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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.
Let us consider some more examples to understand it more clearly.
9. The height of 5 children are 121 cm, 123 cm, 119 cm, 122 cm and 120 cm respectively. Find the average height of the children.
Solution:
Sum of the heights of the children = (121 + 123 + 119 + 122 + 120) cm
= 605 cm
Number of the children = 5
Average height of the children = \(\frac{\textrm{Total height of the children}}{\textrm{Number of the children}}\)
= 605/5
= 121 cm
10. The total cost of 19 books of different costs is ₹ 380. Find their average cost.
Solution:
Total cost of the copies = ₹ 380
Number of copies = 19
Average cost = \(\frac{\textrm{Total cost of copies}}{\textrm{Number of copies}}\)
= 160/8
= ₹ 20
11. The age of 3 students are 13 years, 16 years and 19 years respectively. What is the average age of the students?
Solution:
Sum of the age of students = (13 + 16 + 19) years.
= 48 years
Average age = \(\frac{\textrm{Sum of the ages}}{\textrm{Number of students}}\)
= 48/3
= 16 years
12. Nairitee secured 20 marks in Hindi, 30 marks in English, 27 marks in Social Studies and 35 marks in Maths. Find the average marks secured by her.
Solution:
Total marks secured = 20 + 30 + 27 + 35 = 112
Number of subjects = 4
Average marks = \(\frac{\textrm{Total marks}}{\textrm{Number of subjects}}\)
= 112/4
= 28 marks
Note:
(i) Sum of quantities = Average x Number of quantities
(ii) Number of quantities = \(\frac{\textrm{Sum of quantities}}{\textrm{Average}}\)
Let us consider some different types of Word Problems on Average:
13. Donald’s average marks in English, Hindi, Science and Maths are 86. He secured 98 marks in Maths, 92 marks in Science and 86 marks in Hindi. Find the marks secured by him in English.
Solution:
Average marks = 96
Total marks secured in all the subjects = 86 × 4
= 344
Total marks secured in Maths, Science and Hindi = 98 + 92 + 86
= 276
So, marks secured in English = 344 – 276
= 68
14. Ron, Sam, Sandy and Mike have ₹ 120.40, ₹ 110.30, ₹ 99.30 and ₹ 110.00 respectively. Neil has also some money. If the average of the amount they have, including Neil, is ₹ 110, find the amount Neil has.
Solution:
Total amount Average × Number of boys
= 5 × 110
= ₹ 550
Total amount that Ron, Sam, Sandy and Mike have together
= ₹ 120.40 + ₹ 110.30 + ₹ 99.30 + ₹ 110.00
So, the amount Neil = ₹ 550 – ₹ 440
= ₹ 110
Statistics
Word Problems on Arithmetic Mean
Properties Questions on Arithmetic Mean
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