Comparing fractions without tears: 4 examples

This is the second instalment in a series on fractions. In the last post, we made sense of fractions and their two parts: The numerator and the denominator. This post will look at ways in which we can compare fractions with different denominators. If I say I ate 4 pieces of cake and you say that you ate 5. If the pieces are of different sizes, how can we tell who ate more?

A challenge between siblings

Last time, we met Birthday Bob. Dear reader, meet his sister, Anniversary Aimee. Both siblings love to eat cake but neither has an invitation for the month of October. What to do? They create a friendly contest: Each Saturday, they will bake two identical cakes and compete to see who can eat more cake. Like last time, the cakes will be rectangular to make the cutting and comparing easier.

Same number of pieces, different sizes

On the first Saturday, Bob managed to eat three quarters, $\boldsymbol{\frac{3}{4}}$ , of his cake, whilst Aimee ate three fifths, $\boldsymbol{\frac{3}{5}}$ , of hers. Which sibling ate more cake that day?

Notice the pattern here. Each sibling ate three pieces, that’s what the numerators tell us. Now, the sizes of the pieces are different. The question then becomes: Whose three pieces are larger?

Bob’s cake was cut into fewer pieces than Aimee’s. This makes his pieces larger. Therefore, Bob wins by eating three larger pieces than his sister’s three.

Different number of pieces, different sizes

On the second Saturday of the month, Bob ate three quarters, $\boldsymbol{\frac{3}{4}}$ , again. This time, Aimee ate four fifths, $\boldsymbol{\frac{4}{5}}$ , of her cake. How does this change the comparison?

If you’re finding it tricky to compare the two fractions, let’s flip the problem and ask, which sibling left less of the cake uneaten?

Bob has left one quarter, $\boldsymbol{\frac{1}{4}}$ , of the cake while Aimee ate all but one fifth, $\boldsymbol{\frac{1}{5}}$ , of hers. Now we have a scenario like the one in the previous section. Each sibling has left one piece uneaten. Remember that the more pieces you cut your cake into, the smaller the pieces need to be.

$\boldsymbol{\frac{1}{4}}$ is greater than $\boldsymbol{\frac{1}{5}}$ . Aimee has left the lesser amount and she wins the second Saturday battle!

The siblings make their fractions a little more complicated for each other!

On the third Saturday, things got more complicated. Bob ate two fifths, $\boldsymbol{\frac{2}{5}}$ , of his cake, while Aimee ate eight fifteenths, $\boldsymbol{\frac{8}{15}}$ , of hers! Yep, I have chosen the dweebiest siblings for this cake battle!

Luckily, there is a way to write the two fractions over the same denominator: Let’s further cut Bob’s cake so that the fifths become fifteenths, as can be seen in the figure below:

Figure 1: By cutting each of the five pieces of cake into 3 equal pieces, we now have fifteen pieces in total

We have now turned Bob’s two pieces into six pieces, each a fifteenth of the total cake. The comparison is now between bob’s six fifteenths, $\boldsymbol{\frac{6}{15}}$ , and Aimee’s eight fifteenths, $\boldsymbol{\frac{8}{15}}$ .

Aimee wins again!

Score Update: Aimee 2, Bob 1

When things get even more complicated

Dear reader, the title promised “no tears”. In this section, I may not be able to keep my promise! Either way, I would love to get your feedback on my writing in the comments. Am I making this clear enough? If not, where did I lose you?

On the final Saturday of the month, the two siblings outdid themselves with dweebiness: Bob ate four sevenths, $\boldsymbol{\frac{4}{7}}$ , of his cake and Aimee seven elevenths, $\boldsymbol{\frac{7}{11}}$ , of hers. They deliberately chose to divide their respective cakes into a number of pieces that’s a prime number. You’ve got to love that!

Unlike last time, we cannot further cut Bob’s cake so it has 11 equal pieces. Neither can we cut Aimee’s cake so it has 7 equal pieces. The solution is to further cut each of the cakes so that it has 77 equal pieces, that’s 7 times 11.

Remember that we can make the number of pieces greater by multiplying the denominator by a number, as long as we eat more pieces, multiplying the numerator by the same number.

Bob’s cake

$\boldsymbol{\frac{4 \times 11}{7 \times 11} = \frac{44}{77}}$

Aimee’s cake

$\boldsymbol{\frac{7 \times 7}{11 \times 7} = \frac{49}{77}}$

Now we have equally sized pieces and we can see that Aimee has eaten 5 more of them.