Hello Friends, Quantitative Aptitude is one of the most important section of every competitive exam.
Here we are sharing Shortcuts of Average Chapter of quantitative Aptitude, which is useful for solving average questions in all competitive exams.
Here we are sharing Shortcuts of Average Chapter of quantitative Aptitude, which is useful for solving average questions in all competitive exams.
Average -
An average means the sum of n different items divided by n numbers of items.
The basic formula is
Or
For example,
Ex. If a batsman scores 55, 30 and 44 runs in first, second and third innings respectively, then his average runs in 3 innings is
Sol 1. Average = (55 + 30 + 44)/3= 43
Shortcut Tricks for Average Chapter
1. If the value of each item in a group is increase by the same value x, then the average of the group also increases by x.
For instance, if the income of each person in a group increases by 15, the average income of the group also increase by Rs. 15.
2. If the value of each item in a group is decrease by the same value x, then the average of the group also decreases by x.
For instance, if the income of each person in a group decreases by 20, the average income of the Group also decreases by Rs. 20.
3. If the average age of group of people is x years, then their average age after n years will be (x + n).
This is because with each passing year, each person’s age increases by 1 and vice versa.
4. If the average age of group of people is x years, then their average age n years ago will be (x - n).
5. If the value of each item is multiplied by the same value x, then the average of the group or items will also get multiplied by x.
6. If the value of each item is divided by the same value n, then the average of the group or items will also get divided by n.
7. The average of a group always lies between the smallest value and the longest value in that group.
8. If the average age of n persons increases by x years. Then the total age of n persons increases by (n × x) years.
Ex. Average age of 6 persons is increased by 2 yr when one of them, whose age is 26 yr is replaced by a new man. What is the age of new person ?
Sol. Total age increased = 6 × 2 = 12 yr
Age of new person = (26 + 12) = 38 yr
The increase in the total age of 6 persons is due to the replacement of a person aged 26 yr with a person who is 12 yr older to him.
Sol. Total age increased = 6 × 2 = 12 yr
Age of new person = (26 + 12) = 38 yr
The increase in the total age of 6 persons is due to the replacement of a person aged 26 yr with a person who is 12 yr older to him.
9. If the average age of n persons decreases by x years. Then the total age of n persons decreases by (n × x) years.
10. Suppose a man covers a certain distance at x kmph and an equal distance at y kmph, then the average speed during the whole journey is (2xy/x+y) kmph.
11. Weighted Average Concept :-
When we have two or more groups whose individual averages are known, then to find the combined average of all the elements of all the groups we use Weighted Average Method.
Ex. Suppose in a class, there are 2 students of 20 yr, 3 of 21 yr, 4 of 22 yr and 5 of 23 yr, then their average age is
A = 20 yr, 21 yr, 22 yr & 23 yr.
K = 2 students, 3 students, 4 students & 5 students.
So
11. Weighted Average Concept :-
When we have two or more groups whose individual averages are known, then to find the combined average of all the elements of all the groups we use Weighted Average Method.
A = 20 yr, 21 yr, 22 yr & 23 yr.
K = 2 students, 3 students, 4 students & 5 students.
So
Average Questions with Solutions & Tricks
Ex. 1: The average weight of 4 men is increased by 3 kg when one of them who weigh, 120 kg is replaced by another man. What is the weight of the new man?
Sol. : If the average is increased by 3 kg, then the sum of weights increases by 3 × 4 = 12 kg.
And this increase in weight is due to the extra weight included due to the inclusion of new person.
∴ Weight of new man = 120 + 12 = 132 kg
Ex. 2 : The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15, what is the number of candidates who passed the examination?
Sol. : Let the number of passed candidates be x.
Then total marks = 120 × 35 = 39x + (120 – x) × 15
4200 = 39x + 1800 – 15x
24x = 2400
∴ x = 100
∴ number of passed candidates = 100.
Ex. 3 : The average of 11 results is 50. If the average of first six results is 49 and that of last six is 52, find the sixth result.
Sol. : The total of 11 results = 11 × 50 = 550
The total of first 6 results = 6 × 49 = 294
The total of last 6 results = 6 × 52 = 312
The 6th result is common on both; Sixth result = 294 + 312 – 500 = 56
Ex. 4 : The average age of 8 persons in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. Find the average age of these two women.
Sol. : By the use of average formula,
the total age of two women = 2 × 8 + (35 + 45)
= 16 + 80 = 96 years.
= 16 + 80 = 96 years.
∴ average age of two women = (96 / 2) = 48 years.
∴ average age of two women = 48 years.
Ex. 5 : The average age of a family of 6 members is 22 years. If the age of the youngest member be 7 years, then what was the average age of the family at the birth of the youngest member?
Sol. : Total ages of all members = 6 × 22 = 132 years.
7 years ago, total sum of ages = 132 – (6 × 7) = 90 years.
But at that time there were 5 members in the family.
∴ Average at that time = 90 ÷ 5 = 18 years.
Ex. 6 : The average score of a cricketer in two matches is 27 and in three other matches is 32. Then find the average score in all the five matches.
Sol. : By the use of average formula,
Average in 5 matches
Ex. 7: A man bought 13 shirts of Rs. 50 each, 15 pants of Rs. 60 each and 12 pairs of shoes at Rs. 65 a pair. Find the average value of each article.
Sol. : By the use of average formula,
Average =
Ex. 8 : The average of 11 results is 30, that of the first five is 25 and that of the last five is 28. Find the value of the 6th number.
Sol. : By the use of average formula,
6th number = Total of 11 results – (Total of first five results + Total of last five results)
= 11 × 30 – (5 × 25 + 5 × 28)
= 330 – 265 = 65
Ex. 9 : In a class, there are 20 boys whose average age is decreased by 2 months, when one boy aged 18 years is replaced by a new boy. Find the age of the new boy.
Sol.: By the use of average formula,
Age of new person = Age of removed person – No. of persons × Decrease in average age
Ex. 10 : A batsman in his 17th innings makes a score of 85, and thereby increases his average by 3. What is this average after 17 innings?
Sol. : Let the average after 16th innings be x,
then 16x + 85 = 17 (x + 3) = Total score after 17th innings.
∴ x = 85 – 51 = 34
∴ average after 17 innings = x + 3
= 34 + 3 = 37.
Ex. 11 : A cricketer has completed 10 innings and his average is 21.5 runs. How many runs must he make in his next innings so as to raise his average to 24?
Sol. : Total of 10 innings = 21.5 × 10 = 215
Suppose he needs a score of x in 11th innings; then average in 11 innings
Sol. : Total of 10 innings = 21.5 × 10 = 215
Suppose he needs a score of x in 11th innings; then average in 11 innings
Ex. 12 : There are two sections A and B of a class, consisting of 36 and 44 students respectively. If the average weight of section A is 40 kg and that of section B is 35 kg, find the average weight of the whole class.
Solution: Total weight of (36 + 44) Students = (36 × 40 + 44 ×35) kg = 2980 kg
Therefore average weight of the whole class = (2980/80) kg
Therefore average weight = 37.25 kg.
Ex. 13 : Distance between two stations A and B is 778 km. A train covers the journey from A to B at 84 km per hour and returns back to A with a uniform speed of 56 km per hour. Find the average speed of train during the whole journey.
Solution: required average speed
Ex. 14 : A Batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. Find his average after 17th inning.
Solution: let the average after 17th inning = x
Then average after 16th inning = (x - 3)
Therefore 16 (x - 3) + 87 = 17x
Therefore x = 39
Ex. 15 : There were 35 students in a hostel. Due to the admission of 7 new students the expenses of the mess were increased by Rs.42 per day while the average expenditure per head diminished by Re 1. What was the original expenditure of the mess?
Solution: let the original average expenditure be Rs.x. then,
42 (x - 1) - 35x = 42
= 7x = 84
= x = 12
Therefore original expenditure =Rs.(35 × 12)=Rs.420.