ASVAB · Arithmetic Reasoning · 6 min read
Percents: Discounts, Tax, Interest and Change
- Convert fluently between percents, decimals, and fractions, using non-calculator mental arithmetic techniques to find percentages of numbers.
- Solve real-world consumer transactions involving markdowns, successive discounts, sales tax, and tips.
- Calculate simple interest and percent change accurately using core formulas and designated scratch paper.
Percent Fundamentals and Mental Math Strategies
A percent represents a fraction, ratio, or proportion with a base denominator of 100. The term 'percent' literally means 'per hundred' or 'out of one hundred.' In mathematical word problems, the word 'is' indicates equality, while in a phrase such as '20% of 80,' the word 'of' means multiply, written as 0.20 × 80. To calculate a percent of a number, convert the percentage to a decimal or fraction and multiply it by that number. For a positive starting number, a percent greater than 100% gives a result larger than that number; for example, 150% of 20 is 30.
To convert a percent to a decimal, divide the number by 100 or move the decimal point two places to the left. In reverse, to convert a decimal to a percent, multiply by 100 or move the decimal point two places to the right. When working with fractions, convert a fraction to a percent by dividing the numerator by the denominator and multiplying the resulting decimal by 100. Memorizing key equivalents allows you to bypass tedious division entirely during timed testing.
Because calculators are strictly prohibited on the Arithmetic Reasoning subtest of the ASVAB, fast mental shortcuts are indispensable. To find 10% of any number without a calculator, shift its decimal point one place to the left, and to find 1% of any number, shift its decimal point two places to the left. Once you establish 10%, you can quickly generate other values: five percent of a number can be calculated by finding 10% of the number and dividing that result by 2; fifteen percent can be calculated by adding 10% of the number to 5% of the number; and twenty percent can be calculated by finding 10% of the number and multiplying by 2.
Converting decimals to simplified fractions often speeds up non-calculator arithmetic on percent word problems. For example, using fractional values allows for fast mental cross-cancellation when multiplying by whole numbers on your scratch paper.
| Percent | Fraction | Decimal |
|---|---|---|
| 10% | 1/10 | 0.10 |
| 20% | 1/5 | 0.20 |
| 25% | 1/4 | 0.25 |
| 33 1/3% | 1/3 | 0.333… (repeating) |
| 50% | 1/2 | 0.50 |
| 75% | 3/4 | 0.75 |
Common mistakes
- Adding successive discount percentages together instead of applying the second discount percentage to the reduced price.
- Dividing the amount of change by the new value instead of the original value when calculating percent increase or percent decrease.
- Entering months directly as whole years in the simple interest formula (I = Prt) rather than dividing the number of months by 12.
- Failing to read carefully to distinguish whether the question asks for the dollar amount of the discount/tax or the final total purchase cost.
Key points
- A percent represents a fraction, ratio, or proportion with a base denominator of 100, meaning 'per hundred.'
- Mental math shortcuts allow rapid calculations without a calculator: shift the decimal point one place left for 10% and two places left for 1%.
- For one discount, find the sale price by multiplying the price before that discount by (100% − discount percent); in a basic tax problem, apply tax to the discounted price unless the problem specifies otherwise.
- Simple interest is calculated using I = Prt, where time t must be expressed in years (or months divided by 12), and the ending amount equals principal plus interest.
- Percent change equals the change divided by the original value multiplied by 100%, with increases occurring when the final value is greater and decreases when it is less.
- Arithmetic Reasoning combines with Mathematics Knowledge, Word Knowledge, and Paragraph Comprehension to determine the AFQT score for basic enlistment eligibility, with minimum scores set by each military branch.
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