Question

The average of $6$ numbers is $30$. If the average of the first four is $25$ and that of the last three is $35$, the fourth number is
$\left( {\text{A}} \right)25$
$({\text{B) }}30$
$(C{\text{) }}35$
$(D{\text{) 4}}0$

Answer 1

Here, we have to find the fourth number.
First, we assume a variable for all the $6$ numbers.
Then, we have to form equations.
By substituting one equation to the other, we will get the fourth number.

Formula used: $\dfrac{{{\text{Sum of the total of all observations}}}}{{{\text{Total no}}{\text{.of observation}}}} = {\text{Average}} {\text{value}}$

Complete step-by-step solution:
Let us assume that the $6$ numbers be ${\text{a, b, c, d, e and f}}$.
So we need to find the fourth number that is ${\text{d}}$
It is given that the average of $6$ numbers is $30$.
By using the formula and we can write it as,
$\dfrac{{{\text{a + b + c + d + e + f}}}}{{\text{6}}}{\text{ = }}30$ ,
By taking cross multiplication we get,
$\Rightarrow {\text{a + b + c + d + e + f}} = 30 \times 6$
Let us multiply we get,
$\Rightarrow {\text{a + b + c + d + e + f}} = 180........(1)$
Again it is stated in the question, that the average of the first four numbers is $25$.
By using the formula and we can write it as
$\dfrac{{a + b + c + d}}{4} = 25$
By taking cross multiplication we get,
$\Rightarrow {{a + b + c + d = 25 \times 4}}$
On multiplying the RHS we get,
$\Rightarrow {\text{a + b + c + d = 100}}...........{\text{(2)}}$
Also it is stated in the question that the average of last three numbers is $35$,
By using the formula and we can write it as
$\dfrac{{{\text{d + e + f}}}}{3} = 35$
By taking cross multiplication we get,
${{d + e + f = 35 \times 3}}$
Let us multiply the terms and we get
$\Rightarrow {\text{d + e + f}} = 105...........(3)$
Now, we need to find the fourth number that is the value of ${\text{d}}$
Putting $(2)$ and $(1)$ we get
$1{\text{00 + e + f }} = 180$
On subtracting $100$ on both sides we get,
${\text{e + f}} = 180 - 100$
On subtracting the values we get
$\Rightarrow {\text{e + f}} = 80$
From$(3)$,
${\text{d + e + f}} = 105$
We can substitute ${\text{e + f}} = 80$ in ${\text{d + e + f = 105}}$ to get ${\text{d}}$.
${\text{d}} + 80 = 105$
On subtracting $80$ on both sides we get,
${\text{d}} = 105 - 80$
Let us subtract the values and we get,
${\text{d}} = 25.$
$\therefore$ The fourth number is $25$.

Hence the correct option is A

Note: This type of question is simply a kind of logical reasoning and such questions always contain certain clues.
Here, the clues are, ‘there are 6 numbers’ then ‘average of first four numbers’ and ‘average of last three numbers’. and when we add up 4 and 3, it gives 7.
This indicates that there is a number which is common and by this approach we need to form and substitute the equations.