Equivalent fractions
Equivalent fractions are fractions which mean exactly the same thing but which are written in a completely different form. In the diagram below you can see a circle split into halves and then into quarters. We can see here that one half takes up exactly the same area as two quarters. This means that if I have one half of a pizza and my friend has two quarters of a pizza, we have exactly the same amount of pizza.
\[{1 \over 2} = {2 \times {1 \over 4}} = {2 \over 4}\]
Now every fraction has an infinite number of equivalent fractions, so we will not want to list them all, as this will take quite a while, but it does mean that we can always choose a convenient equivalent fraction when we need one. More of this later.
Calculations with equivalent fraction are pretty straight forward, so long as you remember this simple rule:
If you multiply or divide the numerator by a number, you must do exactly the same to the denominator, and vice versa in order to keep the same value.
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Worked example #1: Equivalent fractions of $\frac{1}{2}$.
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Copy and complete the following list of equivalent fractions of $1 \over 2$.
$${1 \over 2} = {? \over 4} = {? \over 6} = {? \over 8} = {? \over 10} = {? \over 20} = {? \over 50} = {? \over 100}$$ -
\[{1 \over 2} = {2 \over 4} = {3 \over 6} = {4 \over 8} = {5 \over 10} = {10 \over 20} = {25 \over 50} = {50 \over 100}\]Look at the denominators of the first two fractions. Ask yourself what we could multiply 2 by to get 4. 2 has been doubled, so the multiplier is 2. To find an equivalent fraction to one half, we must do the same thing to both top and bottom (× 2). To get from fraction 1 to fraction 3, we will do × 3 and so on.
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\[{1 \over 2} = {2 \over 4} = {3 \over 6} = {4 \over 8} = {5 \over 10} = {10 \over 20} = {25 \over 50} = {50 \over 100}\]
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Worked example #2: Equivalent fractions of $\frac{3}{5}$.
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Fill in the equivalent fractions of $3 \over 5$. $$\frac{3}{5}=\frac{?}{10}=\frac{9}{?}=\frac{12}{?}=\frac{?}{30}=\frac{?}{50}$$
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$$\frac{3}{5}=\frac{6}{10}=\frac{9}{15}=\frac{12}{20}=\frac{18}{30}=\frac{30}{50}$$
Start by looking just at the first 2 fractions, ${3 \over 5} = {? \over 10}$. To get from 5 to 10, we must multiply by 2, so the numerator must be multiplied by 2 also, giving 6.
Now consider fractions 1 & 3, ${3 \over 5} = {? \over 15}$. To get from 3 to 9, we must multiply by 3, so we must do the same to the denominator, giving 15. And so on...
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$$\frac{3}{5}=\frac{6}{10}=\frac{9}{15}=\frac{12}{20}=\frac{18}{30}=\frac{30}{50}$$
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Worked example #3: Cancelling down fractions
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Cancel down the following fractions into their simplest forms
- $\frac{4}{8}$
- $\frac{9}{15}$
- $\frac{36}{45}$
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Examine each fraction’s numerator and denominator to find a common factor between them.
- 4 and 8 have a common factor of 4, so each number in the fraction (numerator and denominator) must be divided by 4 $\frac{4\div4}{8\div4}=\frac{1}{2}$
- 9 and 15 both divide by 3, so we divide each number by 3, $\frac{9\div3}{15\div3}=\frac{3}{5}$
- 36 and 45 have a common factor of 9, $\frac{36\div9}{45\div9}=\frac{4}{5}$
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- $\frac{4}{8}=\frac{1}{2}$
- $\frac{9}{15}=\frac{3}{5}$
- $\frac{36}{45}=\frac{4}{5}$
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Worked example #4: Put fractions in order of size.
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Place the following fractions in order of size, smallest first $$\frac{14}{20},\ \ \frac{4}{5},\ \ \frac{3}{4}$$
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Put all of the fractions over the same denominator, in this case the common denominator will be 20.
${14 \over 20}$ already has the correct denominator.
$$\begin{aligned}\frac{4}{5}=\frac{16}{20} &\\ \frac{3}{4}=\frac{15}{20}\end{aligned} $$
Now that each fraction is written with the same denominator, we can simply look at the numerators of the fractions and put them in order.
$$\frac{14}{20},\ \frac{15}{20}=\frac{3}{4},\ \frac{16}{20}=\ \frac{4}{5}$$ We must now write out the original fractions in order.It is never a good idea to guess these types of question. You may think you have an uncanny knack for judging the size of fractions. Sadly the probabilities are against you. This is certainly a skill which you will acquire over the years as you become more experienced, but do make sure that any method of your own devising, actually works, in every case.
The concept of proof in mathematics is a very important one which will be dealt with in time, but until you are quite highly skilled in this, please show a little humility. For further details on this idea, please read my essay on the Dunning Kruger research.
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$$\frac{14}{20},\ \frac{3}{4},\ \frac{4}{5}$$
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Exercise #1: Put fractions in order of size.
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Find the value of the ? in each of the following. -
1.$${1 \over 3} = {? \over 6}$$
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2.$${3 \over 5} = {? \over 15}$$\[9\]
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3.$${2 \over 9} = {6 \over ?}$$\[27\]
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4.$${4 \over 7} = {16 \over ?}$$\[28\]
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5.$${7 \over 4} = {? \over 36}$$\[63\]
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6.$${8 \over 9} = {? \over 81}$$\[72\]
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7.$$\frac{13}{15} = \frac{?}{60}$$\[52\]
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8.$$\frac{12}{17} = \frac{48}{?}$$\[68\]
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9.$$\frac{7}{8} = \frac{56}{?}$$\[64\]
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10.$$\frac{11}{14} = \frac{?}{56}$$\[44\]
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Exercise #2: Cancelling fractions
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Cancel down the following fractions into their simplest forms
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1.$$\frac{10}{15}$$$$\frac{2}{3}$$
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2.$$\frac{6}{18}$$$$\frac{1}{3}$$
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3.$$\frac{32}{56}$$$$\frac{4}{7}$$
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4.$$\frac{24}{27}$$$$\frac{8}{9}$$
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5.$$\frac{18}{42}$$$$\frac{3}{7}$$
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6.$$\frac{40}{800}$$$$\frac{1}{20}$$
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7.$$\frac{39}{45}$$$$\frac{13}{15}$$
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8.$$\frac{150}{900}$$$$\frac{1}{6}$$
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9.$$\frac{32}{512}$$$$\frac{1}{16}$$
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10.$$\frac{37}{222}$$$$\frac{1}{6}$$
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Exercise #3: Putting sets of fractions in order
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Place each of the following sets of fractions in ascending order (smallest first).
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1.$$\frac{1}{4}, \frac{3}{8}$$$$\frac{1}{4}, \frac{3}{8}$$
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2.$$\frac{4}{5}, \frac{2}{3}$$$$\frac{2}{3}, \frac{4}{5}$$
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3.$$\frac{11}{30}, \frac{2}{5}, \frac{4}{15}$$$$\frac{4}{15}, \frac{11}{30}, \frac{2}{5}$$
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4.$$\frac{1}{2}, \frac{3}{8}, \frac{2}{3}$$$$\frac{3}{8}, \frac{1}{2}, \frac{2}{3}$$
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5.$$\frac{7}{8}, \frac{3}{4}, \frac{6}{7}$$$$\frac{3}{4}, \frac{6}{7}, \frac{7}{8} $$
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6.$$\frac{1}{3}, \frac{2}{5}, \frac{3}{10}$$$$\frac{3}{10}, \frac{1}{3}, \frac{2}{5}$$
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7.$$\frac{5}{6}, \frac{7}{10}, \frac{4}{5}$$$$\frac{7}{10}, \frac{4}{5}, \frac{5}{6}$$
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8.$$\frac{9}{32}, \frac{7}{16}, \frac{3}{4}, \frac{5}{8}$$$$\frac{9}{32}, \frac{7}{16}, \frac{5}{8}, \frac{3}{4}$$
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9.$$\frac{1}{2}, \frac{9}{20}, \frac{3}{5}, \frac{8}{15}$$$$\frac{9}{20}, \frac{1}{2}, \frac{8}{15}, \frac{3}{5}$$
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10.$$\frac{13}{24}, \frac{5}{6}, \frac{7}{12}, \frac{3}{4}$$$$\frac{13}{24}, \frac{7}{12}, \frac{3}{4}, \frac{5}{6}$$
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