What did Ka-yan score in the first test?

Question

Ka-yan sat a series of maths tests, each out of 100, working out the average so far after every one.

From the 2nd test on, each test raised the average by exactly 2.

In the last test she scored 74, and the average of all the tests was then 58.

Accepted Answer

Answer: 42 marks. Before test k there are k − 1 tests averaging A; after, k tests averaging A + 2. Test k = k(A + 2) − (k − 1)A = A + 2k: 2 × k above the average before it. The average rises 2 each time and the margin above it grows 2 too, so each mark is 4 more than the last: the marks go up 4 at a time and the averages 2 at a time, both starting from the first mark. From the first test to the last, the mark rose by 74 − first and the average by 58 − first; the first is twice the second. So 74 − first = 2 × (58 − first). So first = 2 × 58 − 74 = 42; the average rose 58 − 42 = 16, 2 a time, so there were 8 + 1 = 9 tests. Check: marks 42, 46, 50, 54, 58, 62, 66, 70, 74, total 522, average 58 ✓; averages 42, 44, 46, 48, 50, 52, 54, 56, 58 ✓ Common mistake: thinking each mark is only 2 more. To lift the average by 2, each new mark must be further and further above it: the marks climb 4 at a time.

Averages: olympiad question 10

The question

Worked solution