Average: total, number of data, and what happens when the data change - Ganily

The average is the total shared out equally: average = total ÷ number of data. Turn that round and you get the two versions the exam actually uses — total = average × number, and number = total ÷ average. Almost every mark in this topic is won by writing the total down first, because the average on its own cannot be added, subtracted or combined; only totals can. That single habit answers all four hard forms: a missing value (build the total, take away what you know), a value added (old total plus the new one, then share out again), a value removed (the gap between two totals IS the value that went), and two groups combined (add the totals, add the counts, then divide). Watch the trap in that last one: the answer is almost never the average of the two averages.

Worked example

Problem The average mass of 5 boys is 42 kg. A sixth boy joins them, and his mass is 48 kg. Find the new average mass of the boys.

The average is no use on its own, so turn it into a total: 42 × 5 = 210, so the 5 boys weigh 210 kg altogether.

The new boy adds his mass to that total: 210 + 48 = 258 kg.

There are now 6 boys, so share the new total out again: 258 ÷ 6 = 43 kg.

Watch out The new average is not halfway between 42 and 48. Halfway would only be right if there had been one boy to start with — with 5 already there, one new mass barely moves the average.

Tip Total first, every single time. If a question mentions two averages, you will need two totals.

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