 QUESTION

# The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of P is:A. 3500B. 4000C. 4050D. 5000

Hint: We had to make equations for the average income P and Q, O and R, R and P using the formula of average of two amounts i.e $\dfrac{{a + b}}{2}$. And then after solving these equations we will get the required monthly income of P.

Complete step-by-step solution -

Let the monthly income of P is Rs. x
Let the monthly income of Q is Rs. y
Let the monthly income of R is Rs. z
So, as we know that the average of two number a and b is calculated as $\dfrac{{a + b}}{2}$
So, the average income of P and Q will be $\dfrac{{x + y}}{2}$
Average income of Q and R will be $\dfrac{{y + z}}{2}$
Average income of R and P will be $\dfrac{{z + x}}{2}$
So, according to the question,
$\dfrac{{x + y}}{2}$ = 5050 (1)
$\dfrac{{y + z}}{2}$ = 6250 (2)
$\dfrac{{z + x}}{2}$ = 5200 (3)

Now we had to only find the monthly income of P. So, for that we only need to find the value of x.
So, finding the value of y in terms of x from equation 1.
Cross multiplying equation 1. We get,
x + y = 10100
y = 10100 – x (4)
Now, finding the value of z in terms of x from equation 3.
Cross multiplying equation 3. We get,
z + x = 10400
z = 10400 – x (5)
Now to find the value of x we must put the value of y and z from equation 4 and equation 5 to equation 2.
So, equation 2 becomes,
$\dfrac{{\left( {10100 - x} \right) + \left( {10400 - x} \right)}}{2} = 6250$
Cross multiplying above equation to find the value of x.
10100 – x + 10400 – x = 12500
20500 – 2x = 12500
Adding 2x – 12500 to both sides of the above equation. We get,
2x = 8000
x = 4000
So, the monthly income of person P is Rs. 4000
Hence, the correct option will be B.

Note: Whenever we come up with this type of problem then we had first, write the equations for average income of P and Q, Q and R, R and P and then We have to find the value of y and z in terms of x and then put the value of y and z in the equation of average income of y and z to get the required value to get the required value of x. where x, y and z are the monthly income of person P, Q and R.