Question

In a coconut grove (x + 2) tree yields 60 nuts per year, x tree yields 120 nuts per year and (x -2) tree yields 180 nuts per year. If the average yield per year per tree is 100 then find x.
a). 4
b). 3
c). 2
d). 1

Answer 1

Hint: To calculate the value of x, we will use the formula of arithmetic mean of grouped data, that is
Arithmetic mean = $\dfrac{\sum\limits_{i}^{i=n}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i}^{i=n}{{{f}_{i}}}}$ , and here ${{f}_{i}}$ is equal to number of trees yielding particular number of nuts and ${{x}_{i}}$ is the number of particular nuts.

Complete step by step answer:
From the question, we can see that there are (x + 2) such coconut trees which 60 nuts per year, there are x trees which yield 120 nuts per year and there are (x - 2) such trees which yield 180 nuts per year.
And, for the grouped data we know that arithmetic mean = $\dfrac{\sum\limits_{i}^{i=n}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i}^{i=n}{{{f}_{i}}}}$, and here ${{f}_{i}}$ is equal to number of trees yielding particular number of nuts and ${{x}_{i}}$ is the number of particular nuts.
So, we can say that when ${{f}_{1}}$ = (x + 2), ${{x}_{1}}$ = 60. Similarly, when ${{f}_{2}}$= x, ${{x}_{2}}$= 120 and when ${{f}_{3}}$ = (x - 2), ${{x}_{3}}$= 180.
So, we can say that arithmetic mean = $\dfrac{{{f}_{1}}{{x}_{1}}+{{f}_{2}}{{x}_{2}}+{{f}_{3}}{{x}_{3}}}{{{f}_{1}}+{{f}_{2}}+{{f}_{3}}}$= $\dfrac{\left( x+2 \right)\times 60+x\times 120+\left( x-2 \right)\times 180}{x+2+x+x-2}$
So, arithmetic mean = $\dfrac{360x-240}{3x}$
And, we know that average is nothing but the arithmetic mean.
So, we can say that the average yield per year per tree is equal to $\dfrac{360x-240}{3x}$, but from the question we know that average yield per year per tree is 100.
So, $\dfrac{360x-240}{3x}$ = 100
$\Rightarrow 360x-240=300x$
$\Rightarrow 60x=240$
$\therefore x=4$

So, the correct answer is “Option a”.

Note: Students are required to note that ${{f}_{i}}$ is the frequency of each terms(${{x}_{i}}$) in the data set (i.e. number of times a particular term ${{x}_{i}}$ occur) and they should not get confused among ${{f}_{i}}$ and ${{x}_{i}}$. among about which term will be in the denominator of the arithmetic mean. Most of the students write summation of ${{x}_{i}}$in the denominator of the arithmetic mean formula in place of summation of ${{f}_{i}}$.