A percentage is a value divided by 100, shown with the percent symbol (%). 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.
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.)
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
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.