle three months add to 96 − 20 = 76. Therefore January = 88 − 76 = 12 cm.
Two overlapping windows always work this way: subtract to strip out the months they share.
QUICK CHECK
3 years ago the average age of a husband, his wife and their child was 30 years, and 5 years ago the average age of the wife and the child was 20 years. Find the present age of the husband. (SI 24 Aug 2021)
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Answer: 49 years
Three years ago the three of them averaged 30, so their total then was 90. Each has aged 3 years since, so the present total is 90 + 9 = 99.
Five years ago the wife and child averaged 20, so their total then was 40. Each has aged 5 years, so their present total is 40 + 10 = 50. The husband is 99 − 50 = 49 years.
Bring every total forward to today before subtracting — mixing “3 years ago” with “5 years ago” directly is the commonest error here.
QUICK CHECK
Lauren and her spouse had an average age of 27 years when they got married. Now, after five years of marriage, they have a one-year old child. Calculate the current average age (in years) of the family. (SI 18 Aug 2021)
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Answer: 65/3
At marriage the couple's total was 2 × 27 = 54. Five years later each is 5 years older, so their total is 54 + 10 = 64. Add the one-year-old child: total 65 across 3 people, so the average is 65/3 years.
The child is not five years older — he did not exist five years ago. Only the people present at the earlier moment gain the years.
QUICK CHECK
Dinesh, Champa and Bhavna donated money for a flood relief camp in such a way that the average of Dinesh and Champa is ₹2,000, the average of Champa and Bhavna is ₹3,000 and the average of Bhavna and Dinesh is ₹4,000. Find the average share of the three donors. (SI 24 Aug 2021)
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Answer: ₹3,000
Each pair gives a total: D + C = 4000, C + B = 6000 and B + D = 8000. Adding all three, every person is counted twice, so 2(D + C + B) = 18,000 and the three together donated ₹9,000. The average share is 9000 ÷ 3 = ₹3,000.
You never need the individual amounts — adding the pairwise totals is the whole trick.
QUICK CHECK
If the arithmetic mean of x, y and 9 is 12, what is the arithmetic mean of x + 2, y − 6 and 10? (SI 14 Jul 2023)
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Answer: 11
x + y + 9 = 3 × 12 = 36, so x + y = 27. The new three add to (x + 2) + (y − 6) + 10 = (x + y) + 6 = 33, and 33 ÷ 3 = 11.
Shortcut worth keeping: the mean changes by the average of the changes. Here the three changes are +2, −6 and +1 (9 became 10), which total −3, so the mean falls by 1 from 12 to 11.
QUICK CHECK
If the mean of four observations, x + 1, x + 2, x + 3 and x + 4 is 68.5, find the mean of the first two observations. (SI 31 Jul 2023)
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Answer: 67.5
The four observations add to 4 × 68.5 = 274, and they also add to 4x + 10. So 4x = 264 and x = 66. The first two observations are 67 and 68, whose mean is 67.5.
For consecutive values you can skip the algebra: the mean of four consecutive numbers sits halfway between the 2nd and 3rd, and the mean of the first two is one step lower — 68.5 − 1.
QUICK CHECK
The average of all odd natural numbers upto and including 49 is ________. (SI 24 Aug 2021)
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Answer: 25
The odd numbers 1, 3, 5, …, 49 form an arithmetic progression, and the average of any AP is the midpoint of its first and last terms: (1 + 49)/2 = 25.
You can check it: there are 25 such numbers and their sum is 25² = 625, so the average is 625 ÷ 25 = 25. The sum of the first n odd numbers is always n².
CHECK YOURSELF
A student obtained 70%, 72%, 55%, 65% and 75% marks in five subjects in an examination. Find the mean percentage of the marks obtained by him. (SI 29 Jun 2023)
Show answer
Add the five percentages: 70 + 72 + 55 + 65 + 75 = 337, and 337 ÷ 5 = 67.4%. Percentages can be averaged directly only because all five subjects carry the same weight.
Replacement: one swap, one multiplication
Seven questions, all the same shape.
THE REPLACEMENT RULE
New value = old value + (number of members × change in average). If the average rises the new member is heavier; if it falls, lighter. When several members are replaced at once, the same rule applies to their total: new total = old total + n × change in average.
QUICK CHECK
The average weight of 20 sailors in a boat increases by 2.3 kg when one of the crew, whose weight is 52 kg, is replaced by a new woman. Find the weight of the new woman. (SI 29 Jun 2023)
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Answer: 98 kg
Twenty sailors and the average rises 2.3 kg, so the total rises by 20 × 2.3 = 46 kg. All of that rise is supplied by the swap, so the new woman weighs 52 + 46 = 98 kg.
QUICK CHECK
The average weight of 8 persons increases by 2.5 kg when one of them, whose weight is 56 kg, is replaced by a new man. Find the weight of the new man. (SI 2 Jul 2023)
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Answer: 76 kg
8 persons × 2.5 kg = 20 kg added to the total. The new man weighs 56 + 20 = 76 kg. Option D offers 76 g — read the unit before you tick.
QUICK CHECK
The average weight of 10 persons increases by 2.5 kg when a new person comes in place of a person weighing 70 kg. What will be the weight (in kg) of the new person? (SI 20 Jun 2023)
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Answer: 95
10 persons × 2.5 kg = 25 kg. The new person weighs 70 + 25 = 95 kg. Note the trap option 47.5 — that is 70/2 + 12.5, a number the question never asks for.
QUICK CHECK
The mean age of 12 men gets increased by 8 years when three of them, whose ages are 27, 28 and 29 years, are replaced by three women. Find the mean age of the three women. (SI 9 Jul 2023)
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Answer: 60 years
Twelve men and the mean rises 8 years, so the total rises by 12 × 8 = 96 years. The three who left total 27 + 28 + 29 = 84 years, so the three women total 84 + 96 = 180 and their mean is 180 ÷ 3 = 60 years.
The rule works on totals even when more than one member is swapped — just divide at the end.
QUICK CHECK
A man bought 10 silver articles for ₹20,000. If the average cost of 6 silver articles is ₹1,640, then what will be the average cost of the remaining silver articles? (SI 29 Jun 2023)
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Answer: ₹2,540
All ten articles cost ₹20,000. Six of them cost 6 × 1,640 = ₹9,840, so the remaining four cost 20,000 − 9,840 = ₹10,160. Their average is 10,160 ÷ 4 = ₹2,540.
Count the “remaining” group carefully — here it is four, not six.
QUICK CHECK
The average cost of all the mobiles in a shop is ₹50,000. The average cost of 5 brand-X mobiles is ₹70,000 and the average cost of the rest is ₹40,000. What is the total number of mobiles in the shop? (SI 24 Aug 2021)
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Answer: 15
Let there be n mobiles. Their total is 50,000n. Five brand-X mobiles cost 5 × 70,000 = ₹3,50,000 and the other n − 5 cost 40,000(n − 5). So 50,000n = 3,50,000 + 40,000n − 2,00,000, giving 10,000n = 1,50,000 and n = 15.
Alligation gets there faster: the five dear phones are ₹20,000 above the mean and the rest are ₹10,000 below, so dear : rest = 10,000 : 20,000 = 1 : 2, making the rest 10 and the total 15.
Weighted averages
Groups of unequal size never average their averages.
QUICK CHECK
A class of 25 students took a mathematics test. 15 students had an average score of 60. The remaining students had an average score of 80. What was the average score of the whole class? (SI 29 Jul 2023)
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Answer: 68
15 students × 60 = 900 and the other 10 × 80 = 800. Together 1700 marks over 25 students gives 68.
Averaging 60 and 80 to get 70 is the trap: the larger group pulls the answer towards its own average, so the result must sit below the midpoint.
QUICK CHECK
In a class of 32, 12 students averaged a score of 62, 10 students had an average of 58, 7 students had an average of 78 and 3 students had an average test score of 82. Find the average test score for the entire class. (SI 9 Jul 2023)
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Answer: 66.125
Multiply each group by its size: 12 × 62 = 744, 10 × 58 = 580, 7 × 78 = 546 and 3 × 82 = 246. The total is 2116 marks over 32 students, which is 2116 ÷ 32 = 66.125.
Check the group sizes add to the class size before you divide — 12 + 10 + 7 + 3 = 32 ✓.
QUICK CHECK
A trader sells three varieties of wheat A, B and C. He sells 100 kg of variety A at ₹79/kg, 80 kg of variety B at ₹70/kg and 120 kg of variety C at ₹60/kg respectively. What will be the average price (in ₹) per kg of the wheat sold by the trader? (SI 14 Jul 2023)
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Answer: 69
100 × 79 = 7,900; 80 × 70 = 5,600; 120 × 60 = 7,200. Total value ₹20,700 over 300 kg gives ₹69 per kg.
The simple average of 79, 70 and 60 would be 69.67 — close enough to look right, which is exactly why the paper offers it among the options.
QUICK CHECK
Pinky bought 15 books at the rate of ₹12 each, 40 pens at the rate of ₹5 each and 12 pencils at the rate of ₹2 each. Calculate the average price of all the stationery. (SI 14 Jul 2023)
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Answer: ₹6.03
Total spent = 15 × 12 + 40 × 5 + 12 × 2 = 180 + 200 + 24 = ₹404. Total items = 15 + 40 + 12 = 67. The average price is 404 ÷ 67 = ₹6.03.
“Average price of all the stationery” means per item, not per type. Averaging 12, 5 and 2 would give ₹6.33.
QUICK CHECK
The speed of 4 different cars in a car race is given as 200 km/h, 210 km/h, 215 km/h and 225 km/h. Their weighted average speed with weights 1, 2, 3 and 4 is : (SI 29 Jul 2023)
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Answer: 216.5 km/h
Multiply each speed by its weight and divide by the sum of the weights: (200 × 1 + 210 × 2 + 215 × 3 + 225 × 4) ÷ (1 + 2 + 3 + 4) = (200 + 420 + 645 + 900) ÷ 10 = 2165 ÷ 10 = 216.5 km/h.
A weighted average is just a group average where the weights play the part of group sizes.
QUICK CHECK
A bike owner bought petrol at ₹4, ₹5 and ₹6 per litre for three consecutive years. If he spends ₹1,000 every year, find the average price of petrol per litre. (SI 15 Jul 2023)
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Answer: ₹4.86
He spends the same ₹1,000 each year, not the same quantity, so the litres differ: 1000/4 = 250, 1000/5 = 200 and 1000/6 = 166.67 litres. Total spent ₹3,000 for 616.67 litres gives 3000 ÷ 616.67 = ₹4.86 per litre.
This is the harmonic mean. Equal money each year means the cheap petrol buys more litres, so the answer sits below the simple average of 5.
QUICK CHECK
In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs ? (SI 2016)
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Answer: 6.25
Runs so far = 10 × 3.2 = 32. Runs still needed = 282 − 32 = 250, in 40 overs. The required rate is 250 ÷ 40 = 6.25.
Run-rate questions are weighted averages in disguise: overs are the weights.
CHECK YOURSELF
In the first 10 overs of a cricket match, the run rate was only 3 runs per over. What should be the run rate per over in the remaining 40 overs to reach the target of 310 runs? (SI 2 Jul 2023)
Show answer
Same shape, different numbers: 10 × 3 = 30 runs scored, so 310 − 30 = 280 are needed in 40 overs. The required rate is 280 ÷ 40 = 7.
CHECK YOURSELF
The average score of a cricket batsman in three matches is 42 and in four other matches is 56. Find the average score in all seven matches. (SI 29 Jul 2023)
Show answer
3 × 42 = 126 runs and 4 × 56 = 224 runs, a total of 350 over seven matches. 350 ÷ 7 = 50.
Mixtures and alligation
Seven questions. Count one component, never both.
KEY TERM
Track one component through the whole question
Mixture questions go wrong when you try to follow milk and water at the same time. Pick one — usually the one the answer is about — write down how many litres of it there are at each stage, and get the other by subtracting from the total. When a mixture is removed, it carries away both components in the original ratio; when a pure substance is added, only one quantity changes.
remove mixture → both fall→add pure → one rises→total changes only if volume changes
QUICK CHECK
In a 50-litre mixture of milk and water, there is 16% water. How much water should be added to it so as to increase the percentage of water to 40%? (SI 31 Jul 2023)
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Answer: 20 litres
16% of 50 litres is 8 litres of water, so there are 42 litres of milk. Adding w litres of water does not change the milk, so (8 + w)/(50 + w) = 0.4. Then 8 + w = 20 + 0.4w, giving 0.6w = 12 and w = 20 litres.
Alligation is faster: water goes from 16% to 40% by adding 100% water, so existing : added = (100 − 40) : (40 − 16) = 60 : 24 = 5 : 2. The existing 50 litres is 5 parts, so 1 part is 10 and 2 parts is 20.
QUICK CHECK
In a 13-litre mixture, the ratio of milk and water is 3 : 2. If 3 litres of this mixture is replaced with 3 litres of milk, then what will be the ratio of milk and water in the newly formed mixture? (SI 24 Aug 2021)
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Answer: 9 : 4
In 13 litres with milk : water = 3 : 2 there are 7.8 litres of milk and 5.2 of water. Removing 3 litres of mixture takes away 3 × 3/5 = 1.8 litres of milk and 3 × 2/5 = 1.2 of water, leaving 6 and 4. Adding 3 litres of pure milk makes it 9 and 4, so the new ratio is 9 : 4.
The total stays at 13 litres because what was removed was replaced litre for litre.
QUICK CHECK
A wine barrel contains 24 litres of a mixture of wine and water in the ratio 3 : 1. To make the quantities of wine and water equal, 'x' litres of the mixture is taken out and 11 litres of water is added. What will be the value of 'x' (in litres)? (SI 15 Jul 2023)
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Answer: 2
24 litres in the ratio 3 : 1 is 18 litres of wine and 6 of water. Taking out x litres of mixture removes 3x/4 wine and x/4 water. Then 11 litres of water goes in.
Wine = 18 − 3x/4 and water = 6 − x/4 + 11 = 17 − x/4. Setting them equal, 18 − 3x/4 = 17 − x/4, so 1 = x/2 and x = 2 litres.
Check: wine 16.5, water 5.5 + 11 = 16.5 ✓.
QUICK CHECK
The volumes of two bottles are equal. Faruk filled the first bottle half with orange juice and filled four-fifths (4/5) of the second bottle with it. He then fills the bottles up to the brim with sharbat (syrup). He pours the contents into a jug. What will be the ratio of juice to syrup in the jug? (SI 19 Aug 2021)
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Answer: 13 : 7
Take each bottle as 1 unit. The first holds 1/2 juice and 1/2 syrup; the second holds 4/5 juice and 1/5 syrup. Poured together, juice = 1/2 + 4/5 = 13/10 and syrup = 1/2 + 1/5 = 7/10. The ratio is 13 : 7.
Equal volumes mean no weighting is needed — just add the fractions.
QUICK CHECK
Two buckets are filled with milk and water in the ratio of 3 : 1 and 5 : 2. If these mixtures are mixed in the ratio of 1 : 5, the resulting mixture will have water and milk in the ratio of _____. (SI 9 Jul 2023)
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Answer: 47 : 121
Work in water fractions. Bucket 1 is milk : water = 3 : 1, so water is 1/4. Bucket 2 is 5 : 2, so water is 2/7. Mixing them in 1 : 5 gives water = (1 × 1/4 + 5 × 2/7) ÷ 6 = (7/28 + 40/28) ÷ 6 = (47/28) ÷ 6 = 47/168.
Milk is the rest, 121/168, so water : milk = 47 : 121. Note the question asks for water first — the reversed ratio 121 : 47 is not offered, but 47 : 120 is, as a near-miss.
QUICK CHECK
The capacities of three vessels are in the ratio 10 : 12 : 15. These vessels are filled with a mixture of milk and water in such a way that water and milk are in the ratios 1 : 4, 3 : 5 and 5 : 7 respectively in the three vessels. If the contents of these three vessels are poured into a 4th vessel, find the ratio of milk to water in the 4th vessel. (SI 19 Aug 2021)
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Answer: 97 : 51
Take the capacities as 10, 12 and 15 litres.
Vessel 1, water : milk = 1 : 4 → water 10 × 1/5 = 2, milk 8.
Vessel 2, 3 : 5 → water 12 × 3/8 = 4.5, milk 7.5.
Vessel 3, 5 : 7 → water 15 × 5/12 = 6.25, milk 8.75.
Totals: milk 24.25 and water 12.75. Multiplying both by 4 gives 97 and 51, so milk : water = 97 : 51.
Read the stated order carefully: the vessels give water to milk, but the answer wants milk to water.
CHECK YOURSELF
Two alloys P and Q are made of copper and zinc in the ratio of 5 : 2 and 3 : 8, respectively. 14 kg of P, 33 kg of Q and some quantity of pure zinc are mixed and melted together to form a new alloy R. The ratio of copper to zinc in R is 3 : 8. What is the weight (in kg) of the new alloy R? (SI 29 Jun 2023)
Show answer
Alloy P is copper : zinc = 5 : 2, so 14 kg gives 10 copper and 4 zinc. Alloy Q is 3 : 8, so 33 kg gives 9 copper and 24 zinc. Pure zinc adds nothing to the copper, so copper stays at 10 + 9 = 19 kg. For R the ratio is 3 : 8, so if copper is 19 the zinc must be 19 × 8/3 = 152/3 kg. Total R = copper + zinc = 19 + 152/3 = 209/3 = 69 2/3 kg. You can also check the added zinc: 152/3 − 28 = 68/3, and 14 + 33 + 68/3 = 209/3 ✓.
COMMON TRAPS
- Averaging the averages when the groups are different sizes. 15 students at 60 and 10 at 80 give 68, not 70.
- Equal money is not equal quantity. Spending ₹1,000 a year at ₹4, ₹5 and ₹6 gives an average of ₹4.86, not ₹5.
- In an age question, only the people who existed earlier gain the years. A newborn does not.
- Removing a mixture takes away both components in the old ratio; adding a pure substance changes only one. Treating the two the same way is the classic mixture error.
- Answering in the wrong order. If the stem says “water and milk”, give water first — the reversed ratio is always among the options.
QUICK BYTES
- total = average × how many
- New value = old + n × change in average
- Weighted average = Σ(size × average) ÷ Σsize
- Average of an AP = (first + last) ÷ 2
- Sum of first n odd numbers = n²
- Alligation: cheaper : dearer = (dearer − mean) : (mean − cheaper)
- Equal spend, different prices → harmonic mean
- Mean changes by the average of the changes
REMEMBER
Totals in, totals out. The moment you read an average, multiply it by the count and write the total down. Do all the adding, subtracting and swapping on totals, and divide only once, at the very end. Candidates who work in averages throughout are the ones who average the averages by mistake.