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Averages

Problems involving averages are very common on the GRE. They can be classified into four major categories as follows.

The average of N numbers is their sum divided by N, that is, average = $\frac{Sum}{N}$.

Example 1:

 Column A x > 0 Column B The average of x, 2x, and 6 The average of x and 2x

By the definition of an average, Column A equals $\frac{x+2x+6}{3}$ = $\frac{3x+6}{3}$ = $\frac{3\left(x+2\right)}{3}$ = x + 2, and Column B equals $\frac{x+2x}{2}$ = $\frac{3x}{2}$. Now, if x is small, then x + 2 is larger than $\frac{3x}{2}$. But if x is large, then $\frac{3x}{2}$is larger.(Verify this by plugging in x = 1 and x = 100.)

Weighted average: The average between two sets of numbers is closer to the set with more numbers.

Example 2:
If on a test three people answered 90% of the questions correctly and two people answered 80% correctly, then the average for the group is not 85% but rather $\frac{3Â·90+2Â·80}{5}$ = $\frac{430}{5}$ = 86. Here, 90 has a weight of 3â€”it occurs 3 times. Whereas 80 has a weight of 2 â€”it occurs 2 times. So the average is closer to 90 than to 80 as we have just calculated.

Using an average to find a number.

Sometimes you will be asked to find a number by using a given average. An example will illustrate.

Example 3: If the average of five numbers is â€“10, and the sum of three of the numbers is 16, then what is the average of the other two numbers?

(A) â€“33

(B) â€“1

(C) 5

(D) 20

(E) 25

Let the five numbers be a, b, c, d, e. Then their average is 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: $\frac{16+d+e}{5}$ = -10. Solving this equation for d + e gives d + e = â€“66. Finally, dividing by 2 (to form the average) gives $\frac{d+e}{2}$ = -33. Hence, the answer is (A).

Average Speed =

Although the formula for average speed is simple, few people solve these problems correctly because most fail to find both the total distance and the total time.

Example 4:
In traveling 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

(B) $531}{2}$

(C) 55

(D) 56

(E) $571}{2}$

The total distance is 1 â€¢ 50 + 3 â€¢ 60 = 230. And the total time is 4 hours. Hence,

Average Speed= = $\frac{230}{4}=571}{2}$

The answer is (E).
Note, the answer is not the mere average of 50 and 60. Rather the average is closer to 60 because he traveled longer at 60 mph (3 hrs) than at 50 mph (1 hr).