My test results

Let the sum of marks secured in all the subjects in $15$ tests be represented by: $\displaystyle\sum_{i=1}^{15}m_i$

So, $\dfrac{\displaystyle\sum_{i=1}^{15}m_i}{15}=7.2 \\ \displaystyle\sum_{i=1}^{15}m_i =7.2×15 \\=\boxed{108}$

So, the average marks in $15$ tests is: $\dfrac{\displaystyle\sum_{i=1}^{15}m_i}{15} \\=\boxed{\dfrac{108}{15}}$

Now according to the question $\dfrac{108+10x}{15+x}=9 \\ 108+10x=135+9x \\ 10x-9x=135-108 \\ \boxed{x=27}$

First, we need to know what is their total score in the 15 tests they had taken that led them to obtain the average of 7.2. Recall that an average can be calculated by using the formula, $\bar{x} = \frac{\displaystyle\sum _{i=1}^{n}{x_i}}{n},$ where $\bar{x}$ is the average, $x_i$ is the score of each test, and $n$ is the number of tests taken. In the problem, the score of each test taken is not stated. We, however, only need to know the sum of the test scores that led to the average of 7.2. Substituting the needed values, we obtain the sum of the test scores, which is 108.

The problem requires us to find the number of tests should they take in order for their average to be 9, given that they score 10 in each test. Knowing the sum of their test scores beforehand and the number of tests they had taken, we obtain the equation, $\frac{108+10n}{15+n} = 9.$ Using some algebraic manipulation, we find that the value of n is 27. Therefore, they need to take 27 more tests with scores of 10 in order to reach the average of 9. $\Box$

X/15=7.2, (10y+x)/(y+15)=9 Where x=total score before, and y=number of additional tests. I assume that all additional tests y , he must score perfect score of 10.

Easy way to solve using ratios

(because the first 15 tests have an average of 7.2, we can assume that they all are 7.2 due to the fact that it doesn't affect the answer)

If you have an average of 7.2 and want to get an average of 9 by continuous 10. you just have to find the ratio of the 15 tests with 7.2 averages to the new 10 averages that would give you an average of 9.

what ratios of each will give you 9?

lets see, the difference between 7.2 and 9 is 1.8, and the difference between 9 and 10 is 1

this means that the ratio between the amount of tests with a 7.2 average and tests with an average of 10 is 1:1.8.

From this we determine that there are "1.8/1" 10 average tests in terms of 7.2 average tests

1.8 x 15=27

you would need 27 more perfect tests

Let x= no. of extra tests to take, scoring 10 points each.

Then to get an average of 9, Total score / total n of tests = 9

(15x7.2)+10x / 15+x = 9

108+10x / 15+x = 9

108+10x = 9(15+x)

108+10x = 135+9x

10x-9x = 135-108

X=27