## 1. Average

The average of $N$ numbers is their sum divided by $N$, that is, $${Average} = \left(\dfrac{sum}{N}\right)$$

#### Example 1

What is the average of $x$, $2x$, and $6$?

A : $\dfrac{x}{2}$

B : $2x$

C : $\dfrac{x + 2}{6}$

D : ${x + 2}$

E : $\dfrac{x + 2}{3}$

#### Solution:

By the definition of an average, we get:

$\dfrac{x + 2x + 6}{3} = \dfrac{3x + 6}{3}$

$= \dfrac{3 (x + 2)}{3} = x + 2.$

Hence, the answer is ${x + 2}$ or Option (D)

## 2. 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 $\dfrac{3 \times 90 + 2 \times 80}{5} = \dfrac{430}{5} = 86.$

Here, 90 has a weight of 3 => it occurs 3 times.

Whereas 80 has a weight of 2 => 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 -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 $\left(\dfrac{a + b + c + d + e}{5}\right) = 10.$

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: $\left(\dfrac{16 + d + e}{5}\right) = 10.$

Solving this equation for $d + e$ gives $d + e = -66$.

Finally, dividing by 2 (to form the average) gives $\left(\dfrac{d + e}{2}\right) = -33.$

Hence, the answer is A: -33

## 4. Average Speed

$$\text{Average Speed} = \left(\dfrac{\text{Total Distance}}{\text{Total Time}}\right)$$

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 $\dfrac{2xy}{x+y}$ kmph.

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:

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 $1 \times 50 + 3 \times 60 = 230$. And the total time is 4 hours. Hence,

\begin{align*} \text{Average Speed} &=\left(\dfrac{\text{Total Distance}}{\text{Total Time}}\right)\\ &= \dfrac{230}{4} \\ &= 57.5 \end{align*}

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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