GED · Mathematical Reasoning · 6 min read
Percentages, Change, and Averages
- Convert fluently between percentages, decimals, and fractions, and solve practical consumer calculations including sales tax, discounts, tips, and commissions.
- Calculate percent change, markups, and markdowns using standard formulas and efficient one-step multiplier methods.
- Determine and interpret measures of center and spread, mean, median, mode, and range, while analyzing the influence of outliers and managing frequency distributions.
Foundations of Percentages and Everyday Consumer Math
A percentage represents a fraction or ratio with a denominator of 100. Because percentages are based on 100, converting between representations is straightforward. To convert a percentage to a decimal, divide the number by 100 or move the decimal point two places to the left. To convert a decimal to a percentage, multiply by 100 or move the decimal point two places to the right. When working with a fraction, you can convert it to a percentage by dividing the numerator by the denominator and multiplying the resulting value by 100.
In everyday percent problems, the word “of” often means multiply. For instance, calculating 20% of 50 translates directly to multiplying 0.20 by 50. This concept forms the basis of the percentage formula, which states that the part equals the whole multiplied by the percentage in decimal form. Alternatively, you can set up the relationship using the percent proportion, which is expressed as part over whole equals percent over 100. Both methods allow you to solve for any missing piece when given the other values.
Consumer math regularly applies these basic percent calculations to real-world transactions. Sales tax is calculated by multiplying the purchase price by the sales tax rate expressed as a decimal. To find the total purchase cost, add the original price to the sales tax amount. Conversely, when an item is on sale, the discount amount is found by multiplying the original price by the discount rate. Subtracting this discount amount from the original price yields the final sale price.
Similar percentage calculations apply to personal service and workplace earnings. A tip or gratuity is calculated by multiplying the bill subtotal by the tip percentage. In sales roles that pay a flat commission rate, your earnings are determined by multiplying the sales eligible for commission by the commission rate written as a decimal. On the TI-30XS calculator, using the percent key converts an entered percentage directly into a decimal, and the toggle key easily converts displayed values between fraction and decimal formats.
Common mistakes
- Dividing by the new value instead of the original value when calculating percent change, or attempting to calculate percent change when the original value is zero (which makes the percent change undefined).
- Adding or subtracting consecutive percentages directly, such as assuming a 10 percent increase followed by a 10 percent decrease returns to the start, rather than recognizing the final value is lower than the starting value.
- Attempting to find the median before sorting the data set from least to greatest, or failing to take the mean of the two middle numbers when dealing with an even number of values.
- Entering a multi-step mean calculation into the TI-30XS calculator without placing parentheses around the sum, which causes the calculator to divide only the last value entered.
Key points
- A percentage is a ratio with a denominator of 100 that converts to a decimal by dividing by 100 (moving the decimal point two places left) and to a percent by multiplying by 100 (moving two places right).
- In percent calculations, 'of' means multiply; basic consumer applications include sales tax, discounts, sale price, tips, and flat commissions.
- Percent change equals the amount of change divided by the original value multiplied by 100; it represents an increase if the new value is greater or a decrease if the new value is less.
- One-step multipliers allow quick computation of percent changes: multiply by 1 plus the decimal rate for an increase or 1 minus the decimal rate for a decrease.
- The mean is the sum divided by the count, the median is the sorted middle value (or mean of the two middle values for an even count), and the mode is the most frequent value.
- Outliers heavily influence the mean, whereas the median is usually less affected; range measures spread by subtracting the minimum value from the maximum value.
- In frequency tables, compute the mean by dividing the sum of each value multiplied by its frequency by the sum of all frequencies.
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