Averages

The average of ‘n’ numbers is sum of the numbers divided by n. So to find the sum of ‘n’ values we multiply their average with total number of values.

A_n = ( x₁ + x₂ + x₃ …) / n    AND   A_n * n = Sum

Calculation of weighted average:

If we want to calculate the weighted average or combined average of two groups whose individual averages are A1 and A2 and strengths are n1 and n2.

Then W_avg = ( n₁ * A₁ + n₂ * A₂ ) / ( n₁ + n₂ )

Visualize averages:

Suppose we are given 4 numbers 10, 11, 12, 13 and told to calculate the average which is 11.5. This also means that we can replace all numbers with the average and not affect the total viz. 11.5, 11.5, 11.5, 11.5 shall become the new series that shall be same as old one.

This can be used to solve sums. Suppose you have to find a fifth number that increases their average by 0.5.

So new series is 11.5+0.5, 11.5+0.5, 11.5+0.5, 11.5+0.5, 11.5+0.5. So X has to be such that it is 11.5 + 5 * 0.5 = 14

Alligations

Assume two groups with Averages A₁, A₂ and number of members n₁ and n₂. Then the weighted average of the groups Aw is:

A_w = ( n₁ * A₁ + n₂ * A₂ ) / ( n₁ + n₂ )

n₁ * A_w + n₂ * A_w = n₁ * A₁ + n₂ * A₂

n₁ ( A_{w –} A₁ ) = n₂ ( A₂ – A_w)

n₁ / n₂ = ( A₂ – A_w) / ( A_{w –} A₁ )

Note: For convenience when two averages are given we taken smaller value as A₁ and other value as A₂.

Percentage

Calculate Absolute value change or percentage change:

If sales of a year is Rs. 1000 and next year is Rs. 2000 then the absolute value change is Rs. 1000 and the percentage change is:

%change = (Absolute value change) * 100 / Original quantity

Calculate percentage change and percentage point change:

If earlier Yes bank offered 6% interest rate on savings and now offers 4% interest. Then calculate percentage change and percentage point change.

%point change = 6 – 4 = 2%

%change in savings rate = (6 – 4) / 2 * 100 = 100% reduction.

Effect of percentage change on a ratio:

The ratio has a relation with the numerator, when the numerator increases the ratio increases. Thus increase in the numerator only keeping the denominator constant will increase the ratio.

11 / 17 is 10% more than 10 / 17.

Calculation of successive percentage changes:

A’s salary increases by 20% and then decreases by 20%. What is the net percentage change in salary of A.

To calculate this we assume A’s initial salary is 100 which increases by 20% so ( + 20 ) and then becomes 120. Then it decreases by 20% so becomes 96 (-24). Hence net change is +20 – 24 = -4% change or 4% decrease.

A trader gives successive discounts of 10, 20, 10% to find the net percentage change.

To calculate this we should assume original price as 100. Then 10% discount makes it 90 (-10), 20% discount on 90 is 72 (-18) and then 10% discount on 72 is 64.8 (-7.2) so net change is = -10-18-7.2 = -35.2 or 35.2% reduction.