Question

A cricket player scored $180$ runs in the first match and $257$ runs in the second match. Find the number of runs he should score in the third match so that the average of runs in the three matches be $230$.

Answer 1

Here we will be using the formula of calculating the average of any number which is also known as the arithmetic mean and it is calculated by adding a group of numbers given with the count of that particular number. For example, the average of numbers $2$, $3$, $5$ and $7$ will be equals to as below:
${\text{Average}} = \dfrac{{2 + 3 + 5 + 7}}{4}$. By adding the terms in the numerator, we get:
$\Rightarrow {\text{Average}} = \dfrac{{17}}{4}$

Complete step-by-step solution:
Step 1: Let the runs scored in the first match be denoted as
${x_1} = 180$. The runs scored by the player in the second match are denoted as ${x_2} = 257$ and the third match runs are denoted as ${x_3}$ which we need to find.
Step 2: By using the formula of average we get:
${\text{Average}} = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}$
By substituting the value of ${x_1} = 180$and
${x_2} = 257$ in the above expression we get:
$\Rightarrow {\text{Average}} = \dfrac{{180 + 257 + {x_3}}}{3}$
By substituting the value of average which is
$230$ in the above expression we get:
$\Rightarrow 230 = \dfrac{{180 + 257 + {x_3}}}{3}$
Step 3: By doing the addition to the RHS side of the above expression we get:
$\Rightarrow 230 = \dfrac{{437 + {x_3}}}{3}$
Bringing $3$ into the LHS side of the above expression and multiplying it with $230$, we get:
$\Rightarrow 690 = 437 + {x_3}$
By bringing $437$ into the LHS side of the above expression and subtracting it from $690$, we get:
$\Rightarrow {x_3} = 253$

The runs scored by the player in the third match is $253$.

Note: Students need to remember the basic formulas for calculating the average of the terms. There are some important properties of average as below:
When the difference between all the terms is the same then the average equal to the middle term.
If a particular variable $x$ is added in all the terms then, the average will also be increased by $x$
If a particular variable $x$ is subtracted in all the terms then, the average will also be decreased by $x$