### Video Transcript

A student takes five tests ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ. Their mean score for tests ๐ด, ๐ต, and ๐ถ is 48. Their mean score for tests ๐ท and ๐ธ is 38. Calculate the mean score of all five tests.

To answer this question, we can recall that the mean of a set of numbers is the sum of all the numbers divided by how many numbers there are. Weโre told that a student takes five tests, ๐ด through ๐ธ, and that the mean of ๐ด, ๐ต, and ๐ถ is 48 and the mean of ๐ท and ๐ธ is 38. What we need to do here then is worked out the mean of all five. Letโs start by looking at the first three tests where the mean is 48. Using the formula then, we know that the mean is 48. We donโt know the sum of the numbers, but we do know that there are three numbers, since thereโre three tests. To rearrange this then, we can multiply both sides of our equation by three, giving us 48 times three on the left-hand side, which we can evaluate as 144. So now, we know that the sum of the scores in ๐ด, ๐ต, and ๐ถ is 144.

In the same way, we can find the sum of the scores for tests ๐ท and ๐ธ. Weโre given that there mean is 38 and since weโre talking about two tests, then we know that we divide by two. Multiplying both sides of our equation by two will give us that 38 times two is equal to the sum of the numbers. And since this evaluates to 76 on the left-hand side, then we now know that the sum of the scores in test ๐ท and ๐ธ is 76. Itโs worth noting at this point that we donโt know the actual individual scores of each of the tests. We just know the sum. However, we can still continue with our calculation for the mean of all five tests.

Therefore, to find the mean of ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ, we calculate the sum of all the numbers which would be 144 plus 76 and divide by how numbers there are. In this case, there are five tests. So there must be five values. Simplifying then, we have 220 over five or 220 divided by five, which is equal to 44. And so our final answer is that the mean score of all five tests is 44.