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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:

As the month begin with Sunday, so there will be five Sundays in the month. So result will be:
$$= \left (\dfrac{510 \times 5 + 240 \times 25}{30}\right)\\ = \left( \dfrac{8550}{30} \right)\\ = 285$$

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6.

Average of first five multiples of 3 is:

Average:
$$=\dfrac{3(1+2+3+4+5)}{5}\\ = \dfrac{45}{5}\\ = 9$$

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7.

A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500?

Let the sale in the sixth month $$=x$$
Then, $$\dfrac{6435+6927+6855+7230+6562+x}{6}=6500$$
$$\Rightarrow 6435+6927+6855+7230+6562+x=6 \times 6500\\ \Rightarrow 34009 + x = 39000\\ \Rightarrow x = 39000 - 34009 = 4991$$

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8.

A student needed to find the arithmetic mean of the numbers 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14 and $$x$$. He found the mean to be 12. What is the value of $$x$$?

$$\dfrac{\text{3+11+7+9+15+13+8+19+17+21+14+}x}{12} = 12\\ \Rightarrow \dfrac{137 + x}{12} = 12\\ \Rightarrow 137 + x = 144\\ \Rightarrow x = 144 - 137 = 7$$

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9.

The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What is the weight of the new person?

Total increase in weight $$= 8 \times 2.5 = 20$$
If $$x$$ is the weight of the new person, total increase in weight $$= x-65$$
$$\Rightarrow 20 = x - 65\\ \Rightarrow x = 20 + 65 = 85$$

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10.

A motorist travel to a place 150 km away at an average speed of 50 km/hr and returns ar 30 km/hr.His average speed for the whole journey in km/hr is:

$$\dfrac{2xy}{x+y} \text{ km/hr}\\ = \dfrac{2(50)(30)}{50+30} \text{ km/hr}\\ = 37.5 \text{ km/hr}$$