Chapter 1
Number and Quantity
Course: Core Academic Skills For Educators Combined Test 5752
Subject: Core Academic Skills Educators Mathematics 5733
Chapter: Number And Quantity
1. Understanding Percent
Percent, meaning “per hundred,” is a way of expressing a ratio where the whole is divided into 100 equal parts. It is commonly used to describe proportions, comparisons, and changes in quantity. The percent symbol (%) indicates that the value is a fraction of 100. For example, 45% means 45 parts out of 100 total parts.
Percentages are not limited to values between 0% and 100%. Values above 100% indicate that a quantity exceeds the whole. For instance, if a company achieves 120% of its sales goal, it means sales were 20% higher than the target. Negative percentages can represent losses or decreases, such as −15% for a decline in value.
In mathematical terms, a percent is simply a number divided by 100: 45% = 45 ÷ 100 = 0.45.
2. Converting Between Percent, Decimal, and Fraction
Efficient problem-solving often requires switching between these representations:
- To convert a percent to a decimal, divide by 100. Example: 65% becomes 0.65.
- To convert a decimal to a percent, multiply by 100. Example: 0.08 becomes 8%.
- To convert a percent to a fraction, write it over 100 and reduce to simplest terms. Example: 75% becomes 75/100, which simplifies to 3/4.
- To convert a fraction to a percent, convert the fraction to a decimal first, then multiply by 100. Example: 2/5 = 0.4 = 40%.
These conversions are essential in solving applied problems such as tax calculations, probability, and statistical comparisons.
3. Problem-Solving Categories Involving Percent
a) Finding the Percent of a Number
When you know the whole and the percentage, the part is found by multiplying the whole by the percent in decimal form.
Example: To find 25% of 160, convert 25% to 0.25 and multiply: 0.25 × 160 = 40. In a real-world example, a store offering a 25% discount on a 40.
b) Finding the Whole Given a Percent
When the part and percent are known, the whole is found by dividing the part by the percent in decimal form.
Example: If 30 is 20% of a number, divide 30 by 0.20 to find the whole: 30 ÷ 0.20 = 150. In real life, if 150 before discount.
c) Finding the Percent Given a Part and a Whole
When the part and whole are known, the percent is calculated by dividing the part by the whole and multiplying by 100.
Example: If 45 apples are out of 120 total fruits, the percent is (45 ÷ 120) × 100 = 37.5%. A real-world example is determining that 37.5% of surveyed customers prefer a particular product.
4. Applications of Percent
a) Percentage Increase and Decrease
These are used to measure growth or decline relative to an original value.
- Percentage increase = [(new value − original value) ÷ original value] × 100
- Percentage decrease = [(original value − new value) ÷ original value] × 100
Example: If a stock rises from 65, the increase is [(65 − 50) ÷ 50] × 100 = 30%. In practical terms, percentage increase is used in sales growth, investment returns, and performance improvement statistics.
b) Discounts and Markups
Discounts reduce an original price, while markups increase the cost to determine selling price.
Discount amount = original price × discount rate (in decimal form) Sale price = original price − discount amount
Markup amount = cost × markup rate (in decimal form) Selling price = cost + markup amount
Example: An 80 + (0.25 × 100. These calculations are standard in retail pricing and wholesale distribution.
c) Tax and Tip Calculations
Sales tax and tips are additional percentages applied to the base cost.
Tax amount = base price × tax rate Tip amount = base price × tip rate
Example: For a 4, tip is 61.50. In real-world contexts, these are used for budgeting expenses and service industry payments.
d) Simple and Compound Interest
Simple interest measures interest on the original principal only: I = P × r × t Where P = principal, r = annual interest rate (decimal), and t = time in years.
Compound interest includes interest on previous interest earned: A = P × (1 + r/n)^(n × t) Where n is the number of compounding periods per year, and A is the total amount after interest.
Example: For $1,000 at 5% annual interest for 3 years, compounded annually, A = 1000 × (1 + 0.05)^3 = 1157.63.
e) Percent Error
This measures the accuracy of a measured value compared to an accepted value: Percent error = (|measured − actual| ÷ actual) × 100
Example: If the measured length of a table is 98 cm and the actual length is 100 cm, the percent error is (|98 − 100| ÷ 100) × 100 = 2%.
5. Advanced and Technical Considerations
- Successive percentage changes should be calculated by multiplying change factors, not adding percentages. Two successive 10% decreases multiply as 0.9 × 0.9 = 0.81, representing a total decrease of 19%, not 20%.
- When comparing percent changes across different groups, the original or base value matters significantly, as identical changes in amount can represent very different percentage changes depending on the base.
- Annual Percentage Rate (APR) and Annual Percentage Yield (APY) are not the same; APY accounts for compounding and is typically higher than APR when compounding occurs more than once per year.
- Percentages can be applied in statistical data interpretation, probability, and data visualization to compare proportions between categories.
6. Common Mistakes to Avoid
- Confusing a change in percentage points with percent change. A rise from 4% to 6% is an increase of 2 percentage points but a 50% relative increase.
- Forgetting to convert percentages to decimals before multiplying or dividing.
- Misapplying percent formulas to quantities that do not have proportional relationships.
- Treating percentages over 100% as invalid; in many contexts, they represent growth beyond the whole.
7. Real-Life Examples
- In education, calculating pass rates on standardized exams to evaluate performance.
- In finance, determining annual mortgage interest payments.
- In health, calculating body fat percentage using measurement formulas.
- In retail, applying multiple discounts and sales tax to determine a customer’s total payment.
- In government, analyzing changes in unemployment rates over time.
- In sports, calculating an athlete’s shooting accuracy or scoring efficiency.