# Difficult Averages Solved QuestionAptitude Discussion

 Q. In 2011, the arithmetic mean of the annual incomes of Ramesh and Suresh was Rs. 3800. The arithmetic mean of the annual incomes of Suresh and Pratap was Rs. 4800, and the arithmetic mean of the annual incomes of Pratap and Ramesh was Rs. 5800. What is the arithmetic mean of the incomes of the three?
 ✖ A. Rs. 4000 ✖ B. Rs. 4200 ✖ C. Rs. 4400 ✔ D. Rs. 4800

Solution:
Option(D) is correct

Let $a, b$, and $c$ be the annual incomes of Ramesh, Suresh, and Pratap, respectively.

Now, we are given that

The  arithmetic  mean  of  the  annual  incomes  of  Ramesh  and  Suresh  was  Rs. 3800.

Hence, $\dfrac{a+b}{2}=3800$

Multiplying by 2 yields $a+b=2\times 3800=7600$.

The  arithmetic  mean  of  the  annual  incomes  of  Suresh  and  Pratap  was  Rs. 4800.

Hence, $\dfrac{b+c}{2}=4800$

Multiplying by 2 yields $b+c=2\times 4800=9600$.

The  arithmetic  mean  of  the  annual  incomes  of  Pratap  and  Ramesh  was  Rs. 5800.

Hence, $\dfrac{c+a}{2}=5800$

Multiplying by 2 yields $c+a=2×5800=11,600$

Summing these three equations yields:

$(a+b)+(b+c)+(c+a)=7600+9600+11,600$

$2a+2b+2c=28,800$

$a+b+c=14,400$

The average of the incomes of the three equals the sum of the incomes divided by 3:

$=\dfrac{a+b+c}{2}$

$=\dfrac{14,400}{3}$

$=4800$

Edit: Malay Pandey gives better variable names with similar calculation.

Edit 2: Single line solution from Amaila Buzdar, works well here.

## (2) Comment(s)

Amaila Buzdar
()

$\dfrac{3800+4800+5800}{3}=4800$

Malay Pandey
()

$R+S=3800*2$

$S+P=4800*2$

$P+R=5800*2$

NOW ADD ALL 3 EQN, WE GET:-

$2(S+P+R)=14400*2$

$S+P+R=14400$

NOW DIVIDE BY 3 ON BOTH SIDE TO FIND OUT THE AVG OF ALL 3, WE GET:-

$\dfrac{(S+P+R)}{3}=\dfrac{14400}{3}$

AVG INCOME OF S, P & R $= 4800$