French test

Using the formula for the mean

The mean for the first 4 students was found by adding the 4 different scores and then dividing by 4. So the sum of the scores was 4$\times$4 = 16

Similarly, the sum of the last 6 scores was 6$\times$6 = 36

So the total score for the whole group was 16 + 26 = 52, and so the average score for the group was 52$\div$10 = 5.2

Using ratio

The sizes of the groups are in the ratio 4 : 6, or 2 : 3. This means that the impact on the average of the two groups' scores is also in the ratio 4 : 6, or 2 : 3.

So the average score will be closer to the 'last 6' average score than the 'first 4' average score, in the ratio 2 : 3.

The difference between the 'first 4' average of 4 and the 'last 6' average of 6 is 2, so 2 needs to be split in the ratio 2 : 3.

2 : 3 is equivalent to 8 : 12 and to 0.8 : 1.2. 0.8 + 1.2 = 2, so the number that is 0.8 from 6 and 1.2 from 4 is the average for the whole group, which is 5.2