An average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value. Most of us understand the word " average" because everyday use typically refers to numbers or groups which have a Bell Curve or normal distribution - for example people's heights, or their blood pressure measurements.
Formulas:
1. Average:
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Average = |
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Sum of observations |
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Number of observations |
2. Average Speed:
Suppose a man covers a certain distance at x kmph and an equal distance at y kmph.
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Then, the average speed druing the whole journey is |
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2xy |
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kmph. |
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x + y |
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Important Statements and Equations for "Problems based on Ages":
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Examples:
1. If the average of five numbers is -10, and the sum of three of the numbers is 16, then what is the average of the other two numbers?
A: -33 B: -1 C: 5 D: 20 E: 25
Solution:
Let the five numbers be a, b, c, d, e.
Then their average is (a+b+c+d+e/5)=10.
Now three of the numbers have a sum of 16, say, a+b+c=16.
So substitute 16 for a+b+c in the average above: (16+d+e/5)=10.
Solving this equation for d+e gives d+e=−66.
Finally, dividing by 2 (to form the average) gives (d+e/2)=−33
Hence, the answer is A: -33
2. In travelling from city A to city B, John drove for 1 hour at 50 mph and for 3 hours at 60 mph. What was his average speed for the whole trip?
A: 50.0
B: 53.5
C: 55.0
D: 56.0
E: 57.5
The total distance is 1×50+3×60=2301×50+3×60=230. And the total time is 4 hours. Hence,
Average Speed=(Total DistanceTotal Time)=2304=57.5Average Speed=(Total DistanceTotal Time)=2304=57.5
Thus, the answer is Option (E): 57.5
Note, the answer is not the mere average of 50 and 60. Rather the average is closer to 60 because he travelled longer at 60 mph (3 hrs) than at 50 mph (1 hr).
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3. The average of first five multiples of 3 is: |
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A. |
8 |
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B. |
9 |
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C. |
10 |
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D. |
11 |
Solution:
Option(B)
is correct
Basic Formula: 1,2,3... n
If n is odd, the formula is (n+12) th term
The five multiples of 3 is 3,6,9,12,15
(n+12)⇒(5+12)th term
⇒(62)th term = 3rd term
Here 3rd term is 9
4. 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 40kg and that of section B is 35kg, find the average weight of the whole class.
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A. |
30.00 kg |
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B. |
35.00 kg |
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C. |
37.25 kg |
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D. |
42.50 kg |
Solution:
Option(C)
is correct
Total weight of (36+44=80)(36+44=80) Students =(36×40+44×35)=(36×40+44×35) kg = 2980 kg
Therefore average weight of the whole class =(2980/80) kg
Therefore average weight = 37.25kg
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5. The average of 50 numbers is 30. If two numbers, 35 and 40 are discarded, then the average of the remaining numbers is nearly: |
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A. |
28.32 |
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B. |
29.68 |
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C. |
28.78 |
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D. |
29.27 |
Solution:
Option(B)
is correct
Total sum of 48 numbers =(50×30)−(35+40)=(50×30)−(35+40)
=1500−75
=1425
Average =(142548)
= 29.68
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6. The ages of the two persons differ by 20 years. If 5 year ago, the older one be 5 times as old as the younger one, then their present ages, in year are: |
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A. |
25, 5 |
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B. |
30, 10 |
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C. |
35, 15 |
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D. |
50, 30 |
Solution:
Option(B)
is correct
Let the age be x and y years now.
Then, x−y=20 -------- (i)
and (x−5)=5(y−5) ------- (ii)
On solving both equation we get:
x=30 and y=10
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7. A student was asked to divide a number by 6 and add 12 to the quotient. He, however first added 12 to the number and then divided it by 6, getting 112 as the answer. The correct answer should have been: |
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A. |
114 |
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B. |
118 |
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C. |
122 |
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D. |
124 |
Solution:
Option(C)
is correct
Let the number be x, then operations undertook by the student:
=(x+126)=112
⇒x=660
Correct answer:
=660/6+12
=122
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8. Two spinning machines A and B can together produce 3,00,000 m of cloth in 10 hour, if machine B alone can produce the same amount of cloth in 15 hour, then how much cloth can machine A produce alone in 10 hour? |
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A. |
2,00,000 m |
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B. |
1,00,000 m |
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C. |
1,50,000 m |
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D. |
50,000 m |
Solution:
Option(B)
is correct
Machines A and B together will produce 30,000 m of cloth in 1 hour.
Machine B alone can produce 20,000 m cloth in 1 hour.
Therefore, Machine A can produce 10,000 m cloth in 1 hour.
So, in 10 hour Machine A can produce 1,00,000 m of cloth.
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9. The sum of A and B's age is 43 years. 11 year hence, A's age will be 7/6 times B's age then. Find B's present age. |
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A. |
22 years |
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B. |
20 years |
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C. |
24 years |
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D. |
19 years |
Solution:
Option(D)
is correct
Let A's age be x and B's age be y.
x+y=43-------- (i)
(x+11)=76(y+11)
⇒6x−7y=11-------- (ii)
On solving both the equations, we get:
y=19 years
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10. The age of Mr. Chetan in 2002 was 1/90 of his birth year. What is his age in 2006? |
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A. |
30 |
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B. |
28 |
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C. |
26 |
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D. |
22 |
Solution:
Option(C)
is correct
Let age of Chetan in 2002 = x
So,
2002−x/90=x
⇒x=22
So, Chetan’s age in 2006 is,
=22+4=26 yrs
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