Calculating percents

Percents are probably the most common form of ratios we come across in everyday life. In this lesson, we will use what we just learned about percents to solve a few tasks in which percents frequently appear that are very common in real, everyday life.

Examples of calculation with percents

Example 1

The price of gas has risen from $3 per gallon to $4.20 per gallon. How many percents did the gas price rise?

To get to the result, we will first calculate the ratio in decimal form and then convert it into percents. Since percent is a ratio and we are interested how much bigger the new price is in relation to the old price, we will form this ratio:

(4.20 / 3) * 100 = x

This will give us the relative size of the new price in relation to the old price and express it using percents. So we get:

1.41 * 100 = x

X = 141 %

Now since we are only interested in how much bigger is the new price than the old price, we will subtract 100% (the relative size of the old price) from that number and get:

141% – 100% = 41%

The relative increase in price was 41%.

Example 2

You want to buy a mobile phone. The cost of a mobile phone was $400.00, the seller adds on a 20% markup but today they have a special 30% discount on the original sale price. There is also a 5% sales tax you have to pay when you buy the phone. What is the selling price of the mobile phone?

calculating percents

To solve this kind of a task, first you need to know that a markup is the difference between the cost of a good or service and its original selling price. The seller adds a percentage of the original cost to that original cost to form the sales price and that represents the seller’s profit. The sales tax is added as a percent of the price you pay (after markup and discounts).

Now, first we have to calculate the original sale price (OSP). We will do that by adding the markup to the original cost, which will be represented by the symbol OC. We will convert the percents into decimals to make the calculation simpler.

OSP = OC + 0.2 *OC

OSP = 1.2 *OC

OSP = 1.2 * $400 = $480

The original sales price is $480. Now it is time to calculate the discounted price (DP). The discount is 30% of the original sales price, so the discounted price will be:

DP = OSP – 0.3 * OSP

DP = 0.7 * OSP

DP = 0.7 * $480 = $336

Now we are just one step away from the final selling price (FSP). The only thing left to do is to add the sales tax to the discounted price:

FSP = DP + 0.05 * DP

FSP = 1.05 * DP

FSP = 1.05 * $336 = $352.8

Example 3

You won $10,000.00 on the lottery and decided to invest it in a savings account that will bring an annual 3% interest, compounded semiannually for 1.5 years. How much money will you have at the end of that period?

The first thing you should have in mind is that there are two main kinds of interest – simple interest and compound interest. Simple interest is calculated as a percent of the principal (the principal in this case is the $10.000 for viagra or levitra). Compound interest is also a percent of the principal, but it is added to the principal after compounding and the sum then represents the principal for the next calculation. What that actually means, we will explain through this example.

The $10.000 is the basic principal (C0) here. The 3% is the annual compound interest rate (i) and it is compounded semiannually for 1.5 years. Semiannual compounding means that the interest is being compounded (or added to the principal) two times a year or in other words – every 6 months. That also means that we have to adapt the interest rate. So, instead of compounding it 3% after a full year, we will compound it 1.5% every 6 months. And since the full period is 1.5 years, we will do it 3 times. So in the first iteration we will have something like this:

C1 = C0 * (1 + i)

C1 = $10.000 * (1 + 0.015)

C1 = $10.150

In the second iteration, we will use C1 as our principal and it will look like this:

C2 = C1 * (1 + i)

C1 = $10.150 * (1 + 0.015)

C1 = $10.302,25

And in the third and final iteration, we get:

C3 = C2 * (1 + i)

C1 = $10.302,25 * (1 + 0.015)

C1 = $10.456,78

And that is the final result. After 1.5 years at a 3% compound semiannual interest, if you invest $10.000, you will get $10.456,78.

If you wish to practice working with calculations of percents, please feel free to use the worksheets below.