Section 2.3: Percentages, Fractions, and Decimals

Section 2.3: Percentages, Fractions, and Decimals

Fractions, decimals, and percentages are simply three different dialects of the same mathematical language. The EDPT heavily relies on your ability to fluently translate between them, often testing your precision in calculating percentage changes and interest rates.

1. The Core Conversions

Fluency requires memorizing common conversions. This saves valuable time and reduces calculation errors during the exam.

Essential Equivalencies

• 1/2 = 0.50 = 50%
• 1/3 ≈ 0.333 = 33.3%
• 1/4 = 0.25 = 25%
• 1/5 = 0.20 = 20%
• 1/8 = 0.125 = 12.5%
• 1/10 = 0.10 = 10%

Strategic Tip: To convert a decimal to a percentage, move the decimal point two places to the right. To convert a percentage to a decimal, move it two places to the left. To turn a fraction into a decimal, divide the numerator by the denominator.

2. Percentage Translation

Just like algebraic word problems, percentage problems have a translation key.

• "Percent" means "per one hundred" or "divided by 100".
• "Of" means multiply.
• "Is" means equals.
• "What" is your variable (x).

Worked Example 1: 15 is what percent of 75?

Translation: 15 = (x / 100) × 75 Solve for x: 15 / 75 = x / 100 1/5 = x / 100 x = 20. The answer is 20%.

3. Percent Increase and Decrease

This is a notoriously tricky area for many test-takers. The formula for percentage change is universally applicable to markups, discounts, population growth, and depreciation.

The Percentage Change Formula

Percent Change = (New Value - Original Value) / Original Value × 100% Alternatively: Percent Change = (Amount of Change) / Original Value × 100%

Crucial Rule: The denominator is ALWAYS the original, starting value, regardless of whether the value increased or decreased.

Worked Example 2: A store buys a jacket for $40 and sells it for $60. What is the percent markup?

Amount of change = $60 - $40 = $20. Original value = $40. Percent change = (20 / 40) × 100% = 0.5 × 100% = 50%. The markup is 50%.

Sequential Percentage Changes

A common trap is assuming that successive percentage changes are simply additive. They are not. If a price is increased by 20% and then decreased by 20%, you do not return to the original price.

Worked Example 3: A $100 item is marked up by 20%, and then placed on sale for 20% off. What is the final price?

First step: $100 + (0.20 × $100) = $120. Second step: 20% off $120. $120 - (0.20 × $120) = $120 - $24 = $96. The final price is $96, which represents an overall 4% decrease. Always calculate successive changes sequentially.

4. Simple and Compound Interest

Interest problems are applied percentage problems. You must understand the difference between simple and compound interest.

Simple Interest

Interest is calculated only on the principal amount. Formula: I = PRT (Interest = Principal × Rate × Time in years)

Worked Example 4: If you invest $500 at a 4% annual simple interest rate for 3 years, how much interest is earned? I = 500 × 0.04 × 3 = 20 × 3 = $60. Total interest is $60.

Compound Interest

Interest is calculated on the principal AND any previously accumulated interest. Formula for the total amount A: A = P(1 + r/n)^(nt) Where P is principal, r is annual rate (decimal), n is compounding periods per year, and t is time in years. For the EDPT, compound interest is often calculated manually over a short number of periods rather than using the complex formula, because calculators are rarely permitted on military aptitude tests.

Worked Example 5: An account starts with $1,000 and earns 10% interest compounded annually. What is the balance after 2 years?

Year 1: 10% of $1,000 = $100 interest. Balance = $1,100. Year 2: 10% of $1,100 = $110 interest. Balance = $1,100 + $110 = $1,210. The final balance is $1,210.

5. Probability Basics

Probability is essentially a fraction or percentage representing the likelihood of an event. Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes

Independent vs. Dependent Events

• Independent Events: The outcome of one event does not affect the other (e.g., flipping a coin twice). To find the probability of both occurring, multiply their individual probabilities.
• Dependent Events: The outcome of the first event changes the probability of the second (e.g., drawing two cards from a deck without replacement).

Worked Example 6: A bag contains 3 red marbles and 2 blue marbles. What is the probability of drawing two red marbles in a row without replacing the first marble?

Probability of first red marble = 3/5. Since one red marble is removed, there are now 2 red marbles and 4 total marbles left. Probability of second red marble = 2/4 = 1/2. Combined probability = (3/5) × (1/2) = 3/10 or 30%.

Final Review of Strategic Tips

1. Memorize fraction-decimal-percent equivalencies to save time on arithmetic.
2. In percent change problems, always divide by the original starting number.
3. Be wary of successive percent changes; never add or subtract percentages directly unless they apply to the same baseline amount.