Percentages and Real-World Applications

Percentage questions are a staple of the OAR Math Skills Test. They appear both as straightforward calculations and embedded within word problems.

The Three Basic Percentage Problems

Every percentage problem is one of these three types:

Shortcut: The "is/of" Method

• "is" means equals (the part)
• "of" means multiply (the whole)
• "what" is the unknown

Example: 15 is what percent of 60?

15 = x% × 60 → x = 15/60 = 0.25 = 25%

Percent Increase and Decrease

Percent Change Formula

Percent Change = ((New Value - Original Value) / Original Value) × 100

• Positive result = increase
• Negative result = decrease

Example: A recruit's run time improved from 14 minutes to 11 minutes 12 seconds. What is the percent decrease?

• Original = 14 min = 840 seconds
• New = 11 min 12 sec = 672 seconds
• Change = (672 - 840) / 840 × 100 = (-168/840) × 100 = -20%
• The run time decreased by 20%.

Finding the New Value After a Percent Change

Finding the Original Value

If a value increased by 20% to become 360, what was the original?

Original × 1.20 = 360 → Original = 360 / 1.20 = 300

Common mistake: Do not subtract 20% of 360 (72) to get 288. You must divide by 1.20.

Successive Percentage Changes

Warning: You cannot simply add successive percentages.

Example: A price increases by 10% then decreases by 10%. Is the final price the same as the original?

• Start: $100
• After 10% increase: $100 × 1.10 = $110
• After 10% decrease: $110 × 0.90 = $99
• The final price is $99, not $100. A 10% increase followed by a 10% decrease results in a 1% net decrease.

General Rule for Successive Changes

Multiply the multipliers:

• 20% increase then 15% decrease: 1.20 × 0.85 = 1.02 → net 2% increase
• 25% decrease then 25% increase: 0.75 × 1.25 = 0.9375 → net 6.25% decrease

Simple Interest

Formula: I = P × r × t

Where:

• I = interest earned
• P = principal (starting amount)
• r = annual interest rate (as a decimal)
• t = time in years

Example: How much interest does $5,000 earn in 3 years at 4% simple interest?

I = 5000 × 0.04 × 3 = $600

Total after 3 years = $5,000 + $600 = $5,600

Compound Interest

Formula: A = P(1 + r/n)^(nt)

Where:

• A = final amount
• P = principal
• r = annual rate (decimal)
• n = times compounded per year
• t = years

Example: What is $2,000 worth after 2 years at 5% compounded annually?

A = 2000(1 + 0.05/1)^(1×2) = 2000(1.05)² = 2000 × 1.1025 = $2,205

For the OAR, you typically do not need to compute complex compound interest by hand, but understanding the concept and being able to do 1-2 compounding periods is valuable.

Percentage Word Problem Patterns