Representing Fractions Using a Common Denominator

In order to give their mother a bracelet, Jack gives his father $\frac{3}{35}$ of the price of the present, and Sophie gives him $\frac{4}{45}$. Who has given more, Jack or Sophie?
With the help of a calculator, we can see that:  $\frac{3}{35}\approx$ 0.086 and  $\frac{4}{45}\approx$ 0.089. Therefore it is Sophie who has given more.


Is there another method to solve this type of problem?

I. The method

A. Checking whether two numbers written as fractions are equal

Rule: We do not change the value of a quotient if we multiply its numerator and its denominator by the same non-zero number.


In other words, with letters (a, c, and k represent numbers; c ≠ 0 and k ≠ 0) :


Examples: $\frac{2}{3}=\frac{3\times 2}{3\times 3}=...=\frac{7\times 2}{7\times 3}$ and in particular: $5=\frac{5}{1}=\frac{3\times 5}{3}=...=\frac{9\times 5}{9}=...=\frac{17\times 5}{17}$

B. Reducing to a common denominator

Example 1: We want to write the fractions $\frac{3}{5}$ and $\frac{7}{30}$ with the same denominator.


We note that 30 is a multiple of 5 (30 = 5 × 6). Then $\frac{3}{5}=\frac{6\times 3}{6\times 5}=\frac{18}{30}$


 $\frac{3}{5}$ and $\frac{7}{30}$ represent the same numbers as $\frac{18}{30}$ and $\frac{7}{30}$.


We say that we have represented the fractions using a common denominator.


Example 2: Now we want to represent $\frac{3}{12}$ and $\frac{7}{20}$ using a common denominator. This is not as simple as in the previous case. In the table below, we write the multiples of 12 in the top line and the multiples of 20 in the bottom line, until we have the same number, here: 60.

60 is at the same time a multiple of 12 (60 = 5 × 12) and of 20 (60 = 20 × 3). Therefore we choose the number 60 to represent $\frac{3}{12}$ and $\frac{7}{20}$ using a common denominator.

$\frac{3}{12}=\frac{5\times 3}{5\times 12}=\frac{15}{60}$ and $\frac{7}{20}=\frac{{\displaystyle 3\times 7}}{{\displaystyle 3\times 20}}=\frac{21}{60}$


We have represented $\frac{3}{12}$ and $\frac{7}{20}$ using a common denominator.


Note: 60 is the lowest non-zero common multiple of 12 and of 20. We can write LCM(12; 20) = 60.


Example 3: Represent $\frac{5}{14}$ , $\frac{4}{21}$, and $\frac{7}{6}$ using a common denominator.


We will use the same method but this time with three lines in the table.

Reasoning in the same way as for the previous example, we choose 42 for the common denominator. Thus we find:


$\frac{5}{14}=\frac{{\displaystyle 3\times 5}}{{\displaystyle 3\times 14}}=\frac{15}{42}$; $\frac{4}{21}=\frac{{\displaystyle 2\times 4}}{{\displaystyle 2\times 21}}=\frac{8}{42}$; $\frac{7}{6}=\frac{{\displaystyle 7\times 7}}{{\displaystyle 7\times 6}}=\frac{49}{42}$


We have expressed $\frac{5}{14}$, $\frac{4}{21}$, and $\frac{7}{6}$ using a common denominator.


Note: 42 is the lowest non-zero common multiple of 14, 21, and 6. Instead of drawing a table for the multiples, we could break up 14, 21, and 6.


14 = 2 × 7; 21 = 3 × 7; and 6 = 2 × 3.


42, which is equal to 2 × 7 × 3, is a non-zero common multiple of 14, 21, and 6.

II. Applications

A. Comparing two numbers written as fractions

Rule: to compare two numbers written as fractions, we can represent them using a positive common denominator and then compare the new numerators. The two numbers will be arranged in the same order (as the new numerators).


Example 1: We want to compare $\frac{10}{21}$ and $\frac{16}{35}$.


One method is to represent them using a common denominator. We can see that 105 = 21 × 5 and 105 = 35 × 3. Therefore we have: $\frac{10}{21}=\frac{{\displaystyle 5\times 10}}{{\displaystyle 5\times 21}}=\frac{50}{105}$ and $\frac{{\displaystyle 16}}{{\displaystyle 35}}=\frac{{\displaystyle 3\times 16}}{{\displaystyle 3\times 35}}=\frac{48}{105}$.


$\frac{50}{105}$ and $\frac{48}{105}$ have the same positive denominator; compare the two numerators: 50 > 48 therefore: $\frac{50}{105}$ > $\frac{48}{105}$  .


Finally, $\frac{10}{21}$ > $\frac{{\displaystyle 16}}{{\displaystyle 35}}$ .


Example 2: Comparison of $\frac{-7}{10}$ and $\frac{{\displaystyle 5}}{{\displaystyle -7}}$.


We start by making the denominator of the second number positive: $\frac{{\displaystyle 5}}{{\displaystyle -7}}$ = $\frac{-5}{7}$ .


Then we represent them using a common denominator. We find: 70 = 10 × 7 and 70 = 7 × 10. So $\frac{-7}{10}=\frac{{\displaystyle -7\times 7}}{{\displaystyle 10\times 7}}=\frac{-49}{70}$ and $\frac{-5}{7}=\frac{{\displaystyle -5\times 10}}{{\displaystyle 7\times 10}}=\frac{-50}{70}$.


–49 > –50 therefore $\frac{-49}{70}$ > $\frac{-50}{70}$  and finally: $\frac{-7}{10}$ > $\frac{{\displaystyle 5}}{{\displaystyle -7}}$ .

B. Calculating the sum of numbers written as fractions

We want to calculate $\frac{4}{15}-\frac{{\displaystyle 7}}{{\displaystyle 6}}+\frac{11}{20}$.


60 is a common multiple of 15, 6, and 20. Therefore $\frac{4}{15}-\frac{{\displaystyle 7}}{{\displaystyle 6}}+\frac{11}{20}=\frac{{\displaystyle 4\times 4}}{{\displaystyle 15\times 4}}-\frac{{\displaystyle 7\times 10}}{{\displaystyle 6\times 10}}+\frac{11\times 3}{20\times 3}=\frac{16}{60}-\frac{70}{60}+\frac{33}{60}=\frac{21}{60}$.


 $\frac{21}{60}$ can be simplified by dividing the numerator and denominator by 3: $\frac{21}{60}=\frac{7}{20}$.


Finally: $\frac{4}{15}-\frac{{\displaystyle 7}}{{\displaystyle 6}}+\frac{11}{20}=\frac{7}{20}$.