6.1.5 Equivalent Fractions

Activity symbolActivity: Equivalent Fractions

This activity may require you to work out some of the solutions on paper. It is recommended that you use your math notebook to keep track of your work.

(a) Use your knowledge of equivalent fractions to fill in the missing numbers.

  • (i) equation left hand side two divided by three equals right hand side question mark divided by 18
  • (ii) equation left hand side four divided by five equals right hand side question mark divided by 20
Hint symbol

Discussion

The denominator of the fraction on the left is 3. What do you have to multiply it by to get 18? Remember that if you multiply the denominator by a particular number, you must do the same to the numerator.

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Answer

(a)

(i) The 3 is multiplied by 6 to reach 18, so this is the number that is used to multiply the numerator and the denominator. The missing number is 12.

(ii) Similarly, since five multiplication four equals 20, multiply the numerator and the denominator by four.

The missing number is 16.

(b) Which of the following three fractions are equivalent to each other?

  • (i) seven divided by eight
  • (ii) 35 divided by 40
  • (iii) 49 divided by 64
Hint symbol

Discussion

Did you check to see if multiplying the numerator, 7, by any numbers gave the numerator of another of the given fractions? If so, try multiplying the denominator, 8, by the same number and see what denominator is yielded.

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Answer

(b) Multiplying the numerator and the denominator of the first fraction by five gives

Only the two fractions labelled (i) and (ii) are equivalent.

You cannot create 49 divided by 64 from seven divided by eight by multiplying numerator and denominator by the same number.

If you multiply seven divided by eight by 7 top and bottom, which is to multiply it by seven divided by seven, you get 49 divided by 56, which isn’t 49 divided by 64, and if you multiply seven divided by eight by 8 top and bottom (that is, multiply it by eight divided by eight) you get 56 divided by 64, which isn’t 49 divided by 64, either.

(c) Reduce (cancel) the following fractions into their lowest terms.

  • (i) two divided by eight
  • (ii) 10 divided by 30
  • (iii) 27 divided by 81
  • (iv) 60 divided by 12
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Discussion

Look at each fraction individually. What number can divide evenly into both the numerator and the denominator of each? Remember, there could be more than one option.

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Answer

(c)

  • (i) Dividing the top and the bottom by 2 gives equation left hand side two super one divided by eight super four equals right hand side one divided by four.

  • (ii) Dividing the top and the bottom by 10 (or by 5 and then by 2) gives equation left hand side 10 super one divided by 30 sub three equals right hand side one divided by three.

  • (iii) Dividing the top and the bottom by 9 and then by 3 gives equation sequence 27 cubed divided by 81 sub nine equals three super one divided by nine sub three equals one divided by three.

  • Dividing the top and the bottom by 4 and then 3 gives: equation sequence 60 super 15 divided by 12 sub three equals 15 super five divided by three sub one equals five divided by one equals five.

(Of course it does not matter which number you divide by first, and there are even more choices than the ones shown. Reducing is a process.)

If you find it difficult to spot the numbers to divide by, try to work systematically by trying 2, 3, 5, … in turn. You can check these results by using your calculator, as you will see shortly.

Pencast symbol Does the notation of reducing a fraction used in part (c) of this activity look new to you? Click the “View document” link for an explanation.

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6.1.6 Fractions of a Group