# Primary Mathematics/Percentages

## Percentages[edit | edit source]

### Introduction[edit | edit source]

A percentage is a value divided by 100, shown with the percent symbol (%). Percentage comes from two Latin words,per meaning *for each* and cent meaning *100*.
For example, 4% is the same as the decimal value 0.04 or the fraction 4/100 (which could also be reduced to 1/25).

### Uses for percentages[edit | edit source]

Traditionally, percentages are used when dealing with changes in a value, particularly money. For example, a store may have a 20% off sale or a bank may charge 7.6% interest on a loan.

### Defining the base[edit | edit source]

The **base** of a percentage change is the starting value. Many errors occur from using the wrong base in a percentage calculation.

### Example[edit | edit source]

If a store has an $100 item, marks it 20% off, then charges 6% sales tax, what is the final price ?

First, calculate 20% of $100. That's 0.20 × $100 or $20. We then subtract that $20 from the original price of $100 to get a reduced sale price of $80.

Now we add in the sales tax. Here's where the tricky part comes in; what is the base ? That is, do we pay 6% tax on the original $100 price or on the reduced $80 price ? In most places, we would pay tax on the reduced sales price, so $80 is the base. Thus, we multiply $80 × 0.06 to get $4.80 and add that to $80 to get $84.80 for a final price.

Notice that even though we took off 20% and then added back in 6%, this is **not** the same as taking off 14%, since the 20% and the 6% figures each had a different base. If the 6% sales tax did apply to the original full price, however, then both percentages would have the same base and the total reduction in price would, indeed, be 14%, bring the price down from $100 to $86.

### Terms used with percentages[edit | edit source]

- If you take 20%
**off**an amount (or a 20% reduction), that means the new price is 20% less than the original (100%) price, so it's now 80%**of**the original price.

- If you apply 20%
**interest**to an amount, that means the new price is 20% higher than the original (100%) price, or 120%**of**the original price. (Note that this is**simple interest**, we will consider**compound interest**next.)

### Compound interest[edit | edit source]

**Simple interest** is when you apply a percentage interest rate only once.

**Compound interest** is when you apply the same percentage interest rate repeatedly.

For example, let's say a $1000 deposit in a bank earns 10% interest each year, compounded annually, for three years. After the first year, $100 in interest will have been earned for a total of $1100. In the second year, however, there is not only interest on the $1000 deposit, but also on the $100 interest earned previously. This "interest on your interest" is a feature of compounding. So, in year two we earn 10% interest on $1100, for $110 in interest. Add this to $1100 to get a new total of $1210. The 10% interest in the third year on $1210 is $121, which gives us a total of $1331.

For those familiar with powers and exponents, we can use the following formula to calculate the total:

T = P x (1 + I)^{N}

Where:

T = final Total P = initial Principal I = Interest rate per compounding period N = Number of compounding periods

In our example, we get:

T = $1000 x (1 + 10%)^{3}= $1000 x (1 + 0.10)^{3}= $1000 x (1.10)^{3}= $1000 x (1.10 x 1.10 x 1.10) = $1000 x (1.331) = $1331

More complex calculations involving compound interest will be covered in later lessons.