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Question: Ravi obtained 70 and 75 marks in the first two unit tests. Find the minimum marks he should get in t...

Ravi obtained 70 and 75 marks in the first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Explanation

Solution

To solve this problem we need to first know how to find the average of n terms. So suppose we have n terms from a1,a2,a3,...,an{{a}_{1,}}{{a}_{2}},{{a}_{3}},...,{{a}_{n}} and average of these n terms is given by a1+a2+...+ann\dfrac{{{a}_{1}}+{{a}_{2}}+...+{{a}_{n}}}{n} so generally average is given by sum of all the given terms by the total number of that terms. Now in the given question we are given two terms and we have to find the third term such that the minimum average will be 60. We will use the mentioned formula for the average of n terms and find the third term.

Complete step-by-step answer:
To start the solution we should know how to calculate the average of n given terms, average of n given terms is given by, sum of all the given terms divided by the total number of terms.
So average of A1,A2,A3{{A}_{1}},{{A}_{2}},{{A}_{3}} will be given by A1+A2+A33\dfrac{{{A}_{1}}+{{A}_{2}}+{{A}_{3}}}{3}
Now in the given question Ravi obtained 70 and 75 marks in two tests and we have to find the marks that he should get in the third test so that he gets the minimum average of at least 60 marks.
Now, if we take A1=70  and A2=75{{A}_{1}}=70\,\,and\,{{A}_{2}}=75 and minimum average as 60 then we get,
average⁡≥60 A1+A2+A33≥60 70+75+A33≥60 \begin{aligned} & \operatorname{average}\ge 60 \\\ & \dfrac{{{A}_{1}}+{{A}_{2}}+{{A}_{3}}}{3}\ge 60 \\\ & \dfrac{70+75+{{A}_{3}}}{3}\ge 60 \\\ \end{aligned}
Adding terms we get,
145+A33≥60\dfrac{145+{{A}_{3}}}{3}\ge 60
Now by cross multiplying we get,
145+A3≥180145+{{A}_{3}}\ge 180
Taking 145 to the RHS, we get
A3≥180−145{{A}_{3}}\ge 180-145
A3≥35{{A}_{3}}\ge 35
Hence the minimum value of the third test that he should get in order to form a minimum average of 60 is 35.

Note: You need to make sure that when you equate the average as equal to or greater than 60 and not equal to 60 because we have to find the minimum average here. Also remember the mentioned definition and formula to find the average of n terms for the future references.