## Practice Problems on Averages – Set 6

Practice Problems on Averages – Set 6
1. On a journey across Chennai, a tourist bus went at 10 km/h for 30% of the distance, at 40 km/h for 40% of distance and at 30 km/h for the remaining distance. The average speed for the whole journey was:
a) 10 kmph
b) 20 kmph
c) 5 kmph
d) 24 kmph
e) None of these
Solution:
Let the total distance be x km.
We know, Distance = Speed × Time Time = Distance/Speed
Time required to cover 30% of the distance at 10 km/h = 0.3x/10 = 3x/100 hrs
Time required to cover 40% of the distance at 40 km/h = 0.4x/40 = x/100 hrs
Now, distance remaining = x – 0.3x – 0.4x = 0.3x
Time required to cover remaining 0.3x at 30 km/h = 0.3x/30) = x/100 hrs
So, total time required to complete x km distance = (3x/100) + (x/100) + (x/100) = 5x/100 hrs
The average speed for the whole journey = x/(5x/100) = 20 km/h

2. Average age of a class of 25 students is 13 years. The average age of a group of 8 students is 12 years and the average age of another group of 10 students is 11 years. What is the average age of the rest of the students?
a) 10 years
b) 13 years
c) 15 years
d) 16 years
e) 17 years
Solution:
We know that Average = Sum of all quantities/total number of quantities
Sum of ages of students = Average × number of students
The average age of 25 students is 13
Sum of ages of 25 students = Average × number of students
Sum of ages of 25 students = 25 × 13 = 325 years
Average age of group of 8 students is 12.
Sum of ages of 8 students = 12 × 8 = 96 years
Average age of group of 10 students is 11.
Sum of ages of 10 students = 11 × 10 = 110 years
Let the number of remaining students be x
x + 8 + 10 = 25
x + 18 = 25
x = 7 students
Sum of ages of 7 students = (Sum of ages of 25 students) (Sum of ages of 8 students + Sum of ages of 10 students)
Sum of age of 7 students = 325 – (96 + 110) = 325 – 206 = 119 years
Average age of 7 students = Sum of ages of 7 students/7
Average age of 7 students = 119/7 = 17
Average age of remaining students is 17 years

3. The average age of a family of five members is 24. If the present age of the youngest member is 8 years, what was the average age of the family at the time, when the youngest member was about to born?
a) 20 years
b) 16 years
c) 12 years
d) 18 years
e) 22 years
Solution:
Let present ages of five family members be x, x, x3, xand x5.
There average be (x+ x+ x3 + x+ x5)/5 = 24
x+ x2 + x3+ x+x5= 24 × 5 = 120
Since, Age of youngest member x5 is 8 years
x+ x2 + x3+ x+ 8 = 120
x+ x2 + x3+ x= 112.  ----------(1)
Since youngest member has taken birth 8 years ago, sum of ages of family members at that time was
(x- 8) + (x- 8) +(x- 8) +(x- 8)
= x+ x2 + x3+ x– 32
Putting equation one in this equation, we get
8 years ago, sum of ages of family members = 112 – 32 = 80
And average of their ages at that time = 80/4 = 20 years

4. Nine men went to a hotel, 8 of them spent Rs. 3 each over their meals and the ninth spent Rs. 2 more than the average expenditure of all the nine. The total money spent by all of them was
a) Rs.26.54
b) Rs.40.70
c) Rs.29.25
d) Rs.27.75
e) None of these
Solution:
Let us assume the average expenditure of all nine men = Rs. x
The amount spent by 8 men = 8 × 3 = 24 ( it is given that 8 men spent Rs.3 each)
The amount spent by ninth man = x + 2
It is stated that the amount spent by ninth man is 2 more than average of all nine.
Total expenditure of all nine men = 24 + (x + 2)
We know that, Average expenditure = (Total expenditure)/(Total men)
Hence, total expenditure = 9x
On equating total expenditures, we get
24 + x + 2 = 9x 8x = 26 x = 26/8 x = Rs. 3.25
Total expenditure of all nine men = 9 × 3.25 = Rs.29.25
Hence, the expenditure of all nine men is Rs.29.25

5. The monthly salary of a person was Rs. 320 for each of the first three years. He next got annual increment of Rs. 40 per month for each of the following successive 12 years. His salary remained stationery till retirement when he found that his average monthly salary during his service period was Rs. 698. Find the period of his service.
a) 25 years
b) 15 years
c) 40 years
d) 50 years
e) 30 years
Solution:
Total monthly salary for 1st 2 years = 320 + 320 = 640
The salary in next 13 years is in arithmetic progression with a common difference of 40. (Starting from 3rd year)
nth term of A.P = tn = a + (n – 1)d,
Where, a is 1st term and d is the difference.
3rd year’s salary = 1st term = a = 320/month
Salary after 12 years = 13th term = L = 320 + (13 – 1) × 40
= 800 /month
Sum of n terms of A.P = n(a + L)/2
Sum of 13 years of salary/month = 13 × (320 + 800)/2 = 7280
Along with the initial 2 years, Sum of 15 years of salary per month = 640 + 7280 = 7920
Let, he worked for x years more until retirement on 800/month salary.
So, total salary/month for (15 + x) years = 7920 + 800x
Average salary for (15 + x) years = 698
698 = (7920 + 800x)/ (15 + x)
698 × 15 + 698x = 7920 + 800x
2550 =102x
x = 25
So, total years of service = 15 + 25 = 40 years

6. In an exam, the average was found to be 50 marks. After deducting computational errors the marks of 100 candidates had to be changed from 90 to 60 each, and the average came down to 45 marks. The total number of candidates who took the exam was:
a) 300
b) 600
c) 200
d) 150
e) None of these
Solution:
Let number of candidates giving the exam = b
Then, total marks obtained by b candidates = 50b.
Now,
Change in computational marks for 1 candidate = 30,
Change in computational marks for 100 candidates = 3000,
Then, Final marks of b candidates = 45b.
Then according to the condition,
50b 3000 = 45b
5b = 3000
b = 600
Hence, 600 students took the test.

7. In the month of April, Rahul’s average expenditure for the first 10 days was Rs 800 per day. If his average expenditure for the next 10 days decreases by Rs25 and after 10 days the average expenditure again decrease by Rs.25 and he earns Rs 25, 000 per month. Calculate his savings in the month of April.
a) Rs.1500
b) Rs.2000
c) Rs.1850
d) Rs.1950
e) Rs.1750
Solution:
Average expenditure of days 1 to 10 = Rs 800
Total expenditure for days 1 to 10 = Rs 800 × 10 = 8000
Total expenditure for days 11 to 20 = Rs (800 – 25) × 10 = Rs 775 × 10 = Rs 7750
Total expenditure for days 21 to 30 = Rs (775 – 25) × 10 = Rs. 750 × 10 = Rs. 7500
Total expenditure= Rs (7500 + 7750 + 8000) = Rs. 23250
Savings = Income – Expenditure = Rs 25000 – Rs 23250 = Rs. 1750

8. The average age of 30 members in a group is 15. One member aged 20 years left the class, but two new members come in his place the difference of whose ages was 5 years. If the average age of all the boys now in the class still remains 15 years, the age of the younger newcomer is:
a) 20 years
b) 15 years
c) 30 years
d) 10 years
e) 18 years
Solution:
Now after 1 member of 20 year left and two members of age ‘X’ years and
‘x + 5’ year joined. The number of members become 31.
Required average age = Total age of 31 members/31
(450 - 20 + x + x + 5)/31=15 435 + 2x = 465 2x = 30
x = 15 years

9. The sum of five numbers is 290. The arithmetic average of first two numbers is 48.5 and the arithmetic average of last two numbers is 53.5. What is the third number?
a) 89
b) 84
c) 90
d) 87
e) 86
Solution:
Let five numbers are a, b, c, d, e
Average = sum of digits / no. of digits
Given average of first two numbers = 48.5
Sum of first 2 numbers = a + b = 2 × 48.5 = 97
Given Average of last two numbers = 53.5
Sum of last 2 numbers = d + e = 53.5 × 2 = 107
Given, a + b + c + d + e = 290
97 + c + 107 = 290
c = 290 204
c = 86

10. The average of 8 observations was 25.5. It was noticed later that two of those observations were wrongly taken. One observation was 14 more than the original value and the other observation was wrongly taken as 31 instead of 13. What will be the correct average of those 8 observations?
a) 20.5
b) 21.5
c) 22.5
d) 20
e) 21