Chapter 1: Arithmetic & Prealgebra Section 1.2: Fractions Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Numerator and the Denominator The top number is called the numerator. It stands for the number of pieces within the fraction. The bottom number is called the denominator. It indicates the denomination, or size, of each piece. Or, in other words, the number of pieces needed to make a whole. A fraction is written in simplest form (or lowest terms) when the numerator and denominator have no common factor other than 1. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Reducing a Fraction One process of writing a fraction in lowest terms is called dividing out common factors. This is also called cancelling or reducing. When reducing fractions, do not put slashes through different numbers. Instead, factor the numerator and denominator independently and put the slashes through identical factors that appear in the numerator and denominator. Example: Simplify the following fraction. 240 360 = 24 × 10 36 × 10 = 2 × 12 × 10 3 × 12 × 10 = 2 3 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Mixed Numbers A mixed number is a whole number together with a fraction. When writing (or typing) a mixed number in a linear format, be sure to include a single space between the whole number and the fractional part. If 1 2/5 was written as 12/5, we have “twelve-fifths” instead of “one and two fifths” – which are two very different numbers. Fractions less than 1 are called proper fractions and fractions greater than 1 are called improper fractions. Whether a fraction is proper or improper, always be sure to write it in lowest terms. An improper fraction can also be written as a mixed number, with the fractional part being a proper fraction in lowest terms. When we perform arithmetic with mixed numbers, the first step is usually to write all the mixed numbers as improper fractions. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Multiplication with Fractions Multiply the numerators together. Multiply the denominators together. Reduce. - Reducing can be done before or after the multiplications, but it is usually easier to do it before. Example: Multiply and reduce. 14 15 × 25 30 = 2 ∙7 3 ∙5 × 5 ∙5 2 ∙3 ∙5 = 7 3 ∙3 = 7 9 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Division with Fractions Flip the second fraction and multiply. Remember, the first fraction does not get flipped; only the divisor gets flipped. Example: Divide. 5 6 ÷ 35 12 = 5 6 × 12 35 = 5 6 × 2 ∙ 6 5 ∙ 7 = 2 7 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Addition and Subtraction with Fractions To add or subtract fractions, first rewrite the fractions as equivalent fractions with the same denominator. Then add (or subtract) the numerators. If necessary, reduce the answer to lowest terms. Example: Add 5 9 + 3 4 = 20 36 + 27 36 = 47 36 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Converting Fractions to Decimals Putting it as simple as possible… A fraction is a division problem, so divide. Unfortunately, to find the decimal equivalent of 2/5, many people cannot decide if the division is to be 2 ÷ 5 or 5 ÷ 2. Not sure? Actually do both divisions. Since 2/5 is less than a whole, the its decimal equivalent must also be less than a whole. Example: Write 3/8 as a decimal. First note that 3/8 < 1. That means the decimal must also be less than 1. 8 ÷ 3 = 2.66666…, which is greater than 1. So, that cannot be correct. 3 ÷ 8 = 0.375 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Converting Decimals to Fractions To convert a terminating decimal to a fraction, read it, write it, and reduce it. By “read it” we mean to read the correct name for the decimal. 0.5 is not “zero point five;” it is “five tenths.” Once we have the technically correct name, we write the corresponding fraction and then reduce it to lowest terms. Example: Convert 0.35 to a fraction. Read it: 0.35 is “thirty-five hundredths.” Write it: Thirty-five hundredths is 35/100. Reduce it: 35 100 = 5∙7 5∙20 = 7 20 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates