A percent is a mathematical relationship between two quantities where one quantity is a particular portion of the other quantity. This relationship is always defined within the context of a certain number of parts out of 100 parts, where the first number is the percentage.

Percentages can be represented as a number followed by the % sign, a fraction, or a decimal, but they all mean the same thing.

You can determine what percent a quantity is of another quantity in two ways:

**Algebraic Expression**

**Proportion**

*A* is the original quantity. In the algebraic expression, we are seeing what portion of it is equal to *B*.

*B* is the quantity that is some portion of *A*. This is why in the algebraic expression *A* is being multiplied by a percentage.

*x* determines the percentage:

Percentages can also be written as a decimal:

Notice that the proportion is actually the same thing as the algebraic expression. The only difference is that the A, or original quantity, is switched to the other side of the equals sign by dividing both sides by A. The variable A usually represents a larger number than the variable B, but this may not always be the case.

*If you have a decimal in the numerator, move the decimal to the right until you no longer have a decimal, and then add a zero to the denominator for each time you moved the decimal.

Increasing or Decreasing Percentages

In order to increase or decrease a value by a percent, keep these formulas in mind:

Word problems involving percentages will require you to be able to set up equations or proportions that accurately reflect what is being stated in the problem. Make sure you understand to what the percentage in the problem is referring.

**For example**

Store A and Store B both sell pet supplies. In March, Store A sold 30% more cat litter than Store B. If Store A sold 390 bags of cat litter in March, how many bags did Store B sell?

Since Store A sold 30% more than Store B, we can use the percent increase formula such that Store B is the original and Store A is the percent increase.

- 20% of 45 is 30% of what number?
- A class has a ratio of 3 boys to 2 girls. What percentage of the class is girls?
- Bobby made a pitcher of 2 cups of lemonade that contains 30% sugar. He wants to reduce the amount of sugar. If he adds one cup of water without changing the amount of sugar, what will the new percentage of sugar be?
- Rebecca can run a mile in 10 minutes. She wants to increase her speed by 40%. If she meets her goal, how quickly will she be able to run a mile (round to the nearest tenth)?
- What is 10% more of ¼ of 252?

**Answers to Practice Problems**

- 30
- 40%
- 20%
- 7.1 minutes
- 69.3

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