Answer :
To find equivalent fractions for ( \frac{m}{n} ) and ( \frac{p}{q} ) with a common denominator, you need to find the least common multiple (LCM) of the denominators ( n ) and ( q ).
Here’s how you can do it:
1. List the multiples of ( n ) and ( q ).
2. Find the smallest multiple that both lists share. This number is the LCM.
3. Divide the LCM by each denominator to find the number by which to multiply both the numerator and denominator of each fraction to get the equivalent fractions with the common denominator.
For example, if your fractions are ( \frac{2}{3} ) and ( \frac{5}{4} ), you would:
List multiples of 3 (3, 6, 9, 12, …) and 4 (4, 8, 12, 16, …).
Find that 12 is the LCM.
Multiply ( \frac{2}{3} ) by ( \frac{4}{4} ) to get ( \frac{8}{12} ), and multiply ( \frac{5}{4} ) by ( \frac{3}{3} ) to get ( \frac{15}{12} ).
Now both fractions have a common denominator of 12: ( \frac{8}{12} ) and ( \frac{15}{12} ).
Would you like to try this with your specific fractions?
Sure, let's find the rational numbers with a common denominator for the given fractions [tex] \frac{m}{n} [/tex] and [tex] \frac{p}{q} [/tex]
The common denominator of two fractions is the least common multiple (LCM) of their denominators. So, we need to find the LCM of n and q.
Let's denote the LCM of n and q as lcm(n, q).
Now, we can rewrite the fractions with the common denominator:
1. Multiply the numerator and denominator of [tex] \frac{m}{n} [/tex] by q to get the equivalent fraction with the common denominator:
[tex]\frac{m}{n} = \frac{m \cdot q}{n \cdot q} = \frac{m \cdot q}{\text{lcm}(n, q)}[/tex]
2. Similarly, multiply the numerator and denominator of [tex] \frac{p}{q} [/tex] by n:
[tex]\frac{p}{q} = \frac{p \cdot n}{q \cdot n} = \frac{p \cdot n}{\text{lcm}(n, q)}[/tex]
So, the rational numbers [tex]\frac{m}{n}[/tex] and [tex]\frac{p}{q}[/tex] with the common denominator are [tex]\frac{m \cdot q}{\text{lcm}(n, q)}$ and $\frac{p \cdot n}{\text{lcm}(n, q)}[/tex] respectively.