Physical Science - Page 876

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Skills and Reference Handbook > Skills Handbook > Math Skills

Math Skills

Throughout your study of physical science, you will often need to solve math problems. This appendix is designed to help you quickly review the basic math skills you will use most often.

Fractions

Adding and Subtracting Fractions

To add or subtract fractions that have the same denominator, add or subtract the numerators, and then write the sum or difference over the denominator. Express the answer in lowest terms.

Examples

\frac{3}{10} + \frac{1}{10} = \frac{3 + 1}{} = \frac{4}{10} = \frac{2}{5} \frac{5}{7} - \frac{2}{7} = \frac{5 - 2}{} = \frac{3}{7}

To add or subtract fractions with different denominators, find the least common denominator. Write an equivalent fraction for each fraction using the least common denominator. Then add or subtract the numerators. Write the sum or difference over the least common denominator and express the answer in lowest terms.

Examples

\frac{1}{3} + \frac{3}{5} = \frac{5}{15} + \frac{9}{15} = \frac{5 + 9}{15} = \frac{14}{15} \frac{7}{8} - \frac{1}{4} = \frac{7}{8} - \frac{2}{8} = 7 - 2 = \frac{5}{8}

Multiplying Fractions

When multiplying two fractions, multiply the numerators to find the product's numerator. Then multiply the denominators to find the product's denominator. It helps to divide any numerator or denominator by the greatest common factor before multiplying. Express the answer in lowest terms.

Examples

\frac{3}{5} \times \frac{2}{7} = 3 \times 2 = \frac{6}{35} \frac{4}{14} \times \frac{6}{9} = 2 \times 2 \times 2 \times 3 = 2 \times 2 = \frac{4}{21}

Dividing Fractions

To divide one fraction by another, invert and multiply. Express the answer in lowest terms.

Examples

$\frac{2}{5} \div \frac{3}{4} = \frac{2}{5} \times \frac{4}{3} = \frac{2 \times 4}{5 \times 3} = \frac{8}{15} \frac{9}{16} \div \frac{5}{8} = \frac{9}{16} \times \frac{8}{3} = \frac{9 \times 1}{2 \times 5} = \frac{9}{10}$

Ratios and Proportions

A ratio compares two numbers or quantities. A ratio is often written as a fraction expressed in lowest terms. A ratio also may be written with a colon.

Examples

The ratio of 3 to 4 is written as 3 to 4, $\frac{3}{4}$ , or 3: 4.

The ratio of 10 to 5 is written as $\frac{10}{5} = \frac{2}{1}$ , or 2: 1.

A proportion is a mathematical sentence that states that two ratios are equivalent. To write a proportion, place an equal sign between the two equivalent ratios.

Examples

The ratio of 6 to 9 is the same as the ratio of 8 to 12.

$\frac{6}{9} = \frac{8}{12}$

The ratio of 2 to 4 is the same as the ratio of 7 to 14.

$\frac{2}{4} = \frac{7}{14}$

You can set up a proportion to determine an unknown quantity. Use x to represent the unknown. To find the value of x, cross multiply and then divide both sides of the equation by the number that comes before x.

Example

Two out of five students have blue notebooks. If this same ratio exists in a class of twenty students, how many students in the class have blue notebooks?