The mean score of seven students in a maths test was 30 marks. After an eighth student took the test, the mean score of all eight students was 31 marks. Work out the mark that the eighth student achieved on the test.
To work out the mean of a set of data, we find the sum of all the data values — in this case, that’s the sum of all marks — and divide it by how many data values there are — in this case, that’s the number of students. We know that the mean score for the first seven students was 30 marks. So using our formula for the mean, we can deduce that the sum of the first seven marks divided by seven must be equal to 30.
We can work out the sum of the first seven marks by multiplying both sides of this equation by seven. On the left, this will cancel with the factor of seven in the denominator. And on the right, we have 30 multiplied by seven, which is 210. So we know that the sum of the first seven marks was 210.
We also know that the mean score for all eight students is 31 marks. So the sum of all eight marks over eight is equal to 31. By multiplying both sides of this equation by eight, we can work out that the sum of all eight marks is equal to 248. The sum of all eight marks will just be the sum of the first seven marks plus the mark achieved by the eighth student, which we don’t know, so we’ll call it 𝑥.
We do know that the sum of the first seven marks was 210. So we have the equation 210 plus 𝑥 equals 248. To find the value of 𝑥, we need to subtract 210 from each side of this equation, giving 𝑥 equals 38. So we found that the eighth student achieved 38 marks on the maths test.