6.1.3 Equivalent Fractions

[ Why doesn’t this work with zero? You thought about that in Unit 2. ] If you multiply the numerator (the number on top) and the denominator (the number underneath) of any fraction by the same number (except zero), you will get a fraction that is equivalent to the original one:

This diagram shows that one-half is equivalent to two-quarters by multiplying the top and bottom of one-half by two. It also shows that two-quarters are equivalent to six-twelfths by multiplying the top and bottom of two-quarters by three and that six-twelfths are equivalent to sixty one-hundred and twentieths by multiplying the top and bottom of six-twelfths by ten.

Note: You must multiply the numerator and denominator by the same number because it is the same as multiplying by one, so it doesn’t change the value of the fraction.

You can also generate equivalent fractions by dividing the numerator (top number) and the denominator (bottom number) of the fraction by the same number (again, not zero).

This diagram shows that forty-sixtieths is equivalent to four-sixths (by dividing top and bottom by ten) and that four-sixths is equivalent to two-thirds (by dividing the top and bottom by two).

Pencast symbol

Would you have reduced this fraction using different steps? This process of dividing the numerator (the number above the line) and the denominator (the value below the line) by the same number is known as “cancelling,” “reducing,” or “simplifying.” If there is no whole number that can be divided into both the numerator and the denominator, the fraction is said to be in lowest terms, or fully reduced. Answers are usually left in this form, as they are easier to visualize and understand (click on “View document”).

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6.1.4 Pizza Math