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. 3500 B. 4000 C. 4050 D. 5000
ANSWER
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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.