ParaPro Math Study Guide: Percentages Practice

Understanding Percentages

Percentages represent parts per hundred and are a way to express proportions or fractions with a denominator of 100. As a paraprofessional, you’ll encounter percentages when discussing grades, test scores, attendance rates, and various statistical data in educational settings.

What is a Percentage?

The word “percent” means “per hundred” (from Latin “per centum”). A percentage is a number expressed as a fraction of 100.

The symbol % represents percent
25% means 25 out of 100, or 25/100, or 0.25
Percentages provide a standardized way to compare quantities

Converting Between Percentages, Decimals, and Fractions

Converting Percentages to Decimals

Remove the percent symbol (%)
Divide by 100 (or move the decimal point two places to the left)

Example 1: Convert 45% to a decimal

Step 1: Remove the percent symbol from 45%

Step 2: Divide by 100: 45 ÷ 100 = 0.45

Therefore, 45% = 0.45

Example 2: Convert 6.5% to a decimal

Step 1: Remove the percent symbol from 6.5%

Step 2: Divide by 100: 6.5 ÷ 100 = 0.065

Therefore, 6.5% = 0.065

Converting Decimals to Percentages

Multiply by 100 (or move the decimal point two places to the right)
Add the percent symbol (%)

Example 3: Convert 0.72 to a percentage

Step 1: Multiply by 100: 0.72 × 100 = 72

Step 2: Add the percent symbol: 72%

Therefore, 0.72 = 72%

Example 4: Convert 0.035 to a percentage

Step 1: Multiply by 100: 0.035 × 100 = 3.5

Step 2: Add the percent symbol: 3.5%

Therefore, 0.035 = 3.5%

Converting Percentages to Fractions

Remove the percent symbol (%)
Write the number over 100 as a fraction
Simplify the fraction if possible

Example 5: Convert 75% to a fraction

Step 1: Remove the percent symbol from 75%

Step 2: Write as a fraction with denominator 100: 75/100

Step 3: Simplify by dividing both numbers by their GCF (25): 75/100 = 3/4

Therefore, 75% = 3/4

Example 6: Convert 12.5% to a fraction

Step 1: Remove the percent symbol from 12.5%

Step 2: Write as a fraction with denominator 100: 12.5/100

Step 3: Express 12.5 as 125/10: 125/1000

Step 4: Simplify by dividing both numbers by their GCF (125): 125/1000 = 1/8

Therefore, 12.5% = 1/8

Converting Fractions to Percentages

Convert the fraction to a decimal by dividing the numerator by the denominator
Multiply the decimal by 100
Add the percent symbol (%)

Example 7: Convert 3/5 to a percentage

Step 1: Divide 3 by 5: 3 ÷ 5 = 0.6

Step 2: Multiply by 100: 0.6 × 100 = 60

Step 3: Add the percent symbol: 60%

Therefore, 3/5 = 60%

Example 8: Convert 7/8 to a percentage

Step 1: Divide 7 by 8: 7 ÷ 8 = 0.875

Step 2: Multiply by 100: 0.875 × 100 = 87.5

Step 3: Add the percent symbol: 87.5%

Therefore, 7/8 = 87.5%

Common Percentage Conversions

Percentage	Decimal	Fraction
10%	0.1	1/10
20%	0.2	1/5
25%	0.25	1/4
33.33…	0.333…	1/3
50%	0.5	1/2
66.67…	0.667…	2/3
75%	0.75	3/4
80%	0.8	4/5
100%	1.0	1

Finding the Percentage of a Number

Convert the percentage to a decimal (divide by 100)
Multiply the decimal by the number

Example 9: Find 35% of 80

Step 1: Convert 35% to a decimal: 35% = 0.35

Step 2: Multiply: 0.35 × 80 = 28

Therefore, 35% of 80 is 28

Example 10: What is 125% of 60?

Step 1: Convert 125% to a decimal: 125% = 1.25

Step 2: Multiply: 1.25 × 60 = 75

Therefore, 125% of 60 is 75

Note: Percentages greater than 100% represent values larger than the original.

Finding What Percentage One Number is of Another

Divide the first number by the second number
Multiply by 100
Add the percent symbol (%)

Example 11: What percentage of 50 is 12?

Step 1: Divide 12 by 50: 12 ÷ 50 = 0.24

Step 2: Multiply by 100: 0.24 × 100 = 24

Step 3: Add the percent symbol: 24%

Therefore, 12 is 24% of 50

Example 12: What percentage of 25 is 30?

Step 1: Divide 30 by 25: 30 ÷ 25 = 1.2

Step 2: Multiply by 100: 1.2 × 100 = 120

Step 3: Add the percent symbol: 120%

Therefore, 30 is 120% of 25

Finding the Original Number When a Percentage is Known

Set up an equation using the percentage as a decimal
Solve for the unknown number

Example 13: 15 is 30% of what number?

Step 1: Set up an equation: 15 = 0.3 × n

Step 2: Solve for n by dividing both sides by 0.3: n = 15 ÷ 0.3 = 50

Therefore, 15 is 30% of 50

Example 14: 42 is 120% of what number?

Step 1: Set up an equation: 42 = 1.2 × n

Step 2: Solve for n by dividing both sides by 1.2: n = 42 ÷ 1.2 = 35

Therefore, 42 is 120% of 35

Percentage Increase and Decrease

Finding Percentage Increase

Calculate the amount of increase: New Value – Original Value
Divide the increase by the original value
Multiply by 100 and add the percent symbol (%)

Example 15: A student’s test score improved from 60 to 78. What is the percentage increase?

Step 1: Calculate the increase: 78 – 60 = 18

Step 2: Divide by the original value: 18 ÷ 60 = 0.3

Step 3: Multiply by 100: 0.3 × 100 = 30%

Therefore, the percentage increase is 30%

Finding Percentage Decrease

Calculate the amount of decrease: Original Value – New Value
Divide the decrease by the original value
Multiply by 100 and add the percent symbol (%)

Example 16: School enrollment decreased from 450 students to 382 students. What is the percentage decrease?

Step 1: Calculate the decrease: 450 – 382 = 68

Step 2: Divide by the original value: 68 ÷ 450 = 0.151

Step 3: Multiply by 100: 0.151 × 100 = 15.1%

Therefore, the percentage decrease is 15.1%

Finding a New Value After a Percentage Change

For an increase:

Multiply the original value by (1 + percentage/100)

For a decrease:

Multiply the original value by (1 – percentage/100)

Example 17: A book costs $20. The price increases by 15%. What is the new price?

Step 1: Convert 15% to a decimal: 15% = 0.15

Step 2: Add 1: 1 + 0.15 = 1.15

Step 3: Multiply by the original value: $20 × 1.15 = $23

Therefore, the new price is $23

Example 18: A laptop originally priced at $800 is on sale for 20% off. What is the sale price?

Step 1: Convert 20% to a decimal: 20% = 0.20

Step 2: Subtract from 1: 1 – 0.20 = 0.80

Step 3: Multiply by the original value: $800 × 0.80 = $640

Therefore, the sale price is $640

Compound Percentage Changes

When multiple percentage changes apply sequentially, they don’t simply add together. Each percentage change applies to the result of the previous change.

Example 19: A house valued at $200,000 increases in value by 10% in the first year and then by another 10% in the second year. What is the new value?

Step 1: Calculate the value after the first year: $200,000 × 1.10 = $220,000

Step 2: Calculate the value after the second year: $220,000 × 1.10 = $242,000

Note: The total increase is $42,000, which is 21% of the original value, not 20%.

Example 20: A store marks up an item by 40% and then offers a 20% discount. What is the final price as a percentage of the original price?

Step 1: After 40% markup: Original × 1.40

Step 2: After 20% discount: Original × 1.40 × 0.80 = Original × 1.12

Therefore, the final price is 112% of the original price (a 12% increase)

Percentage Applications in the Classroom

Scenario 1: Grade Calculations

A student scored 42 points out of 50 on a test. What percentage grade did the student earn?

Step 1: Divide points earned by total possible points: 42 ÷ 50 = 0.84

Step 2: Multiply by 100 to convert to a percentage: 0.84 × 100 = 84%

The student earned an 84% on the test.

Scenario 2: Attendance Rates

In a class of 25 students, 22 were present on Monday. What was the attendance percentage?

Step 1: Divide the number present by the total number of students: 22 ÷ 25 = 0.88

Step 2: Multiply by 100 to convert to a percentage: 0.88 × 100 = 88%

The attendance rate was 88%.

Scenario 3: Budget Planning

A school receives a $150,000 grant. If 35% is allocated to technology, 25% to staff development, and the rest to curriculum materials, how much money is allocated to curriculum materials?

Step 1: Calculate the amount for technology: $150,000 × 0.35 = $52,500

Step 2: Calculate the amount for staff development: $150,000 × 0.25 = $37,500

Step 3: Calculate the remaining amount: $150,000 – $52,500 – $37,500 = $60,000

Step 4: Calculate this as a percentage: $60,000 ÷ $150,000 × 100 = 40%

$60,000 (40% of the grant) is allocated to curriculum materials.

Tips for Teaching Percentages

Connect to real-world contexts: Use examples involving sales, discounts, grades, statistics, and other familiar situations.
Use visual models: 100-square grids, circle models, and number lines help students visualize percentages.
Emphasize the relationship between fractions, decimals, and percentages.
Focus on conceptual understanding before introducing formulas and procedures.
Practice mental math with common percentages like 10%, 25%, 50%, and 75%.
Provide plenty of examples involving percentages greater than 100% and less than 1%.

Common Percentage Misconceptions and Errors

Adding percentages directly: Students may think that 10% + 20% = 30% in all contexts, not understanding compound percentage changes.
Percentage vs. percentage points: Students may confuse an increase of 5 percentage points (e.g., from 10% to 15%) with a 5% increase.
Ignoring the reference value: Students may not understand that the same percentage can represent different absolute values depending on the whole.
Decimal placement errors: When converting between percentages and decimals, students may move the decimal point in the wrong direction.
Mixing up percentage increase/decrease formulas: Students may divide by the new value instead of the original value.

Key Points to Remember

Percentages represent parts per hundred and provide a standardized way to express proportions.
To convert from a percentage to a decimal, divide by 100 (move the decimal point two places left).
To convert from a decimal to a percentage, multiply by 100 (move the decimal point two places right).
To find the percentage of a number, multiply the number by the percentage expressed as a decimal.
Percentage increase/decrease compares the amount of change to the original value.
When working with percentage changes, be careful to use the correct reference value (usually the original value).

Percentage