Rivalry2

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Original 2026-01-01

Modified 2026-02-21

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

2 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:

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

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

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

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

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

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

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

10 Then,

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

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

13 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,

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

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

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

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

18 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,

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

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

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

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

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

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