Percentages
What is a Percentage?
A percentage is a way of expressing a number as a fraction of 100. The word “percent” literally means “per hundred,” which indicates that a percentage is a relative value representing parts of a whole divided into 100 equal parts.
Understanding the Basics:
- 1% of a number is equivalent to 1/100 of that number.
- This means that any percentage can be understood as a fraction with 100 as the denominator. For instance:
- 29% means 29 parts out of 100, or 29/100 of the total value.
- 81% is 81/100 of the total value.
- 300% represents 300/100, which is equal to thrice the original value.
Calculating Percentages:
To calculate a percentage of a number, simply multiply the number by the percentage (expressed as a decimal or fraction). This allows you to determine what portion of the whole is represented by the percentage.
Example 1: To find 25% of 60
25% x 60 = 25/100 x 60 = 25 x 60 ÷ 100 = 15
Example 2. To find 40% of 80
40% x 80 = 40/100 x 80 = 40 x 80 ÷ 100 = 32
Converting Percentages to Decimals
To convert a percentage into its decimal form, simply shift the decimal point two places to the left and remove the percent sign. This process changes the percentage into a decimal that represents the same value.
For example:
- 75% becomes 0.75 when you move the decimal two places to the left.
- 5% becomes 0.05 after the conversion.
Converting Decimals to Percentages
To convert a decimal into a percentage, shift the decimal point two places to the right and then add a percent sign. This process transforms the decimal into a percentage that represents the same value.
For example:
- 0.85 becomes 85% when you move the decimal two places to the right.
- 0.07 becomes 7% after the conversion.
Converting Fractions to Percentages
To convert a fraction into a percentage, start by converting the fraction to a decimal. Once you have the decimal, you can then easily convert it into a percentage by moving the decimal point two places to the right and adding a percent sign.
For example:
- 1/4 becomes .25 and can be converted 25% by moving the decimal two places to the right
- 4/10 becomes .4 and can be converted to 40% by moving the decimal two places to the right.
Operations on Percent
Addition and Subtraction of Percent
- Adding and subtracting percentages is done just like with regular numbers; simply perform the addition or subtraction, then add the % symbol to your answer.
Multiplication and Division of Percent
- When multiplying or dividing percentages, start by converting the percentages into fractions with 100 as the denominator, like 38% is 38/100. After converting, proceed with the multiplication or division as you would with any fraction.
- In word problems, the term “of” typically indicates a multiplication operation, helping you identify where to apply this process.
- For example. 30% of 40 is the same is 30% x 40. To solve this, 30/100 x 40 = 12
- On the other hand, the word “is” can be understood as representing equality (=)
- For example, 20% of a number is 10. This is the same as 20% of x (n) is 10. To solve for this, 20/100 x (n) = 10. n = 10 ÷ 20/100. n = 10 x 100/20 = 50
Percent, Rate and Base
Percentage
- The percentage represents the ratio of a part to the whole.
- This value can be expressed as a decimal, fraction, or in percentage (%) form.
- It shows what portion of the base is being considered.
Rate
- The rate is the specific percentage or ratio used to determine the part of the base.
- It is often expressed as a percentage (%) and is used to calculate how much of the base is taken or considered.
Base
- The base is the original total or whole number from which a part (percentage) is taken.
- It serves as the reference point for calculating the percentage, and it is the value on which the rate is applied.
Together, these concepts help in understanding and solving problems involving percentages, where the base is the total amount, the rate is the percentage applied to that amount, and the percentage represents the resulting portion of the base.
Solving for Percentage, Rate and Base
Percentage (P) = (R) x (B)
Rate (R) = Percentage (P) / Base
Base (B) = Percentage (P) / Rate (R)
Clue words to identify which is Percentage, Rate or Base
Identifying Percentage, Rate and Base in Word Problems
Identifying the Base, Percentage, and Rate in Word Problems
To accurately interpret word problems and determine which values represent the base, percentage, or rate, it’s essential to recognize the key clue words within the problem. Common clues include:
“Is”: This word typically indicates equality and helps you identify the number that is equal to the percentage of the base. It usually corresponds to the part or the percentage.
“Of”: This word usually signifies multiplication and points to the base, the whole quantity from which a percentage is taken.
“What”: This term represents the unknown value you’re trying to find. It could refer to the base, percentage, or rate, depending on the context of the problem.
Example:
1. “What is 20% of 50?”
x = 20% x 50. To solve this, x = .2 x 50 = 10
2. “What part of 10 is 4?”
(n) x (10) = 4. To solve this, n = 4 ÷ 10. n = .4 or 40%
3. “30% of what is 6?”
30% x (n) = 6. To solve this, (.3) (n) = 6. Then, n = 6 ÷ .3 = 20
4. “0.75 of 8 is what?”
.75 x 8 = n. To solve this, n = .75 x 8 = 6
By understanding these key terms, you can effectively break down word problems and determine the base, percentage, or rate with confidence.
Percentage, Rate and Base Exercises
1. What is 15% of 80?
2. What part of 25 is 5?
3. 40% of what is 16?
4.0.6 of 50 is what?
5.What is 25% of 120?
6. What part of 60 is 18?
7. 50% of what is 10?
8. 0.25 of 32 is what?
9. What is 10% of 150?
10. What part of 200 is 50?
Answers
1. 12
2. 20%
3. 40
4. 30
5. 30
6. 30%
7. 20
8. 8
9. 15
10. 25%
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