Percent of Change: Solving Percentage Word Problems

percentofchange“Percent of change” is a term most often used in mathematical word problems. It is used specifically when students are being asked to calculate the difference between two numerical values, in terms of percentage. A few different methods exist for calculations like these, and with enough practice, people can calculate percent of change without pen, paper, or calculator. The ability to understand percent of change is a valuable life skill, and it will come in handy more often than many other math lessons encountered during your school career. If you’re unfamiliar with percent of change, this guide will help you to deconstruct percentage values, and solve the word problems involving them.

Percentages

Understanding the nature of percentage is the first step to breaking down the concept of percent of change. Math students are taught very early on that our number system operates from a base of ten. When counting, every time you accumulate a group of ten, you move up to a different numerical group. For example, when counting to ten, you start over, in a sense, once you get to eleven, which is just another way of saying ‘ten plus one more.’ In this sense, the number twenty-two is really another way of saying ‘two sets of ten, plus two more.’ In the same way, ten groups of ten becomes one set of a hundred, ten groups of a hundred becomes a thousand, and so on, infinitely, and all the number groups continue to operate on a base of ten.

The concept of percentage was developed as a way of expressing ratios, based on a scale of 0 to 100, with 100 percent (100%) being equal to a whole. Percentages can be expressed as whole numbers, like 50%, fractions, like 1/2 , or decimals, like .5. Percentages are always expressed so that 100 percent is equal to the entire amount or number being discussed. Each of these expressions has the same numerical meaning, but the kind of calculations for which they are used can vary a great deal. Using one object, a pie for example, here are the three most commons ways in which we can express a percentage:

If somebody eats one pie, leaving no left overs, and not sharing with anybody else, he or she has eaten 100% of the pie. If he or she eats half of the pie, he or she has consumed 50% of it, and if he or she eats only a quarter of the pie, he or she has eaten 25% of the pie. Since 50 is one half of 100, 50% means one half of whatever you’re talking about, and since 25 is one quarter of 100, 25% means one quarter of whatever you’re talking about.

The second way to express a percentage is as a decimal, where the number 1 is equal to 100%, and anything less than 100% is a decimal, or a number between 0 and 1. If you eat an entire pie, the way to express it as a decimal is 1.0 pie, or simply 1 pie. If you eat half of the pie, then you’ve eaten .50 of the pie, because .50 is the decimal that represents half of the number 1. If you eat a quarter of the pie, you’ve eaten .25 of it, because .25 is the decimal that represents a quarter of the number 1.

The third way to express a percentage is as a fraction. In terms of fractions, a quarter is ¼, a half is ½, and a whole is 1/1, or simply 1. These fractions are equivalent to 25%, 50%, and 100%. The fractions are simplified forms of 25/100, 50/100, and 100/100.

To sum up:

One quarter = ¼ = 25/100 = .25 = 25%

One half = ½ = 50/100 = .50 = 50%

One whole = 1/1 – 100/100= 1.0 = 100%

Expressing and Calculating a Percentage

As you can see, the base number of 100 is very important when switching between fractions, decimals, and percentages. So, how do you work with numbers that aren’t 1, 10, or 100? Calculating percentages becomes a bit difficult depending on the numbers you’re working with. Let’s look at an example with the number 20. How we would answer a question that asks us to find 15% of 20?

To find 15% of 20, we can use both decimals and fractions. To use decimals to answer the question, convert the percentage that you’re looking for into a decimal:

15% = .15

Then multiply that decimal by the whole number you’re working with. The product will be your answer:

.15 x 20 = 15% of 20

.15 x 20 = 3

15% of 20 = 3

To find the answer using fractions, convert the percentage that you’re looking for into a fraction.

15% = 15/100

Then multiply that fraction by the whole number you’re working with. The product will be your answer, just like the decimal method:

15/ 100 x 20 = 15% of 20

15/ 100 x 20/1 = 300/ 100

300/100= 3/1 = 3

15% of 20 = 3

This method is different because it requires multiplying a fraction by a whole number. This is accomplished by turning the whole number into a fraction, then multiplying the numerators and denominators to create a new fraction. In this example, the whole number 20 becomes the fraction 20/1.

Determining Percent of Change

When a mathematical word problem asks you to determine a percent of change, it will present you with two numerical values, and ask you to determine the difference between the values in terms of the first value. In other words, if A is the first value and B is the second value, the percent of change is the value of B-A, written as a percentage of A.

For example, how would we find the percent of change between 25 and 30? First, we need the numerical difference between the numbers. In this case, 30 – 25 = 5. Then, we need to use the following formula to find the percent of change:

amount of change =  percent change

­­­original amount               100

All that’s left to do is substitute the values we have into this equation, and cross-multiply the fractions:

5     =   percent change

25           100

5 x 100 = 25 x percent change

500 = 25 x percent change

500 / 25 = percent change

percent change = 20

According to our calculations, the difference between 25 and 30 is 20% of 25. In other words, Increasing 25 by 20% will give us 30.

Real World Applications

The most common application of percent change in every day life occurs when shopping. Almost all retail sales are based on a percentage change in the price of a product or item. If you know how to calculate the percent of change, you can decipher what an item will cost you when it’s on sale. For example, if a $40 jacket is on sale for 15% off, you can use the percent change formula to find the new price:

amount change =  15% change

$40                             100

amount change x 100 = 15 x 40

amount change x 100 = 600

amount change = 600/100

amount change = 6

The amount change is $6, so the sale price of the $40 jacket is 40 – 6, or $34.

As long as you can fill in three of the terms in the percent of change formula, you can find the fourth term using cross-multiplication. You can find the change in numerical amount, the change in percent, the original amount, or the new amount.

This percent of change formula will help you decipher word problems in an academic sense, but as you can see, it will also enable you to understand the use of percentages in the real world. Understanding how this and other mathematical formulas work will allow  you to keep track of your finances using your own math skills.