Like Fraction
2026-02-21 20:43 Diff
https://brightchamps.com/en-us/math/numbers/like-fraction

According to this property, the grouping of fractions in addition and multiplication does not change the result. 

Associative property of addition: The addition of three like fractions, such as \(\frac{p}{q}\), \(\frac{c}{q}\), and \(\frac{d}{q}\), will be like:

\(\frac{p}{q} + \left(\frac{c}{q} + \frac{d}{q}\right) = \left(\frac{p}{q} + \frac{c}{q}\right) + \frac{d}{q} \)

For example, \(\frac{2}{4} + \left(\frac{5}{4} + \frac{3}{4}\right) = \frac{2}{4} + \frac{8}{4} = \frac{10}{4} \)
Then, 

\(\left(\frac{2}{4} + \frac{5}{4}\right) + \frac{3}{4} = \frac{7}{4} + \frac{3}{4} = \frac{10}{4} \)

So, \(\frac{2}{4} + \left(\frac{5}{4} + \frac{3}{4}\right) = \left(\frac{2}{4} + \frac{5}{4}\right) + \frac{3}{4} \).

Associative property of subtraction: The subtraction of like fractions does not follow the associative property. If \(\frac{p}{q}\), \(\frac{c}{q}\), and d/q are the three fractions, then, 

\(\frac{p}{q} - \left(\frac{c}{q} - \frac{d}{q}\right) \neq \left(\frac{p}{q} - \frac{c}{q}\right) - \frac{d}{q} \)

For instance, \(\frac{7}{4} - \left(\frac{5}{4} - \frac{3}{4}\right) = \frac{7}{4} - \frac{2}{4} = \frac{5}{4} \)

Then, 

\(\left(\frac{7}{4} - \frac{5}{4}\right) - \frac{3}{4} = \frac{2}{4} - \frac{3}{4} = -\frac{1}{4} \)

So, \(\frac{7}{4} - \left(\frac{5}{4} - \frac{3}{4}\right) \neq \left(\frac{7}{4} - \frac{5}{4}\right) - \frac{3}{4} \) 

Associative property of multiplication: Like fractions’ multiplication is associative. If \(\frac{p}{q}\),\(\frac{c}{q}\), and \(\frac{d}{q}\) are the three fractions, then,

  \(\frac{p}{q} \times \left(\frac{c}{q} \times \frac{d}{q}\right) = \left(\frac{p}{q} \times \frac{c}{q}\right) \times \frac{d}{q} \)

For example, \(\frac{2}{4} \times \left(\frac{5}{4} \times \frac{3}{4}\right) = \frac{2}{4} \times \frac{15}{4} = \frac{30}{4} \) 

Then, 
\(\left(\frac{2}{4} \times \frac{5}{4}\right) \times \frac{3}{4} = \frac{10}{4} \times \frac{3}{4} = \frac{30}{4} \)

Therefore, \(\frac{2}{4} \times \left(\frac{5}{4} \times \frac{3}{4}\right) = \left(\frac{2}{4} \times \frac{5}{4}\right) \times \frac{3}{4} \)

Associative property of division: The division of like fractions is not associative. If p/q, c/q, and d/q are the three fractions, then, 

\(\frac{p}{q} \div \left(\frac{c}{q} \div \frac{d}{q}\right) \neq \left(\frac{p}{q} \div \frac{c}{q}\right) \div \frac{d}{q} \)

For instance, \(\frac{2}{3} \div \left(\frac{4}{3} \div \frac{1}{3}\right) = \frac{2}{3} \div 4 = \frac{1}{6} \) 
Then, 

\(\left(\frac{2}{3} \div \frac{4}{3}\right) \div \frac{1}{3} \), first we find the value of \(\frac{2}{3} \div \frac{4}{3} \)

\(\frac{2}{3} \times \frac{3}{4} = \frac{2}{4} = \frac{1}{2} \)

\(\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} \)

So, \(\frac{2}{3} \div \left(\frac{4}{3} \div \frac{1}{3}\right) \neq \left(\frac{2}{3} \div \frac{4}{3}\right) \div \frac{1}{3} \)