Question

For each month of a given year except December, a worker earned the same monthly salary and donated one-tenth of that salary to charity. In December, the worker earned N times his usual monthly salary and donated one-fifth of his earnings to charity. If the worker's charitable contributions totaled one-eighth of his earnings for the entire year, what is the value of N?

A) 8/5

B) 5/2

C) 3

D) 11/3

E) 4

Answer 1

D) 11/3

Explanation:

Let x represent monthly salary.

We have been given that for each month of a given year except December, a worker earned the same monthly salary and donated one-tenth of that salary to charity.

Salary earned in 11 months would be
`11x`.

Money donated in 11 months would be
`(11x)/(10)`.

Further we are told that in December, the worker earned N times his usual monthly salary and donated one-fifth of his earnings to charity.

Salary for December would be
`Nx`.

Money donated in December would be
`(Nx)/(5)`.

The worker's charitable contributions totaled one-eighth of his earnings for the entire year that is
`(1)/(8)\cdot (11x+Nx)`.

`(11x)/(10)+(Nx)/(5)=(1)/(8)\cdot (11x+Nx)`

Dividing whole equation by x, we will get:

`(11x)/(10x)+(Nx)/(5x)=(1)/(8x)\cdot x(11+N)`

`(11)/(10)+(N)/(5)=(1)/(8)\cdot (11+N)`

`(11)/(10)+(N)/(5)=(11)/(8)+(N)/(8)`

Combine like terms:

`(N)/(5)-(N)/(8)=(11)/(8)-(11)/(10)`

`(N)/(5)*40-(N)/(8)*40=(11)/(8)*40-(11)/(10)*40`

`8N-5N=11*5-11*4`

`3N=55-44`

`3N=11`

`(3N)/(3)=(11)/(3)\\\\N=(11)/(3)`

Therefore, the value of N is 11/3 and option D is the correct choice.