Question

Andrew has taken 5 of the 8 equally weighted tests in his U.S. Maths class this semester he has an average score of exactly 78.0 points. How many points does he need to earn on the ${{6}^{th}}$ test to bring his average score up to exactly 80 points?
(a) 90
(b) 88
(c) 82
(d) 80
(e) 79

Answer 1

Hint: Let us assume the sum of the score in 5 tests of Andrew is x. We know that the formula for average is equal to: $\dfrac{\text{Sum of all observations}}{\text{Total number of observations}}$. Using this formula, we can find the sum of the score of 5 tests. Now, let us assume that in the ${{6}^{th}}$ test, he scored y points. Then again we are going to use the formula for average and hence, will find the points he will get in ${{6}^{th}}$ test.

Complete step-by-step solution:
We have given that the average score of Andrew in 5 tests is 78 points. We know the formula for average is equal to:
$Average=\dfrac{\text{Sum of all observations}}{\text{Total number of observations}}$
Now, let us assume that sum of the points that Andrew scored in 5 tests is x then substituting the sum of observations as x, number of observations as 5, and average as 78 in the above formula we get,
$78=\dfrac{x}{5}$
Multiplying 5 on both the sides of the above equation we get,
$\begin{align}
  & 78\times 5=x \\
 & \Rightarrow 390=x \\
\end{align}$
From the above, we got the summation of points in 5 tests of Andrew as 390.
Now, let us assume that if Andrew will earn y points in the ${{6}^{th}}$ test then his average will become 80 points then substituting average as 80, sum of observations as $x+y$ and total number of observations as 6 in the given average formula we get,
\[\begin{align}
  & \Rightarrow 80=\dfrac{x+y}{6} \\
 & \Rightarrow 80\times 6=x+y \\
 & \Rightarrow 480=x+y \\
\end{align}\]
Substituting the value of x as 390 in the above equation we get,
$\begin{align}
  & 480=390+y \\
 & \Rightarrow 480-390=y \\
 & \Rightarrow 90=y \\
\end{align}$
Hence, Andrew needs 90 points to make the average score 80 points and the correct option is (a).

Note: The mistake that could be possible in the above problem is that when we have taken the average of 6 terms then you might forget to add the sum of the score of 5 tests so make sure you won’t make this mistake. Another mistake which could be possible is that you might even be in place of writing 6 in a total number of observations you might write 5 so here also make sure you won’t make such blunders.