## Averages and Percentages

Formula

The average of a list of items $$x_1, x_2, x_3 . . . x_n$$ is given by the sum of the values divided by n. Hence, the average of 34, 45, 67, 43, 78 is equal to
average = (34 + 45 + 67 + 43 + 78) / 5 = 267 / 5 = 53.4.

Tip

To quickly calculate average of a list of items try to guess where the average is likely to lie. For e.g. for the list 34, 45, 67, 43, 78 the average is likely to be around 50. It does not matter what number you choose as long as it is easy to calculate the difference of the numbers with the given number.

Then calculate the sum of the differentials around this assumed average.
In this case the differentials are 34-50, 45-50, 67-50, 43-50, 78-50 i.e. -16, -5, 17, -7, 28. Thus the sum of the differentials= 17. The point here is that the different differentials cancel each other out and we get a sum close to zero.

Then divide the sum of differentials by n and add it to the assumed average to get the real average.
Hence, real average = 50 + 17/5 = 50 + 3.4 = 53.4.

So if we have to calculate the average of 103, 102, 96, 99, 120 we can quickly calculate it as 100 + 20/5 = 104.

Formula

The weighted average of n terms equals $$\frac{w_1*x_1+ w_2*x_2+. . .+ w_n*x_n}{ w_1+ w_2+ . . . w_n}$$

Formula

Percentage Change = (Final Value – Initial Value)/Initial Value * 100

Shortcuts

For two successive changes of of x% and y%, the total % change is (x + y + xy/100)%

Formula

If a population has two type of entities say X and Y and average of total population is a and of X is b then average of Y is equal to
average of Y = (a(X+Y)-bX)/Y

For e.g. if a classroom has 20 boys and 10 girls and the average weight of the class is 30kg and the average weight of the boys is 32kg then the average weight of the girls is equal to
average = (30*30 - 20*32)/10 = (900-640)/10 = 26 kg.

Shortcuts

If a population has two type of entities say X and Y and average of total population is a and the average of X is $$a_x$$ and Y is $$a_y$$ then
$$\frac{\left | a-a_x \right |}{\left | a-a_y \right |} =\frac{ y}{x}$$. Hence, for earlier e.g.
$$\frac{\left | 30-32 \right |}{\left | 30-a_y \right |} =\frac{ 1}{2}$$.
Hence, |30 - $$a_y$$| = 4. As 32>30, $$a_y$$<30. Thus the average of y = 26.

Shortcuts

If A is x% more than B then B is x%/(1+x%) less than A.

For e.g.
If A is 20% more than B then B is 20%/120% = 16.66% less than A.

Solved Example

Ravi scored an average of 56% in all the twelve courses in school. In five of the courses he scored an average of 70%. What is his average in the other seven courses?

Using the shortcut,
|56-70| / |56-x| = 7/5 . Hence, |56-x| = 14*5/7= 10. As 56<70, 56>x to balance the surplus. Hence, x=56 - 10 = 46.

Solved Example

The sum of the number of boys and girls in a school is 150. If the number of boys is x, then the number of girls becomes x% of the total number of students. The number of boys is:

If girls are x% of the total number of students, boys will be 1-x% of the students. Hence, x/150 = 1-x%. Hence, x = 150 - 150x/100. Hence, 2.5x =150. Hence, x=60.

Thus, there are 60 boys.