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

How to solve average word problems?

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

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

** **

Worked-out problems based on average:

1. The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.

**Solution:**

Mean weight of 7 boys = 56 kg.

Total weight of 7 boys = (56 × 7) kg = 392 kg.

Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg

= 338 kg.

Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)

= (392 - 338) kg

= 54 kg.

Hence, the weight of the seventh boy is 54 kg.

**2**. A cricketer has a mean
score of 58 runs in nine innings. Find out how many runs are to be scored by
him in the tenth innings to raise the mean score to 61.

**Solution: **

Mean score of 9 innings = 58 runs.

Total score of 9 innings = (58 x 9) runs = 522 runs.

Required mean score of 10 innings = 61 runs.

Required total score of 10 innings = (61 x 10) runs = 610 runs.

Number of runs to be scored in
the 10th innings

= (total score of 10 innings) - (total score of 9 innings)

= (610 -522) = 88.

Hence, the number of runs to be scored in the 10th innings = 88.

**3.** The mean of five numbers is 28. If one of the numbers is excluded, the mean gets reduced by 2. Find the excluded number.

**Solution: **

Mean of 5 numbers = 28.

Sum of these 5 numbers = (28 x 5) = 140.

Mean of the remaining 4 numbers = (28 - 2) =26.

Sum of these remaining 4 numbers = (26 × 4) = 104.

Excluded number

= (sum of the given 5 numbers) - (sum of the remaining 4 numbers)

= (140 - 104)

= 36.

Hence, the excluded number is 36.

**4**. The mean weight of a
class of 35 students is 45
kg. If the
weight of the teacher be included, the mean weight increases by 500 g. Find the weight of the teacher.

**Solution: **

Mean weight of 35 students = 45 kg.

Total weight of 35 students = (45 × 35) kg = 1575 kg.

Mean
weight of 35 students and the teacher (45 + 0.5) kg = 45.5 kg.

Total weight of 35 students and the teacher = (45.5 × 36) kg = 1638 kg.

Weight of the teacher = (1638 - 1575) kg = 63 kg.

Hence, the weight of the teacher is 63 kg.

**5. **The average height of 30
boys was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the
correct mean.

**Solution: **

Calculated average height of 30 boys = 150 cm.

Incorrect sum of the heights of
30 boys

= (150 × 30)cm

= 4500 cm.

Correct sum of the heights of 30 boys

= (incorrect sum) - (wrongly copied item) + (actual item)

= (4500 - 135 + 165) cm

= 4530 cm.

Correct mean = correct sum/number of boys

= (4530/30) cm

= 151 cm.

Hence, the correct mean height is 151 cm.

**6. ** The mean of 16 items
was found to be 30. On
rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively.
Find the correct mean.

**Solution:**

Calculated mean of 16 items = 30.

Incorrect sum of these 16 items = (30 × 16) = 480.

Correct sum of these 16 items

= (incorrect sum) - (sum of incorrect items) + (sum of actual items)

= [480 - (22 + 18) + (32 + 28)]

= 500.

Therefore, correct mean = 500/16 = 31.25.

Hence, the correct mean is 31.25.

**7. ** The mean of 25 observations
is 36. If the mean of the first
observations is 32 and that of
the last 13 observations is 39,
find the 13th observation.

**Solution:**

Mean of the first 13 observations = 32.

Sum of the first 13 observations = (32 × 13) = 416.

Mean of the last 13 observations = 39.

Sum of the last 13 observations = (39 × 13) = 507.

Mean of 25 observations = 36.

Sum of all the 25 observations = (36 × 25) = 900.

Therefore, the 13th observation = (416 + 507 - 900) = 23.

Hence, the 13th observation is 23.

**8. ** The aggregate monthly expenditure of a family was $ 6240 during the first 3 months, $ 6780 during the next 4 months and $ 7236 during the last 5 months of a year. If the total saving during
the year is $ 7080, find the
average monthly income of the family.

**Solution: **

Total expenditure during the
year

= $[6240 × 3 + 6780 × 4 + 7236 × 5]

= $ [18720 + 27120 + 36180]

= $ 82020.

Total income during the year = $ (82020 + 7080) = $ 89100.

Average monthly income = (89100/12) = $7425.

Hence, the average monthly income of the family is $ 7425.

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