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Equal and comparing fractions

Numbers · level 3, lesson 6 of 7

A question to start

Two friends order the same size of pizza. Ana's is cut into 3 slices and she eats 2. Ben's is cut into 6 slices and he eats 4. Who ate more?

Ben ate more slices. But his slices were smaller. In fact, they ate exactly the same amount: \(\tfrac23\) and \(\tfrac46\) are two names for the same number. This lesson is about why that happens, how to find all the names of a fraction, and how to tell which of two fractions is bigger even when they look nothing alike.

The idea

Fraction strips

Here is a set of fraction strips: strips of paper all the same length, one whole, each cut into equal parts.

1halvesthirds quarterssixthseighths

Look down the dashed lines. The orange \(\tfrac23\) ends at exactly the same place as the orange \(\tfrac46\). The blue \(\tfrac12\), \(\tfrac24\) and \(\tfrac48\) all end at the same place too. Fractions that are the same amount, the same point on the number line, are called equivalent fractions ("equivalent" means "equal in value").

Why cutting pieces in two keeps the amount the same

Take \(\tfrac23\). Now cut every third in half. What happens?

  • The whole now has twice as many pieces: \(3 \times 2 = 6\). So the pieces are sixths.
  • The part you had also has twice as many pieces: \(2 \times 2 = 4\).
  • But you did not add or remove any pizza. You only cut. The amount is the same.

So \(\tfrac23 = \tfrac{2 \times 2}{3 \times 2} = \tfrac46\). Cut every piece into 3 instead, and you get \(\tfrac{6}{9}\). Into 4, and you get \(\tfrac{8}{12}\). In general:

Multiplying the numerator and the denominator by the same number gives an equivalent fraction.

\[\frac23 = \frac46 = \frac69 = \frac{8}{12} = \frac{10}{15} = \cdots\]

A fraction has infinitely many names, one for every way of cutting the pieces smaller.

Be careful: it must be the same number on top and bottom, and it must be multiplying (or dividing). Adding does not work: \(\tfrac{2+1}{3+1} = \tfrac34\), and \(\tfrac34\) is bigger than \(\tfrac23\) (look at the strips).

Going the other way: simplifying

We can also glue pieces together. In \(\tfrac{6}{8}\), glue the eighths together in pairs. There are now 4 pieces in the whole (quarters), and you have 3 of them:

\[\frac68 = \frac{6 \div 2}{8 \div 2} = \frac34\]

Dividing the top and bottom by the same number is called simplifying. A fraction is in simplest form (or lowest terms) when no whole number bigger than 1 divides both the top and the bottom. \(\tfrac34\) is in simplest form; \(\tfrac68\) is not, because 2 divides both 6 and 8.

To simplify quickly, divide by the biggest number that goes into both. \(\tfrac{12}{18}\): 6 goes into both, so \(\tfrac{12}{18} = \tfrac23\). If you only spot 2 at first, that is fine: \(\tfrac{12}{18} = \tfrac69\), then 3 goes into both again: \(\tfrac69 = \tfrac23\). You get there in two steps.

Comparing fractions

Which is bigger? There are four good ways, and the trick is to pick the easiest.

1. Same denominator: compare numerators. \(\tfrac58\) and \(\tfrac38\) are both eighths, so they are the same size of piece. 5 pieces beat 3 pieces: \(\tfrac58 > \tfrac38\).

2. Same numerator: compare the size of the pieces. \(\tfrac35\) and \(\tfrac37\) both have 3 pieces. Fifths are bigger than sevenths (fewer people sharing). So 3 fifths beat 3 sevenths: \(\tfrac35 > \tfrac37\). With the same numerator, the smaller denominator wins.

3. Make a common denominator. For \(\tfrac23\) and \(\tfrac34\), find a size of piece that both can be cut into. Twelfths work, because 12 is in the 3 times table and the 4 times table:

\[\frac23 = \frac{8}{12} \qquad \frac34 = \frac{9}{12}\]

Now they are both twelfths, and 9 beats 8. So \(\tfrac34 > \tfrac23\), by just \(\tfrac{1}{12}\).

2/33/4 8/129/12

A quick way to find a common denominator is to multiply the two denominators (\(3 \times 4 = 12\)). It always works, though sometimes a smaller one works too (for halves and quarters, quarters are enough).

4. Use a benchmark. A benchmark is a fraction you know well, like \(\tfrac12\) or 1, used to compare others. Which is bigger, \(\tfrac37\) or \(\tfrac59\)?

  • \(\tfrac37\): half of 7 is \(3\tfrac12\), and 3 is less than that. So \(\tfrac37\) is less than \(\tfrac12\).
  • \(\tfrac59\): half of 9 is \(4\tfrac12\), and 5 is more. So \(\tfrac59\) is more than \(\tfrac12\).

So \(\tfrac37 < \tfrac12 < \tfrac59\), and \(\tfrac59\) is bigger. No common denominator needed!

Another benchmark: how far from 1? Which is bigger, \(\tfrac78\) or \(\tfrac{9}{10}\)? \(\tfrac78\) is \(\tfrac18\) short of a whole; \(\tfrac{9}{10}\) is \(\tfrac1{10}\) short. A tenth is a smaller gap than an eighth, so \(\tfrac{9}{10}\) is closer to 1, and bigger.

Worked examples

1. Find the missing number. \(\tfrac34 = \tfrac{\square}{20}\).

The denominator went from 4 to 20, which is times 5. Do the same to the top: \(3 \times 5 = 15\). So \(\tfrac34 = \tfrac{15}{20}\).

2. Simplify. Write \(\tfrac{18}{24}\) in simplest form.

6 divides both 18 and 24: \(\tfrac{18 \div 6}{24 \div 6} = \tfrac34\). 3 and 4 have no common factor bigger than 1, so we are done.

3. Compare with a common denominator. Which is bigger, \(\tfrac56\) or \(\tfrac79\)?

\(6 \times 3 = 18\) and \(9 \times 2 = 18\), so eighteenths work. \(\tfrac56 = \tfrac{15}{18}\) and \(\tfrac79 = \tfrac{14}{18}\). So \(\tfrac56\) is bigger, by \(\tfrac{1}{18}\).

4. Put in order. Order from smallest: \(\tfrac12\), \(\tfrac25\), \(\tfrac{7}{10}\).

Use tenths: \(\tfrac12 = \tfrac{5}{10}\) and \(\tfrac25 = \tfrac{4}{10}\). Order: \(\tfrac25\), \(\tfrac12\), \(\tfrac{7}{10}\).

5. Benchmark. Which is bigger, \(\tfrac{4}{9}\) or \(\tfrac{6}{11}\)?

\(\tfrac49\) is less than a half (half of 9 is \(4\tfrac12\)). \(\tfrac{6}{11}\) is more than a half (half of 11 is \(5\tfrac12\)). So \(\tfrac{6}{11}\) is bigger.

Try it

  1. Write three fractions equivalent to \(\tfrac35\).
  2. Fill in: (a) \(\tfrac23 = \tfrac{\square}{12}\) (b) \(\tfrac{5}{\square} = \tfrac{20}{24}\) (c) \(\tfrac{\square}{7} = \tfrac{12}{28}\)
  3. Simplify: (a) \(\tfrac{10}{15}\) (b) \(\tfrac{16}{20}\) (c) \(\tfrac{21}{28}\) (d) \(\tfrac{24}{36}\).
  4. Which is bigger? (a) \(\tfrac47\) or \(\tfrac67\) (b) \(\tfrac25\) or \(\tfrac28\) (c) \(\tfrac34\) or \(\tfrac58\)
  5. Use \(\tfrac12\) to decide: which is bigger, \(\tfrac{5}{12}\) or \(\tfrac{4}{7}\)?
  6. Order from smallest to biggest: \(\tfrac23\), \(\tfrac12\), \(\tfrac56\), \(\tfrac14\).
  7. Leo says \(\tfrac23 = \tfrac34\) "because I added 1 to the top and bottom". Explain why Leo is wrong.
  8. Harder: find a fraction that lies between \(\tfrac13\) and \(\tfrac12\).
Answers
  1. For example \(\tfrac{6}{10}\) (times 2), \(\tfrac{9}{15}\) (times 3), \(\tfrac{12}{20}\) (times 4).
  2. (a) \(3 \times 4 = 12\), so \(2 \times 4 = 8\): \(\tfrac{8}{12}\). (b) \(5 \times 4 = 20\), so \(24 \div 4 = 6\): \(\tfrac56\). (c) \(28 \div 4 = 7\), so \(12 \div 4 = 3\): \(\tfrac37\).
  3. (a) Divide by 5: \(\tfrac23\). (b) Divide by 4: \(\tfrac45\). (c) Divide by 7: \(\tfrac34\). (d) Divide by 12: \(\tfrac23\).
  4. (a) Same denominator: \(\tfrac67\). (b) Same numerator, fifths are bigger pieces: \(\tfrac25\). (c) \(\tfrac34 = \tfrac68\), which is more than \(\tfrac58\): \(\tfrac34\).
  5. Half of 12 is 6, and 5 is less, so \(\tfrac{5}{12} < \tfrac12\). Half of 7 is \(3\tfrac12\), and 4 is more, so \(\tfrac47 > \tfrac12\). So \(\tfrac47\) is bigger.
  6. In twelfths: \(\tfrac{8}{12}\), \(\tfrac{6}{12}\), \(\tfrac{10}{12}\), \(\tfrac{3}{12}\). Order: \(\tfrac14\), \(\tfrac12\), \(\tfrac23\), \(\tfrac56\).
  7. Adding the same number to top and bottom does not keep a fraction the same. In twelfths, \(\tfrac23 = \tfrac{8}{12}\) and \(\tfrac34 = \tfrac{9}{12}\), which are different. Only multiplying (or dividing) top and bottom by the same number keeps the amount, because that is cutting (or gluing) every piece the same way.
  8. Use a common denominator: \(\tfrac13 = \tfrac26\) and \(\tfrac12 = \tfrac36\). There is no whole number between 2 and 3, so cut smaller: \(\tfrac13 = \tfrac{4}{12}\) and \(\tfrac12 = \tfrac{6}{12}\). So \(\tfrac{5}{12}\) is between them. (Many others work too, such as \(\tfrac25\): \(\tfrac13 = \tfrac{5}{15}\), \(\tfrac25 = \tfrac{6}{15}\), and \(\tfrac25 < \tfrac12\) since 2 is less than half of 5.)

Common mistake

Adding to the top and bottom. \(\tfrac12\) and \(\tfrac23\) are not equivalent, even though you can get one from the other by adding 1 to the top and the bottom. Only multiplying or dividing both by the same number keeps a fraction the same.

Thinking bigger numbers mean a bigger fraction. \(\tfrac{5}{12}\) has bigger numbers than \(\tfrac12\), but it is smaller. Compare the amounts, not the size of the numbers written down.

Where it leads