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1. ACT math intro
2. ACT Math
2.1 Pre-algebra
2.2 Elementary algebra
2.3 Intermediate algebra
2.3.1 Essential modeling
2.3.2 Properties of logarithms
2.3.3 Prime factorization
2.3.4 Radicals and rational exponents
2.3.5 Solving a system of equations
2.3.6 Factorization of quadratics and cubics
2.3.7 Solving quadratic equations
2.3.8 Counting problems, permutations, and combinations
2.3.9 Percent change
2.3.10 Expected value of x
2.3.11 Imaginary and complex numbers
2.4 Plane geometry
2.5 Coordinate geometry
2.6 Trigonometry
3. ACT English
4. ACT Reading
5. ACT Science
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2.3.9 Percent change
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2. ACT Math
2.3. Intermediate algebra

Percent change

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In a previous section, Percentages, ratios, proportions, decimals, fractions, we talked about how to use percentages as decimals. In this section, we will be expanding on this skill. We will discover how to find out how much a number will change when we apply a percentage to it, especially by using algebra.

Percent increase and decrease

You have undoubtedly heard the phrase “50 percent increase.” What does this mean, exactly? Perhaps our first thought is that a 50% increase is our original number multiplied by 0.5. However, this is not quite right. We have to pay very close attention to the words increase and decrease.

Percent increase

When finding a percent increase, we must first recognize that in order for the number to increase, the number has to be at least 100%. So, to find this increase we will actually add our percentage of increase to a base of 100%. Let’s use the idea of a 50% increase in the example below.

Given that x=20, what is the value of x after a 50% increase?

Adding our 50% to a base of 100% the actual value we will multiply by our number is 150% or 1.5:

20∗1.5=30

Percent decrease

Finding the percent decrease of a number will also use a base percentage of 100%. Instead of adding our percentage, we will simply subtract it from the starting 100%. Let’s do a 30% decrease.

Given that x=12, what is the value of x after a 30% decrease?

(spoiler)

Subtract our 30% from the initial 100%:

100%−30%=70%

Now, find the new value by multiplying the decimal form of the percentage by the value of x:

(0.7)∗12=8.4

To help solidify this idea in your mind, take a close look at these percent changes of 10.

  • 10 decreased by 100%=0
  • 10 decreased by 50%=5
  • 100% of 10=10
  • 10 increased by 50%=15
  • 10 increased by 100%=20

% change in algebra

In algebra, it is common for you to be asked to find the percent change given one number and the value it is changed to. Let’s jump straight into an example.

When Jack started his job, he earned $7.00 an hour. Now, he earns $17.50 an hour. What is the % change in his wage?

We must set up an equation that represents this change. What we know is that the starting number is 7, which is multiplied by some percentage to result in 17.50. So how would that equation look?

7∗x=17.50

It is as simple as that! We just need to find x, which is 715​ or 2.5.

Now, this is a percent change of 250%, but what is the percent increase?

(spoiler)

% increase=150%

There is also a formula: % change=(old)[(new)−(old)]​

Let’s do the example above but use the formula instead. We should get the same answer.

(spoiler)

% increase​=(17.50−7.00)/7.00=10.50/7.00=1.5=150%​

This will also work if the new value is lower than the old value. In this case, you will always get a negative % change value, which is the same as a decrease.

% of another number

This form of percent change will require a bit more technique. We will actually be relating two different numbers and percent changes. Here is an example.

50% of 16 is 20% of what number?

What do you think we should do here? We definitely need to make an algebraic equation, so let’s start with that. 50% of 16 can be written as (0.5)∗16 and 20% of some number can be written as (0.2)∗x. Altogether, we get:

(0.5)∗16=(0.2)x

Now solve for x!

(spoiler)

x=8/(0.2)=40

Key points

% increase. Using a base of 100%, add the percent by which you are increasing your number, then multiply the decimal by the original value to get the increased value.

% decrease. Using a base of 100%, subtract the percent by which you are decreasing your number, then multiply the decimal by the original value to get the decreased value.

% change in algebra. Look for the numbers you are given to create an algebraic equation. Find out how much the number changes to get from the initial to final numbers, then you will find the percent change of the original number. From there, you can determine the percent increase or decrease by using your base value of 100%.

% change formula. % change=(old)[(new)−(old)]​

% of another number. You will need to create an algebraic equation where one number and its percentage is on either side of the equal sign. Then, solve for the missing variable using order of operations.

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