Comparing Fractions

Comparing Fractions exercise

• Compare the fractions with the same denominator.

  Hints

  When we compare fractions with the same denominator, we need to look at the numerator.

  Remember,

  • < means less than
  • > means greater than.

  For example, $\dfrac{2}{5}$ is less than $\dfrac{3}{5}$ because $2 < 3$.

  There are two correct choices.

  Solution

  The two correct comparisons are:

  • $\frac{1}{4}$ < $\frac{3}{4}$

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  • $\frac{7}{9}$ > $\frac{4}{9}$

  ${}$

  To compare fractions with the same denominator, we look at the numerator.

  Here, 1 < 4 therefore, $\frac{1}{4}$ < $\frac{3}{4}$ and 7 > 4 therefore, $\frac{7}{9}$ > $\frac{4}{9}$.

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  These two choices are incorrect:

  • $\frac{2}{7}$ > $\frac{5}{7}$ 2 is actually less than 5 so it should read $\frac{2}{7}$ < $\frac{5}{7}$.
  • $\frac{8}{11}$ < $\frac{3}{11}$ 8 is actually greater than 3 so it should read $\frac{8}{11}$ > $\frac{3}{11}$.

• Compare the fractions with the same numerator.

  Hints

  Remember, for fractions with the same numerators, we need to look at the denominators. In this case, the fraction with the smaller denominator is larger. This is because a smaller denominator means that the whole has been shared into fewer equal parts.

  In this example, we can see that $\frac{1}{3}$ is greater than $\frac{1}{5}$ because the whole has been shared into fewer parts so one of those parts is larger.

  You could draw fraction bars to help you compare the fractions above.

  Solution

  1.$\frac{1}{3}$ > $\frac{1}{6}$

  2.$\frac{2}{7}$ < $\frac{2}{3}$

  3.$\frac{5}{8}$ < $\frac{5}{6}$

  4.$\frac{1}{5}$ = $\frac{1}{5}$

  5.$\frac{9}{10}$ > $\frac{9}{14}$

• How do we compare fractions with different denominators and numerators?

  Hints

  To compare fractions with different numerators and denominators, we make the bottom number of each fraction the same.

  When converting $\frac{1}{3}$ to a fraction with a denominator of 12, we multiply the denominator of 3 by 4. We then also need the multiply the numerator by 4 too.

  To compare $\frac{4}{12}$ and $\frac{9}{12}$, imagine a pie split into 12 pieces. One has 4 pieces left and one has 9 pieces left.

  Which one has more pieces left?

  Solution

  When we compare fractions with different denominators and numerators, we must find a common denominator This could be the lowest common multiple or LCM.

  If we were comparing $\frac{1}{3}$ and $\frac{3}{4}$, we would need to find a common denominator of 3 and 4. To do this, we can multiply 3 x 4 to get 12.

  Next, rewrite each fraction with a denominator of 12.

  $\frac{1}{3}$ = $\mathbf{\frac{4}{12}}$ and $\frac{3}{4}$ = $\mathbf{\frac{9}{12}}$.

  Now we can compare these fractions. $\frac{4}{12}$ is less than $\frac{9}{12}$ therefore

  $\frac{1}{3}$ is less than $\frac{3}{4}$.

• Can you order the fractions?

  Hints

  Can you find a common denominator for all of the fractions?

  A common denominator for all of the fractions is 24.

  Convert all of the fractions so they have 24 as a denominator and then compare.

  Draw visual representations of each fraction if this helps you.

  Solution

  The correct order of the fractions from smallest to greatest is:

  1.$\frac{1}{12}$ = $\frac{2}{24}$

  2.$\frac{1}{6}$ = $\frac{4}{24}$

  3.$\frac{6}{24}$

  4.$\frac{5}{12}$ = $\frac{10}{24}$

  5.$\frac{1}{2}$ = $\frac{12}{24}$

  6.$\frac{2}{3}$ = $\frac{16}{24}$

  7.$\frac{22}{24}$

  ${}$

  When we give all of the fractions a denominator of 24, we can compare and sort them more easily.

• Compare the fractions.

  Hints

  Look at the two fractions. Are the denominators or the numerators the same?

  Remember, the numerator is the top number and the denominator is the bottom number.

  In the fractions above, the denominators are the same so we can compare the numerators.

  Use the drawings to help too, which circle has more parts shaded?

  Solution

  $\frac{1}{5}$ < $\frac{4}{5}$

  In these two fractions, the denominators are the same so we compare the numerators. 1 < 4, therefore $\frac{1}{5}$ < $\frac{4}{5}$.

  We can also use the drawings and see that fewer parts are shaded in the circle representing $\frac{1}{5}$.

• Can you compare these fractions?

  Hints

  When comparing two fractions with the same denominators, compare the numerators.

  If the fractions have different numerators and denominators, find a common denominator and compare.

  Remember, for fractions with the same numerators, we need to look at the denominators. In this case, the fraction with the smaller denominator is larger. This is because a smaller denominator means that the whole has been shared into fewer equal parts.

  You can also make the numerators the same and compare.

  For example, we could make $\frac{1}{3}$ have a numerator of 4 by multiplying the numerator and denominator by 4 which equals $\frac{4}{12}$.

  We can now see that $\frac{4}{10}$ is greater than $\frac{1}{3}$ as $\frac{4}{10}$ is greater than $\frac{4}{12}$.

  You can draw visual representations of each fraction to help you compare.

  For example, we can use these drawings to compare $\frac{1}{7}$ to $\frac{1}{4}$.

  Solution

  Here we can see the fractions sorted into the correct groups.