Average as a total in disguise
Average = sum / count. The useful rearrangement is the other one:
sum = average * count
Almost every average question is solved by turning averages into sums, doing simple addition or subtraction, and turning back.
Two facts worth remembering:
- The average of consecutive numbers is the middle one. For 1 to 99 it is 50. For an even count it is the average of the two middle terms.
- The average of the first
nnatural numbers is(n + 1) / 2.
Deviation method. For numbers clustered around a value, subtract an assumed average, average the small leftovers, and add back. For 102, 106, 99, 103: assume 100, deviations are 2, 6, -1, 3, sum 10, average 2.5, so the answer is 102.5. Much faster than adding four three-digit numbers.
Alligation
Alligation answers one question: in what ratio must two things at different prices be mixed to get a required average price?
The rule, with c = cheaper value, d = dearer value, m = mean value of the mixture:
quantity of cheaper : quantity of dearer = (d - m) : (m - c)
Say it in words: each ingredient's share is proportional to the other one's distance from the mean. The ingredient closer to the mean is present in larger quantity, which is the sense check.
The most common error is writing the ratio the wrong way round — (m - c) : (d - m). Check it against common sense every time. If the mean sits close to the cheap price, most of the mixture must be the cheap thing.
Alligation is not only about price. It works for any weighted average: average marks of two sections, average age of two groups, concentration of two solutions.
Removal and replacement
A vessel holds V litres of pure liquid. You remove x litres and top up with water, and repeat this n times. The liquid remaining is
V * (1 - x/V)^n
The formula works because each round keeps the same fraction, (V-x)/V, of whatever liquid was there.
Worked problem 1
The average age of 30 students is 14 years. When the teacher's age is included, the average becomes 15 years. Find the teacher's age.
Sum of students' ages = 30 * 14 = 420
New count = 31, new average = 15
New sum = 31 * 15 = 465
Teacher's age = 465 - 420 = 45 years
The quick way: adding one person lifted 30 students by 1 year each, which needs 30 extra years, plus the teacher's own place in the new average of
- So 15 + 30 = 45.
Worked problem 2
In what ratio must rice at Rs 40 per kg be mixed with rice at Rs 60 per kg so that the mixture costs Rs 45 per kg?
c = 40, d = 60, m = 45
Cheaper : dearer = (d - m) : (m - c) = (60 - 45) : (45 - 40) = 15 : 5 = 3 : 1
Check with real quantities: 3 kg at 40 costs 120, 1 kg at 60 costs 60, total 180 for 4 kg, and 180 / 4 = 45. Correct.
Sense check: 45 is much nearer 40 than 60, so there should be more of the cheap rice. There is, three times as much.
Worked problem 3
A vessel contains 40 litres of milk. 4 litres are removed and replaced with water. This is done a second time. How much milk is left?
V = 40, x = 4, n = 2
Fraction kept each time = (40 - 4) / 40 = 36/40 = 0.9
Milk left = 40 (0.9)^2 = 40 0.81 = 32.4 litres
Water = 40 - 32.4 = 7.6 litres
Verify the long way. After round one: 36 litres of milk, 4 of water. The mixture is 90% milk, so removing 4 litres of mixture removes 3.6 litres of milk. 36 - 3.6 = 32.4. Correct.
If the second removal in this problem gave you 32 litres, you removed 4 litres of milk instead of 4 litres of mixture. After the first top-up the vessel no longer holds pure milk. This is the trap the question is built around.
Quick revision
- sum = average * count, always.
- Alligation ratio is the cross difference, and the closer ingredient wins.
- Replacement keeps a fixed fraction each round, so use powers.
Next: time, speed and distance.