# Primary Mathematics/Percentages

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## Percentages

### Introduction

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

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

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

### Example

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

• 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

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.

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