Question

If the sum of four consecutive odd integers is 400, what is the value of first odd integer?
a)95
b)96
c)97
d)98

Answer 1

Hint: This is a general problem of linear equations in one variable. It can be approached by assuming the first number as a variable and successive odd numbers by adding 2. After this, form an equation by adding the four odd numbers assumed and equation their sum to 400. Solve this equation to get your answer.

Complete step-by-step answer:

Let us assume the first odd number to be x. Consecutive odd numbers have a difference of two(1, 3, 5,... and so on). So the next three odd numbers will be $x+2, (x+2)+2$ and $(x+2+2)+2$. The four numbers are-
$x, x+2, x+4$ and $x+6$

It is given that the sum of these four numbers is 400, so it can be represented as-
$\mathrm x+\left(\mathrm x+2\right)+\left(\mathrm x+4\right)+\left(\mathrm x+6\right)=400$
$4\mathrm x+12=400\\$
$4\mathrm x=400-12=388\\$
$\mathrm x=\dfrac{388}4\\$
$\mathrm x=97$

We assumed that x is the value of the first odd integer, which comes out to be 97, which is the answer.

The correct option is C. 97

Note: Since we know that the answer is an odd positive integer, we can eliminate options B. 96 and D. 98 which are even numbers, making the problem simpler. To make the calculation easier we can assume another value of the first odd number.
Let the first odd number be x-2, then subsequent numbers will be $x-2+2, x-2+4, x-2+6$. Hence, the equation will be-
$(x-2) + x + (x+2) + (x+4)=400$
$4x + 4 = 400$
$4x = 396$
$x = 99$
We have to find the first odd number which is assumed as $x-2$. So, its value will be-
$ x-2 = 99-2$
$ = 97$