# Percents

## What is Percent?

Percent is a fraction in which the denominator is 100.

In French, “cent” means one hundred.

So, the fraction 20/100 can also be expressed as 20 per 100 or 20 per cent or 20%

∴ When we say that the population of India increased by 20% in the last decade, what we mean is that for every 100 Indians ten years ago, there are now 20 more, or 120 in all.

## Conversions

### Converting a fraction to a percent

To convert a fraction to a Percent, simply multiply the fraction by 100 and append the % symbol at the end.

Example: Converting the fraction ¼ to a percent

¼ × 100% = (100/4)% = 25%

### Converting a percent to a fraction

To convert a percent to a fraction, remove the % sign and divide by 100

Example: Converting 20% to a fraction

20% = 20/100 = 1/5

### Converting a percent to a decimal

A percent value can also be expressed a decimal. This notation is sometimes used in calculations.

To convert a percent to a decimal, remove the % sign, divide by 100, and then work out the answer as a decimal value.

Example: Converting 20% to its decimal equivalent

20/100 = 0.20

This means that finding 20% of a value is the same as multiplying the value by 0.20

Likewise, when we say that a quantity increases by 20%, we are adding 0.2 times the old value to the original.

i.e. (1 + 0.20), which means that we are effectively multiplying by 1.20

Therefore, a 40% increase means 1.40 times original, a 90% increase means 1.90 times original, a 270% increase means 3.70 times original (in the last instance, we are doing 1 + 2.70 = 3.70)

On the other hand, if there is a decrease of say 30%, we are subtracting 0.30 times a value from itself,

which in effect is (1 – 0.3 ) = 0.70 times the original value.

Also remember that a 40% decrease is the same as calculating 60% of, or 0.60 times, the original.

So, a 65% decrease means 0.35 times the original value, and a 99% decrease means 0.01 times original value.

Example:

A man purchased a house for $400,000, but had to sell it later at 15% lower value. How much did he sell the house for ?

Solution :

**Method 1 **

15% of 400,000 = 0.15 × 400,000 = 60,000

Therefore, selling price of the house = $ 400,000 – $ 60,000 = $ 340,000

**Method 2**

15% less means 85% of, or 85/100 times the value.

Therefore, selling price = (85/100) × $ 400,000 = $ 340,000

## Using standard fractions instead of percents

For some commonly used percent values, given below are the equivalent fractions that can be applied to calculate the new value after a given percentage increase or decrease on the old value. You may memorize these results in order to perform such percent calculations faster.

**To increase by Multiply by **

16.66% 7/6

20% 6/5

25% 5/4

33.33% 4/3

40% 7/5

50% 3/2

60% 8/5

66.66% 5/3

100% 2

**To decrease by Multiply by **

16.66% 5/6

20% 4/5

25% 3/4

33.33% 2/3

40% 3/5

50% 1/2

60% 2/5

66.66% 1/3

For example:

An item costs $72. The price of the item increases by 16.66%. Find the new price.

Solution:

Instead of multiplying by 1.1666 (which is a 16.66% rise), we multiply by the fraction given in the table i.e. 7/6

∴ New price = 72 × 7/6 = 84

## Can we offset +ve and -ve percent changes?

*Question*

On Monday, the price of gold increases 10%. On Tuesday, the price of gold drops 10%.

Are we back to Monday’s opening price?

*Answer*

No. Even though the % increase and % decrease are the same, they are not applied to the same value.

Let’s assume Monday’s opening price of gold is $1000 per ounce.

The increase of 10% is on this value of $1000 i.e. an increase of $100, which makes the value $1100.

Now, the decrease of 10% is on this new value $1100, which works out to $110, giving a closing price of $1100 - $110 = $990

From the above example, we realize that an **increase of *** x%* cannot be offset by a subsequent

**decrease of**

*on the new value.*

**x%**## Working out a % decrease to offset a % increase

Example:

The price of apples increases by 25%. What should be the % decrease in consumption of apples so that the total amount spent remains unchanged?

Solution:

The total amount spent on apples is the product of two quantities, the cost per apple and the quantity purchased.

Total spent = Cost of an apple × Quantity of apples purchased

As the cost per apple increases by 25%, it is obvious that the other term in the product, i.e. the quantity purchased will have to decrease so that total spending remains unchanged.

We can make use of a formula for this purpose.

To offset an x% increase in one quantity that constitutes the product, we decrease the other by

[x/(100 + x)] × 100%

∴ To offset a 25% increase in one quantity, we reduce the other quantity by

[25/125] × 100% = 20%

## Commutative property of percents

*Question*

Is a% of b = b% of a?

*Answer*

Yes. a% of b = [a/100]× b. This can be written as a × b/100 = [b/100] × a = b% of a

## Profit and Loss

Profit = Selling Price (SP) – Cost Price (CP) [when SP exceeds CP]

Loss = Cost Price – Selling Price [when CP exceeds SP]

% Profit = Profit / Cost Price × 100

% Loss = Loss / Cost Price × 100

## Applying Discounts

Mark-up Price is the listed price or the showroom price.

Sometimes discounts are given on the mark-up price, so that the actual selling price is lower than the listed price. Percentage discount is always calculated on the mark-up price (the original selling price)

Example:

A shopkeeper marks-up an item costing $200 by 40% and then offers a discount of 25%. Find his profit.

Solution:

Mark-up price = 200 × 1.4 = 280

Discount is 25%, calculated on mark-up price = 0.25 × 280 or ¼ × 280 = $70

∴ Selling price = 280 – 70 = 210

∴ Profit = 210 – 200 = $10

When successive discounts are given, each discount is applied on the discounted price so far.

Example:

The mark-up price of an item is $600. Two successive discounts of 25% and 10% are offered. What is the final selling price ?

Solution:

1st discount = 25% on 600 = $150

Discounted price after 1st discount = 600 – 150 = 450

2nd discount = 10% on $450 = $45

Final selling price = 450 – 45 = $405

[Note : This does not work out to the same value as a single discount of 25% + 10% = 35% on the mark-up price]