Arithmetic

A

Adding and Subtracting Fractions

To add or subtract fractions that have the same denominators, add or subtract the numerators according to the rules for integers, and express the result as a fraction of the denominator. The result is normally reduced to lowest terms. For example,

Only fractions with equal denominators may be added or subtracted as they stand. If the denominators of fractions to be added are unequal, we must find a common denominator. In the expression ’ + •, for example, the denominators 3 and 4 are different. One quick way to obtain a common denominator for two fractions is to multiply their denominators. In this case that multiplication gives us 12. Thus, to add ’ and •, we should change the fractions into their equivalents with 12 as a common denominator. To do so, multiply both the numerator and denominator of each fraction by the denominator of the other fraction:

Multiplying both the numerator and denominator by the same number does not change the overall value of a fraction. Now we can add

Subtracting fractions requires the same procedure:

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B

Multiplying Fractions

Multiplying two fractions, ¡ and ¢, is straightforward. Simply multiply numerators together and multiply denominators together:

For example,

The answer has been reduced to its lowest term, in this case ˜.

The rules for multiplying signed (positive or negative) fractions are the same as those for multiplying signed integers. The same is true for the rules governing addition, subtraction, and division of signed fractions:

C

Dividing Fractions

The division of fractions is most easily understood in terms of reciprocals. Every number (except 0) has a reciprocal, or another number such that the product of the number and its reciprocal equals one. The reciprocal of 3, for example, is €. Zero (0) has no reciprocal, because no number can be multiplied by it to equal 1. Any number multiplied by zero equals zero.

To divide ¡ by ¢, multiply ¡ by the reciprocal of ¢:

Division is equivalent to multiplying by the reciprocal—that is, y ÷ ‚ is the same as y × because both the numerator and the denominator can be multiplied by the same nonzero number without affecting the overall value of the fraction. Multiply both numerator and denominator in the graphic above by ¤:

This division is equivalent to multiplying the first number, ¡, by the reciprocal of the second number—by ¤. The reciprocal of a fraction is simply the fraction flipped upside down. Here is an example that uses actual numbers:

Dividing a whole number by a fraction works the same way:

V

Operations With Decimals

The idea of place values can be extended to accommodate fractions. Instead of writing 1‘ (one and two-tenths), we can use a decimal point (.) to represent the same fraction as 1.2. Just as places to the left of the decimal represent units, tens, hundreds, and so on, those to the right of the decimal represent places for tenths (s), hundredths (t), thousandths ( ), and so forth. In a decimal number such as 8.632 the numbers to the right of the decimal point represent

This number is read “eight and six hundred thirty-two thousandths” or “eight point six three two.” Zeroes are often added to the left of the decimal point when a number is less than one. For example, we can write y as either .5 or 0.5.

A

Adding and Subtracting Decimals

Decimals allow us to add and subtract numbers that include fractions just as we add and subtract integers. But we must be careful to always align the decimal point so that tens are under tens, units under units, tenths under tenths, and so on, ensuring that each value is being added to or subtracted from a similar value at every step. For example, to add 365.289 and 32.4, align the decimals and then add the numbers beginning at the right and moving to the left:

Extra zeros to the right of a number do not change the number. Filling in the number of places with zeroes can help ensure that the same number of places exists to the right of the decimal point for all numbers being added or subtracted:

The decimal point in the sum falls directly beneath the decimal points in the numbers being added.

Subtraction with decimals proceeds in much the same way as addition: We can use zeros to ensure that the numbers’ place values line up, then subtract the numbers as usual.

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