Equivalent fractions and comparison

More pieces can name the same amount

Mark one half of a strip. Cut each half into two equal parts. There are now four quarters in the whole and two quarters marked:

Cutting each old part changes both counts. Multiply numerator and denominator by the same nonzero whole number: 2/3 = 4/6 = 8/12.

Changing only the numerator changes the amount. Two thirds is not four thirds.

Check your understanding 1 Change the partition, keep the amount

Write a fraction equal to 2/3 with a different denominator. Explain how both counts change. More than one answer is possible.

Your equivalent fraction and explanation

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One answer is 4/6: split each third into two equal parts. Both counts multiply by two. Another is 6/9, multiplying both by three. Copying 2/3 would name the same amount, but would not show the different partition requested here.

Reverse the splitting to simplify

For 8/12, group the twelve small parts into sets of four. Eight marked parts make two groups, and the whole has three groups. So 8/12 = 2/3.

A fraction is in simplest form when its numerator and denominator have no common whole-number factor greater than one. Divide both by a common factor until none remains.

Use the given numerator or denominator

For a missing numerator in 3/5 = Box/10, five becomes ten by multiplying by two. Do the same to three: the missing numerator is six. For a given numerator in 2/3 = 6/(Box), two becomes six by multiplying by three. The denominator must also multiply by three: the missing denominator is nine, giving 6/9.

Check your understanding 2 Which part has been supplied?

Write an equivalent fraction to 3/4 with numerator 9. Then write an equivalent fraction to 1/2 with denominator 8. Write each complete fraction, not only its missing part.

Your two complete equivalent fractions

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For the first, three becomes nine by multiplying by three, so four becomes twelve: 3/4 = 9/12. For the second, two becomes eight by multiplying by four, so one becomes four: 1/2 = 4/8.

3/4 has the right value for the first question, but it does not use the requested numerator. Both the value and the supplied part matter.

Compare equal-sized pieces

Compare 2/3 and 3/4 using equal wholes. Split the whole into twelfths:

Nine twelfths is greater than eight twelfths, so 3/4 > 2/3, or 2/3 < 3/4. A larger numerator alone would not decide the comparison before the pieces matched.

Unlike fractions can also be equal. 2/4 and 3/6 both name one half, so 2/4 = 3/6. Different denominators alone do not tell you whether to write < or >.

Check your understanding 3 Order unlike fractions

Put 1/2, 5/6 and 2/3 in increasing order, then decreasing order. Show equivalent fractions with a shared denominator. You can separate ordered fractions with commas, or use < for increasing order and > for decreasing order.

Your two orders and equivalent fractions

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Use sixths: 1/2 = 3/6, 2/3 = 4/6 and 5/6 stays unchanged. Increasing order is 1/2 < 2/3 < 5/6. Decreasing order reverses it: 5/6 > 2/3 > 1/2. Each list contains all three given values. This compares numbers of the same-sized part.

Check a proposed shortcut

Mia says 6/10 = 3/5 because subtracting three from the numerator and five from the denominator gives three and five. Is that a useful rule?

Check your understanding 4 Hold the marked amount fixed

A strip has twelve equal parts, with nine marked. Name the marked fraction in simplest form. How many of four equal parts of the same whole would be marked?

Your simplified fraction and selected parts

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9/12 = 3/4 because both counts divide by three. Three of the four larger quarter-parts would be marked. You changed the partition, not the marked amount.