# Word Problems on Arithmetic Mean

Here we will learn to solve the three important types of word problems on arithmetic mean (average). The questions are mainly based on average (arithmetic mean), weighted average and average speed.

How to solve average (arithmetic mean) word problems?

To solve various problems we need to follow the uses of the formula for calculating average (arithmetic mean)

Average = (Sums of the observations)/(Number of observations)

Follow the explanation to solve the word problems on arithmetic mean (average):

1. The heights of five runners are 160 cm, 137 cm, 149 cm, 153 cm and 161 cm respectively. Find the mean height per runner.

Solution:

Mean height = Sum of the heights of the runners/number of runners

= (160 + 137 + 149 + 153 + 161)/5 cm

= 760/5 cm

= 152 cm.

Hence, the mean height is 152 cm.

2. Find the mean of the first five prime numbers.

Solution:

The first five prime numbers are 2, 3, 5, 7 and 11.

Mean = Sum of the first five prime numbers/number of prime numbers

= (2 + 3 + 5 + 7 + 11)/5

= 28/5

= 5.6

Hence, their mean is 5.6

3. Find the mean of the first six multiples of 4.

Solution:

The first six multiples of 4 are 4, 8, 12, 16, 20 and 24.

Mean = Sum of the first six multiples of 4/number of multiples

= (4 + 8 + 12 + 16 + 20 + 24)/6

= 84/6

= 14.

Hence, their mean is 14.

4. Find the arithmetic mean of the first 7 natural numbers.

Solution:

The first 7 natural numbers are 1, 2, 3, 4, 5, 6 and 7.

Let x denote their arithmetic mean.

Then mean = Sum of the first 7 natural numbers/number of natural numbers

x = (1 + 2 + 3 + 4 + 5 + 6 + 7)/7

= 28/7

= 4

Hence, their mean is 4.

5. If the mean of 9, 8, 10, x, 12 is 15, find the value of x.

Solution:

Mean of the given numbers = (9 + 8 + 10 + x + 12)/5 = (39 + x)/5

According to the problem, mean = 15 (given).

Therefore, (39 + x)/5 = 15

⇒ 39 + x = 15 × 5

⇒ 39 + x = 75

⇒ 39 - 39 + x = 75 - 39

⇒ x = 36

Hence, x = 36.

More examples on the worked-out word problems on arithmetic mean:

6. If the mean of five observations x, x + 4, x + 6, x + 8 and x + 12 is 16, find the value of x.

Solution: Mean of the given observations

= x + (x + 4) + (x + 6) + (x + 8) + (x + 12)/5

= (5x + 30)/5

According to the problem, mean = 16 (given).

Therefore, (5x + 30)/5 = 16

⇒ 5x + 30 = 16 × 5

⇒ 5x + 30 = 80

⇒ 5x + 30 - 30 = 80 - 30

⇒ 5x = 50

⇒ x = 50/5

⇒ x = 10

Hence, x = 10.

148 + 153 + 146 + 147 + 154

7. The mean of 40 numbers was found to be 38. Later on, it was detected that a number 56 was misread as 36. Find the correct mean of given numbers.

Solution:

Calculated mean of 40 numbers = 38.

Therefore, calculated sum of these numbers = (38 × 40) = 1520.

Correct sum of these numbers

= [1520 - (wrong item) + (correct item)]

= (1520 - 36 + 56)

= 1540.

Therefore, the correct mean = 1540/40 = 38.5.

8. The mean of the heights of 6 boys is 152 cm. If the individual heights of five of them are 151 cm, 153 cm, 155 cm, 149 cm and 154 cm, find the height of the sixth boy.

Solution:

Mean height of 6 boys = 152 cm.

Sum of the heights of 6 boys = (152 × 6) = 912 cm

Sum of the heights of 5 boys = (151 + 153 + 155 + 149 + 154) cm = 762 cm.

Height of the sixth boy

= (sum of the heights of 6 boys) - (sum of the heights of 5 boys)

= (912 - 762) cm = 150 cm.

Hence, the height of the sixth girl is 150 cm.

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