Next Generation Arithmetic Study Guide for the ACCUPLACER Test

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Percents

The term percent means per one hundred, so \(63\%\) is the same as \(63\) out of \(100\) or \(.63\) (\(63\) hundredths), or \(\frac{63}{100}\). You’ll need to know that you can substitute any of these for the other when working with percents in problems. The following are lists of skills that you should be able to do regarding percent.

Change a % to a Fraction and Back

To change a percent to a fraction, use the percent as the numerator and \(100\) as the denominator. Simplify the fraction if needed. For example:

\(25\% = \frac{25}{100}\), which can be simplified to \(\frac{1}{4}\).

To change a fraction to a percent, change the fraction to an equivalent fraction with \(100\) as the denominator. The numerator is your percent. For example, \(\frac{3}{5}\) is equivalent to \(\frac{60}{100}\), so \(\frac{3}{5}\) is the same as \(60\%\).

Change a % to a Decimal and Back

To change a percent to a decimal, remember that a number such as \(33\) can be written as \(33.0\). Write the percent with a decimal point and then move the decimal point two places to the left. (This is the same as dividing by \(100\).) For \(33\%\), first, write it as 33.0 and then move the decimal two places to the left. The decimal form of \(33\%\) is \(0.33\).

To convert a decimal number to a percent, move the decimal two places to the right and use the % sign: \(0.47\) is the same as \(47\%\).

Change a Fraction to a Decimal and Back

To change a fraction to a decimal, first see if can be reduced to an equivalent fraction with smaller numbers. For example, \(\frac{10}{16}\) can be reduced to \(\frac{5}{8}\). This makes the next step easier. A fraction represents a division problem, so to convert to the decimal do a decimal division of \(\text{numerator} \div \text{denominator}\), or in this case, \(5 \div 8\). The answer to this division is the decimal form, \(0.625\).

To convert a decimal to a fraction, remember that the decimal number is given as an amount in tenths, hundredths, etc., depending on the last place of the given number. For example, \(0.55\) is \(\frac{55}{100}\). This can be reduced to \(\frac{11}{20}\), which is the correct fraction form of \(0.55\).

Calculate % Increase or Decrease

The percent increase or decrease is calculated by dividing the amount of change by the original amount. The calculation is usually done with decimal numbers and the result must then be converted to a percentage.

You can also set up a fraction with the amount of change in the numerator and the original amount in the denominator. For example, if a price was originally \(\$80\) and increases to \(\$100\), the amount of change is \(\$20\) and the original amount is \(\$80\).

As a fraction, this is \(\frac{20}{80}\) which is equivalent to \(\frac{1}{4}\). This is the same as \(0.25\) or \(25\%\).

Or do \(20 \div 80\) which equals \(0.25\) or \(25\%\). It is worthwhile to remember a few fraction-to-decimal to percent equivalents, such as:

\(\frac{1}{4} = 0.25 = 25\%\)
\(\frac{2}{4} = 0.5 = 50\%\)
\(\frac{3}{4} = 0.75 = 75\%\)
\(\frac{1}{3} = 0.33 = 33\%\)
\(\frac{2}{3} = 0.67 = 67\%\)
\(\frac{1}{5} = 0.2 = 20\%\)
\(\frac{2}{5} = 0.4 = 40\%\)
\(\frac{3}{5} = 0.6 = 60\%\)
\(\frac{4}{5} = 0.8 = 80\%\)
\(\frac{1}{10} = 0.1 = 10\%\)
\(\frac{3}{10} = 0.3 = 30\%\)
\(\frac{7}{10} = 0.7 = 70\%\)
\(\frac{9}{10} = 0.9 = 90\%\)

Determine the % of a Number

To determine the percent of a number, first convert the percent to a fraction or decimal. Calculations are always done with the decimal or fractional form of the percent. Then multiply by the number. For example, to find \(55\%\) of \(200\), you can multiply either \(0.55 \times 200\) or \(\frac{55}{100} \times 200\). The answer is \(110\).

Apply % to Real-Life Contexts

The most common real-life example of percentages is in prices. Sales are usually given as a percent. A \(20\%\) sale means \(20\%\) off of the usual price. If you want to know how much you are saving, multiply \(0.2\) or \(\frac{2}{10}\) by the original price.

To find the sale price, subtract the amount you saved from the original price. For example, if a \(\$50\) item is on sale at \(20\%\) off, you save \(0.2 \times \$50\), or \(\$10\), and the sale price is \(\$50 - \$10\), or \(\$40\).

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