TEAS 7 · Mathematics · 5 min read
Percent Problems: Change, Discounts, Tax and Tips
- Convert between percentages, decimals, and fractions, and solve percent scenarios using the fundamental percent equation and percent proportion.
- Calculate percent change, percent increase, and percent decrease using initial baseline values.
- Compute consumer math applications involving markdowns, discounts, markups, sales tax, tips, and multi-step combinations.
Foundations of Percentages and the Core Equations
A percentage represents a ratio, fraction, or number of parts per 100. Understanding this fundamental concept allows you to move seamlessly between different numerical forms. To convert a percentage to a decimal, divide the percentage value by 100 or move the decimal point two places to the left. In reverse, to convert a decimal to a percentage, multiply by 100 or move the decimal point two places to the right and append a percent sign. When working with fractions, convert a fraction to a percentage by dividing the numerator by the denominator and multiplying the result by 100.
Word problems often require translating verbal statements directly into algebraic operations. In percentage word problems, the word 'of' indicates multiplication, while the word 'is' indicates an equals sign. Recognizing these verbal markers makes setting up equations straightforward and prevents structural errors during calculations.
Two standard mathematical tools are used to solve these questions: the fundamental percentage equation and the percent proportion. The fundamental percentage equation is Part equals Percent (in decimal form) multiplied by Whole. Alternatively, the percent proportion is expressed as Part divided by Whole equals Percent divided by 100. In these relationships, the whole represents the base amount, reference value, or 100 percent of the quantity, while the part represents the specific portion of the whole corresponding to the given percentage.
Common mistakes
- Using the new value as the denominator when calculating percent change instead of the original starting value.
- Simply adding successive discount rates together rather than applying each subsequent discount rate to the already reduced intermediate price.
- Calculating sales tax on the original list price instead of the discounted sale price when an item is both discounted and taxed.
- Subtracting the tax percentage directly from a final taxed total rather than dividing the total by (1 plus sales tax rate) to find the pre-tax price.
Key points
- A percentage is a ratio or number of parts per 100, where 'of' indicates multiplication and 'is' indicates an equals sign.
- Core percentage calculations use Part equals Percent (in decimal form) multiplied by Whole, or Part divided by Whole equals Percent divided by 100.
- Percent change is the amount of change divided by the original value multiplied by 100 percent, and it is undefined if the original value is zero.
- The sale price equals the original price multiplied by (1 minus discount rate), while retail price with markup equals cost multiplied by (1 plus markup rate).
- Successive discounts cannot be added together directly; each discount applies to the newly reduced intermediate price.
- Total cost with sales tax equals the subtotal multiplied by (1 plus sales tax rate), and pre-tax price equals the total divided by (1 plus sales tax rate).
- Tips are applied to the pretax subtotal, discounts precede sales tax calculations, and currency amounts round up if the thousandths digit is 5 or greater.
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Sample lesson: Fractions, Decimals and Percents: Converting Between Forms25 practice questions on this exam
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