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# Average

1.

The average monthly income of A and B is Rs. 5050. The average monthly income of B and C is Rs. 6250 and the average monthly income of A and C is Rs. 5200. What is the monthly income of A?

Let monthly income of A = a
monthly income of B = b
monthly income of C = c

$$a + b = 2 \times 5050~\cdots(Equation 1)\\ b + c = 2 \times 6250 ~\cdots(Equation 2)\\ a + c = 2 \times 5200 ~\cdots (Equation 3)\\ (Equation 1) + (Equation 3) - (Equation 2)\\ \Rightarrow a + b + a + c - (b + c) = (2 \times 5050) + (2 \times 5200) - (2 \times 6250)\\ \Rightarrow 2a = 2(5050 + 5200 - 6250)\\ \Rightarrow a = 4000$$

i.e., Monthly income of A = 4000

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2.

A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is:

Since the month begins with a Sunday, to there will be five Sundays in the month.
Required average
$$\left( \dfrac {510 \times 5 + 240 \times 25}{30} \right)\\ =\dfrac{8550}{30}\\ =285$$

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3.

If the average marks of three batches of 55, 60 and 45 students respectively is 50, 55, 60, then the average marks of all the students is:

Required average
$$\left( \dfrac {55 \times 50 + 60 \times 55 + 45 \times 60}{55+60+45} \right)\\ =\left( \dfrac {2750 + 3300 + 2700}{160} \right)\\ =\dfrac{8750}{160}\\ =54.68$$

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4.

A batsman makes a score of 87 runs in the 17th match and thus increases his average by 3. Find his average after 17th match.

Let the average after 17th match is $$x$$
Then the average before 17th match is $$x-3$$
So,
$$16(x-3) + 87 = 17x\\ \Rightarrow x = 87 - 48 = 39$$

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5.

A library has an average of 510 visitors on Sundays and 240 on other day. The average number of visitors in a month of 30 days starting with Sunday is:

$$= \left (\dfrac{510 \times 5 + 240 \times 25}{30}\right)\\ = \left( \dfrac{8550}{30} \right)\\ = 285$$