Fractions and Decimals
Fractions and Decimals
Fractions
A fraction represents parts of a whole. The top number of a fraction, called the numerator, indicates how many equal-sized parts are present. The bottom number of a fraction, called the denominator, indicates how many equal-sized parts make a whole.
Fractions have several forms:
- proper fraction: the numerator is less than the denominator
- improper fraction: the numerator is greater than or equal to the denominator
- mixed number: the combination of a whole number and a fraction
Improper fractions can be converted to mixed numbers by dividing. (In fact, the fraction bar is also a division symbol.)
\(\frac{14}{3}=14÷3=4=4\frac{2}{3}\)
To convert a mixed number to a fraction, multiply the whole number by the denominator of the fraction, and add the numerator. The result becomes the numerator of the improper fraction; the denominator remains the same.
\(5\frac{2}{3}=\frac{(5\ ×\ 3)+2}{3}=\frac{17}{3}\)
To multiply fractions, multiply numerators and multiply denominators. Reduce the product to lowest terms. To divide fractions, multiply the first fraction by the reciprocal of the second fraction. When multiplying and dividing mixed numbers, the mixed numbers must be converted to improper fractions.
\(\frac{a}{b}×\frac{c}{d}=\frac{ac}{bd}\)
\(\frac{a}{b}÷\frac{c}{d}=(\frac{a}{b})(\frac{d}{c})=\frac{ad}{bc}\)
The reciprocal of a fraction is just the fraction with the top and bottom numbers switched.
Adding or subtracting fractions requires a common denominator. To find a common denominator, multiply the denominators of the fractions. Then, add the numerators and keep the denominator the same to find the sum.
\(\frac{a}{b}+\frac{c}{b}=\frac{a\ +\ c}{b}\)
\(\frac{a}{b}−\frac{c}{b}=\frac{a\ −\ c}{b}\)
Decimals
In the base-10 system, each digit (the numeric symbols 0 – 9) in a number is worth ten times as much as the number to the right of it. For example, in the number 321 each digit has a different value based on its position: 321 = 300 + 20 + 1. The value of each place is called place value.
Place Value Chart | 1,000,000 | 106 | millions | ||||||
|---|---|---|---|---|---|---|---|---|---|
100,000 | 105 | hundred thousands | |||||||
10,000 | 104 | ten thousands | |||||||
1,000 | 103 | thousands | |||||||
100 | 102 | hundreds | |||||||
10 | 101 | tens | |||||||
1 | 100 | ones | |||||||
. | decimal | ||||||||
1/10 | 10-1 | tenths | |||||||
1/100 | 10-2 | hundredths | |||||||
To convert a decimal number to a fraction, write the digits in the numerator and write the place value of the final digit in the denominator. Reduce to lowest terms, if necessary.
\(0.096=\frac{96}{1000}=\frac{96\ ÷\ 4}{1000\ ÷\ 4}=\frac{23}{250}\)
To convert a fraction to a decimal, divide the numerator by the denominator.
\(\frac{5}{8}=5÷8=0.625\)