Equivalent fractions I

Counting how many things there are in a collection of some kind is usually not too difficult. It can become difficult if we are trying to count, for example, the number of leaves on a large tree. This is because the number is likely to be large and it would be easy to make a mistake.

But, even when the number is large and the counting is difficult we feel certain that there is an exact number that says how many things there are.

A different situation arises when we wish to talk about the length of something or its weight or its temperature. We have a problem because the length is likely to be somewhere in-between two whole numbers of metres, and the weight is nearly always between two whole numbers of kilograms and the temperature is hardly ever exactly a whole number of degrees.

For this kind of measurement, mathematicians in ancient times invented the idea of fractions. This was to make it possible to be precise about measurements where there was something left over after counting off the whole numbers of the measuring unit.

We would like to be able to add measurements that have fractional parts just as we can add whole numbers but, at first, it is not easy to see how this can be done. To make any headway at all, we must first have a clear understanding of what a fraction is and how it is notated.

Eleven statements about fractions

1	The fractional part is the part left over after the whole number of measurement units has been counted off.
2	If we put together two of the same fractional part and find that we then have a full unit, we say the fractional part was $\frac{1}{2}$12​ (one-half), meaning it took exactly two pieces to make one whole.
3	If we can make a whole unit by putting together three equal sized parts, then we say the fractional part was $\frac{1}{3}$13​ (one-third), meaning it took three pieces to make one whole.
4	A more complicated example: If by putting together five pieces of a particular size we can make exactly three units, we say the pieces correspond to the fraction $\frac{3}{5}$35​ (three-fifths), meaning it takes five of them to make three whole units. The size of the part can be thought of as three parts out of a unit that has been split into five equal pieces. Look at the diagram below to help clarify this idea.
5	A tricky example: If four equal sized pieces make exactly two units, the pieces represent the fraction $\frac{2}{4}$24​, meaning four pieces make two units. Now, if four pieces make two units then we must need just two pieces to make one unit. But, this is what happens for pieces corresponding to the fraction  $\frac{1}{2}$12​. Apparently, the size of a fractional part can be represented in more than one way. The two fractions $\frac{2}{4}$24​ and $\frac{1}{2}$12​ are called equivalent fractions.
6	When fractions are notated in this way we call the upper number the numerator and the lower number the denominator: $\frac{\text{numerator}}{\text{denominator}}$numeratordenominator​.
7	We can express the idea of a fraction with a symbol like $\frac{a}{b}$ab​ where the letters $a$a and $b$b can be replaced by any numbers.
8	If the numerator $a$a is less than the denominator $b$b, then the fraction is less than one whole unit because it takes $b$b of the pieces to make a smaller number $a$a of whole units.
9	If $a$a and $b$b are the same number, then the fraction is equivalent to $\frac{1}{1}$11​ which says $1$1 of the parts is needed to make $1$1 unit. That is, the part is equal to a unit. We write $\frac{a}{a}=1$aa​=1 for any number $a$a.
10	The idea of a fraction is extended to include the possibility that the numerator could be greater than the denominator. This would be the case, for example, if a fraction were written $\frac{3}{2}$32​. This says $2$2 groups of the 'part' makes $3$3 units. Fractions of this kind are called improper fractions but they can be just as useful as the proper kind.
11	Comparing the sizes of fractional parts can be difficult but we can do it with the help of equivalent fractions. Fractions can be compared easily if their denominators are the same. So, we look for fractions equivalent to the ones we are trying to compare, that have the same denominator.

Comparing fractions

Imagine that a measurement of two samples of something has been made. One sample has $10$10 whole measurement units and a fractional part represented by $\frac{2}{3}$23​. That is, three groups of the fractional part make two wholes. The other sample has $10$10 whole measurement units and a fractional part represented by $\frac{5}{8}$58​. That is, eight groups of the fractional part make five units.

If we were able to put the two samples side-by-side we might be able to see which was the bigger. However, with just the measurements it is hard to tell which is larger out of $\frac{2}{3}$23​ and $\frac{5}{8}$58​.

We use the idea of equivalent fractions. In the first sample, if $3$3 groups of the fractional part make $2$2 units, then $24$24 groups of the fractional part must make $16$16 units. (We multiplied both numbers by $8$8.) Also, in the second sample, if $8$8 groups of the fractional part make $5$5 units, then $24$24 groups of the fractional part must make $15$15 units. (We multiplied both numbers by $3$3.) Therefore, the first sample is bigger than the second.

We are using the fact that the two fractions we are comparing are equivalent to $\frac{16}{24}$1624​ and $\frac{15}{24}$1524​. These fractions can be compared because their denominators are the same.

But how did we know to choose $24$24 as the denominator for each of the fractions?

The key to the problem is to realise that for any fraction, we can make another fraction that is equivalent to it by multiplying (or dividing) both its numerator and its denominator by the same number. 

Example 1

Make $13$13 more fractions that are equivalent to $\frac{1}{2}$12​ and $13$13 more fractions that are equivalent to $\frac{2}{3}$23​. Which pairs of equivalent fractions could you use to compare the sizes of the fractions $\frac{1}{2}$12​ and $\frac{2}{3}$23​?

$\frac{1}{2}$12​

$\frac{2}{4}$24​

$\frac{3}{6}$36​

$\frac{4}{8}$48​

$\frac{5}{10}$510​

$\frac{6}{12}$612​

$\frac{7}{14}$714​

$\frac{8}{16}$816​

$\frac{9}{18}$918​

$\frac{10}{20}$1020​

$\frac{11}{22}$1122​

$\frac{12}{24}$1224​

$\frac{13}{26}$1326​

$\frac{14}{28}$1428​

$\frac{2}{3}$23​

$\frac{4}{6}$46​

$\frac{6}{9}$69​

$\frac{8}{12}$812​

$\frac{10}{15}$1015​

$\frac{12}{18}$1218​

$\frac{14}{21}$1421​

$\frac{16}{24}$1624​

$\frac{18}{27}$1827​

$\frac{20}{30}$2030​

$\frac{22}{33}$2233​

$\frac{24}{36}$2436​

$\frac{26}{39}$2639​

$\frac{28}{42}$2842​

We look for pairs of fractions with the same denominator. So, we could use
$\frac{1}{2}\equiv\frac{3}{6}$12​≡36​ and $\frac{2}{3}\equiv\frac{4}{6}$23​≡46​ or
$\frac{1}{2}\equiv\frac{3}{12}$12​≡312​ and $\frac{2}{3}\equiv\frac{8}{12}$23​≡812​ or
$\frac{1}{2}\equiv\frac{9}{18}$12​≡918​ and $\frac{2}{3}\equiv\frac{12}{18}$23​≡1218​ and so on.

We probably already knew that $\frac{2}{3}$23​ was greater than $\frac{1}{2}$12​ but the same technique can be used in more difficult cases.

 

Finding the difference between fractional parts

Using the equivalences $\frac{2}{3}\equiv\frac{16}{24}$23​≡1624​ and $\frac{5}{8}\equiv\frac{15}{24}$58​≡1524​, we saw that $\frac{2}{3}$23​ is greater than $\frac{5}{8}$58​. But, how much greater?

We want to know the difference between the two fractions, which is the same thing as the subtraction expression: $\frac{2}{3}-\frac{5}{8}$23​−58​. 

This is equivalent to $\frac{16}{24}-\frac{15}{24}$1624​−1524​.

The difference between the number of units that can be made from $24$24 pieces of each size is $16-15=1$16−15=1 unit. So, 24 of these small differences have added up to 1 whole unit. But, this is what occurs with the fraction $\frac{1}{24}$124​.

This is a round-about way of saying that if the denominators are the same we only need to find the difference between the numerators. We can write the working as follows:

$\frac{2}{3}-\frac{5}{8}$23​−58​	$=$=	$\frac{16}{24}-\frac{15}{24}$1624​−1524​
	$=$=	$\frac{16-15}{24}$16−1524​
	$=$=	$\frac{1}{24}$124​

 

Example 2

Find the difference between the fractions $\frac{3}{4}$34​ and $\frac{7}{9}$79​.

Since $9\times4=36$9×4=36, we can multiply the numerator and denominator of the first fraction by $9$9 and the numerator and denominator of the second fraction by $4$4 to get equivalent fractions with the denominator $36$36.

So, we write
$\frac{3}{4}\equiv\frac{27}{36}$34​≡2736​ and
$\frac{7}{9}\equiv\frac{28}{36}$79​≡2836​

Thus, $\frac{7}{9}$79​ is greater than $\frac{3}{4}$34​ and the difference is 

$\frac{7}{9}-\frac{3}{4}$79​−34​	$=$=	$\frac{28}{36}-\frac{27}{36}$2836​−2736​
	$=$=	$\frac{28-27}{36}$28−2736​
	$=$=	$\frac{1}{36}$136​

 

 

Adding fractions

The addition of fractional parts is easy when the denominators are the same. Therefore, we convert fractions to equivalent forms with the same denominator when we wish to add them.

This often results in a fraction that is greater than $1$1 unit. So, we need to see how this is dealt with.

Suppose we wish to add fractions $\frac{1}{2}$12​ and $\frac{1}{6}$16​. This can be done if we write the addition using an equivalent fraction for $\frac{1}{2}$12​.

$\frac{1}{2}+\frac{1}{6}$12​+16​	$=$=	$\frac{3}{6}+\frac{1}{6}$36​+16​
	$=$=	$\frac{4}{6}$46​
	$=$=	$\frac{2}{3}$23​

We divided the numerator and denominator of $\frac{4}{6}$46​ by $2$2 in order to get the equivalent fraction $\frac{2}{3}$23​. 

This last fraction is said to be in lowest form because no further numbers other than $1$1 can be divided into both the numerator and the denominator.

Example 3

Find the sum of $\frac{3}{4}$34​ and $\frac{1}{3}$13​.

$\frac{3}{4}+\frac{1}{3}$34​+13​	$=$=	$\frac{9}{12}+\frac{4}{12}$912​+412​
	$=$=	$\frac{9+4}{12}$9+412​
	$=$=	$\frac{13}{12}$1312​

We have arrived at an improper fraction - a number greater than $1$1.

Knowing what we know about equivalent fractions and the addition process, we see that we can write

$\frac{13}{12}$1312​	$=$=	$\frac{12+1}{12}$12+112​
	$=$=	$\frac{12}{12}+\frac{1}{12}$1212​+112​
	$=$=	$1+\frac{1}{12}$1+112​

We have split the improper fraction into a whole number part and a proper fraction part. Normally, we write this simply as $1\frac{1}{12}$1112​ and we call it a mixed fraction.

 

 

Worked Examples

Question 1

Question 2

Question 3