In mathematics, a fraction represents equal parts of a whole thing or whole group. The bottom number, called the *denominator*, represents the total number of equal parts making up the corresponding whole. The top number, called the *numerator*, indicates how many of the total equal parts there are. The numerator and denominator are separated by a line called a *fraction bar*.

*Equivalent Fractions*

There are infinite ways to express a fraction, for example is the same as , , , and so on. To find an equivalent fraction, simply multiply or divide the numerator and denominator by the same number. (This works accurately because you are multiplying or dividing by , , , etc. and these numbers are all equal to 1.) This knowledge can be very useful when you do operations with fractions or reduce them to lowest terms.

*Simplifying or Reducing to Lowest Terms*

If you are asked to simplify a fraction or *write it in “lowest terms”*, you’ll need to find the *greatest common factor* of the numerator and denominator. This is the largest number by which both the numerator and denominator can be evenly divided. For example, to simplify the fraction , do this:

factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36

The greatest factor of both numbers is 12, so divide the numerator and denominator by 12 to reduce the fraction:

*Mixed Numbers*

The term *mixed number* refers to a number that includes a whole number and a fraction, such as or .

*Improper Fractions*

An *improper fraction* is one whose numerator is larger than its denominator, such as or . Most tests require you to change improper fractions into mixed numbers. To do so, divide the numerator by the denominator, producing the whole number and using any remainder as the numerator of the fraction.

becomes

becomes

To *add or subtract fractions with like denominators*, add or subtract the numerators as usual and leave the denominator as is. Simplify, if necessary.

To *add or subtract fractions with different denominators*, first rewrite the problem, using equivalent fractions that share the same (least) common denominator. Then proceed as usual.

To *multiply fractions*, multiply their numerators by one another. The resulting product serves as the numerator in the answer. Then multiply their denominators by one another as well, and use the resulting product as the answer’s denominator. Remember to simplify the answer to lowest terms, if applicable.

To *divide fractions*, multiply the dividend by the reciprocal of the divisor by flipping the second fraction upside down and then following the usual steps for multiplying fractions. Again, simplify to lowest terms if appropriate.

To *convert a fraction to a decimal*, divide the numerator by the denominator.

If the problem reads, “Write as a decimal,” divide 3 by 4. The answer is 0.75 [.]

The order of fractions that have the *same* denominator can be determined by ignoring their denominators and arranging them in accordance with the values of their numerators.

The fractions would be ordered in sequence like this:

If the denominators are *not* the same, you will need to create equivalent fractions with the same denominator, before comparing them, like this:

become

Then, they are easily compared by placing the numerators in order of value.

A quick way to compare two fractions whose denominators are *not* the same is to cross multiply. Multiply the numerator of the first fraction by the denominator of the second fraction and write the resulting product next to the first fraction. Then multiply the numerator of the *second* fraction by the denominator of the *first* fraction and write the resulting product next to the second fraction. The larger product identifies the fraction with the greater value.

For example:

Compare and

and

So,

Decimals are numbers that have fractional parts, separated from whole parts by a decimal point. Hence, they express values that fall in between whole numbers. Here are some examples:

(read “13 thousandths”)

(read “28 and 4 hundredths”)

(read “5 tenths”)

To *add or subtract decimals*, position their digits one above the other so that all corresponding place values are aligned. If desired, insert zeros so the numbers are all the same length as well. Then add or subtract using column addition or subtraction as normal, making sure to maintain the decimal point in the same position in the resulting sum or difference.

So, would be written:

To *multiply decimals*, begin by multiplying the numbers as usual, ignoring all decimal points. Then count the combined number of decimal places in the numbers you multiplied, and insert a decimal point in the product such that the number of decimal places in the answer matches the total number of decimal places you counted. For example:

1.2 × 0.11 = 0.132

First, rewrite vertically:

There are a total of 3 decimal points in the 2 numbers being multiplied, so move the decimal point 3 spaces to the left, after finding the numerical answer.

To *divide decimal numbers*, if there is a decimal point in the divisor, move it all the way to the right. Then move the decimal point in the dividend to the right that same number of places. Finish by dividing, as usual, remembering to insert the decimal point in the quotient directly above the decimal point in the dividend. This would mean that 81 ÷ 0.9 can be rewritten vertically, like this:

Move the decimal point to the right of the 9, and, correspondingly, one place to the right in the dividend, adding a 0 to make this possible. Then, solve as regular long division, being sure to place the quotient’s decimal point directly above the decimal point in the dividend.

A decimal can be converted into a faction by placing it over a fraction bar and using the place value of its right-most digit as the denominator. For example, in the decimal 1.4910, the right-most digit (zero) is in the ten-thousandths place, so 1.4910, written as a fraction, is or , once it is reduced to lowest terms.

To convert a decimal into a percent, simply move its decimal point two places to the right and follow the last number with a percent sign. Hence, , written as a percent, is .

A percent is an amount expressed as part of 100 and is often designated by the sign. To find the percent of a value, multiply it by the percent’s decimal equivalent, which is obtained by moving the decimal point two places to the left and dropping the percent sign. For example:

of

To add or subtract percents, simply combine or subtract them as you would any number, making sure that digits in corresponding places are aligned, so .

To add or subtract a given percentage to or from a number, multiply the number by the percent’s decimal equivalent and then add or subtract the result from the original number, as appropriate. That means the cost of a $65 item that is marked 40% off would be , or [.]