Average Question is an important topic in GMAT problem solving and data sufficiency. Let us start with the Basics.
Average (Arithmetic Mean)
Average of n numbers a1, a2, a3, a4, a5....an
(An) = (a1+a2+.....an) /n
Example: Find the average of 34, 56, 75 and 83
a1 = 34
a4 = 83
Total Number of Elements (n) = 4
Average (An) = (a1+a2+a3+a4)/n
= (34 + 56 + 75 + 83)/4 = 62
Shortcut to Remember: An x n = a1+a2+.......an
Let us straight away apply this shortcut
Q) The average of four numbers is 20. If one of the numbers is removed, the average of the remaining numbers is 15. What number was removed?
Four Numbers = a1, a2, a3, a4
An = 20
An x n = a1 + a2 + a3 + a4
20 x 4 = a1 + a2 + a3 + a4 -> Statement 1
Let us assume that a4 has been removed
n = 3
15 x 3 = a1 + a2 + a3 -> Statement 2
Statement 1 – Statement 2
A4 = 35
Answer = D
So the point to remember here is Arithmetic Mean and Average are the same.
In GMAT, you are unlikely to encounter easy questions like the one above. A more probable question type is Weighted Average Question.
When we have groups of numbers whose individual averages are known then Weighted Averages will help us find the average of the entire group.
Boring theory! Let us dive into some formulas.
Suppose we have N groups with averages a1,a2,a3,a4 ....aN and n1,n2,n3.....nN represents the number of elements in each group then
Weighted Average = ( n1*a1 + n2*a2 + n3*a3 + ....nN*aN )/(n1+n2...nN)
Shortcut to Remember: Wn x nN = n1*a1+n2* a2+.......nN*aN
Try this question:
Q) The average rainfall for the week was 2.5cm. The first 2 days was 2.5 cm, the next 4 days was 2cm. What was the average rainfall for the last day of the week?
Wn x nN = n1*a1+n2* a2+.......nN*aN
Wn = 2.5cm
nN = 7
n1 = 2
a1 = 2.5cm
n2 = 4
a2 = 2cm
2.5 x 7 = 2 x 2.5 + 4 x 2 + 1 x a3
17.5 = 5+8+a3
Average rainfall for the last day of the week = 4.5 cm
Want a little more tricky type?
Q) John scored 45 out of 50 in his last two tests. In his first test, he topped the class with 50 out of 50. What is his average percentage score?
Hint: The only tricky part is to convert the scores to percentages before solving the question
GMAT Average Age Question
This is a common question seen in GMAT. You will be given the average age of a group of people, x years ago and another average x – n years ago.
Remember this rule
If average age of a group of people is x years
n years from now the average age of a group of people would be x + n
n years before the average age of a group of people would be x - n
Ready for a Question?
Q) The average age of a husband, his wife and son 3 years ago was 29 years and that of his wife and son 5 years ago was 21 years. What is the husband’s present age?
Answer: We don’t have the value of x but we have x-n
X – 3 = 29
X = 32
Average age of Husband Wife and Son, now is 32 Years
X = (Husband + Wife + Son)/3
A (Husband, Wife and Son) = 32
Husband + Wife + Son = 32 x 3 = 96 --> Statement 1
Let the average age of Wife and Son be Y
Y – 5 = 21
Y = 26
A (Wife, Son) = 26
Wife + Son = 26 x 2 = 52 --> Statement 2
Statement 1 – Statement 2
Husband = 96 – 52 = 44
Shortcut to Solve GMAT Weighted Average
If Leo bought two black shirts for $55 each and three blue shirts for $65 each, what is the average that Leo paid for the 5 shirts?
Average Leo Paid for the 5 Shirts = (2 x 55 + 3 x 65)/ 5 = 61
Where is the freaking shortcut? :)
Here is the deal.
In Weighted average question if the first average is lower than the second number, you don't have to do all the calculations.
Put the first average as 0
Second Number is greater than first by 10
Weighted Average (3 x 10)/5 = 6
Actual Weighted Average = Add 6 to 0 (or the first average) = 6 + 55 = 61 (Voila)
Weighted Average or Average Leo Paid for the shirts = 61
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