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Match each fraction on the left with an equivalent fraction on the right. Some options on the right will be used more than once.

\[
\begin{array}{ll}
\frac{4}{7} & \frac{9}{10} \\
\frac{18}{20} & \frac{30}{25} \\
\frac{6}{5} & \frac{12}{21} \\
\frac{81}{90} & \\
\end{array}
\]

Answer :

To match each fraction on the left with an equivalent fraction on the right, we'll follow a series of steps to simplify each fraction and find their equivalents. Here's a detailed breakdown:

1. Understand the Problem: We have a list of fractions on the left and need to find their equivalent fractions on the right. Some options on the right may be used more than once.

2. List the Fractions:
- Left side:
- [tex]\(\frac{4}{7}\)[/tex]
- [tex]\(\frac{18}{20}\)[/tex]
- [tex]\(\frac{6}{5}\)[/tex]
- [tex]\(\frac{81}{90}\)[/tex]
- Right side:
- [tex]\(\frac{9}{10}\)[/tex]
- [tex]\(\frac{30}{25}\)[/tex]
- [tex]\(\frac{12}{21}\)[/tex]

3. Simplify the Fractions:
- Simplify the left-side fractions:
- [tex]\(\frac{18}{20}\)[/tex]: The greatest common divisor (GCD) of 18 and 20 is 2. So, [tex]\(\frac{18}{20} = \frac{9}{10}\)[/tex].
- [tex]\(\frac{6}{5}\)[/tex] is already in its simplest form.
- [tex]\(\frac{81}{90}\)[/tex]: The GCD of 81 and 90 is 9. So, [tex]\(\frac{81}{90} = \frac{9}{10}\)[/tex].
- [tex]\(\frac{4}{7}\)[/tex] is already in its simplest form.
- Simplify the right-side fractions:
- [tex]\(\frac{9}{10}\)[/tex] is already in its simplest form.
- [tex]\(\frac{30}{25}\)[/tex]: The GCD of 30 and 25 is 5. So, [tex]\(\frac{30}{25} = \frac{6}{5}\)[/tex].
- [tex]\(\frac{12}{21}\)[/tex]: The GCD of 12 and 21 is 3. So, [tex]\(\frac{12}{21} = \frac{4}{7}\)[/tex].

4. Match the Equivalent Fractions:
- [tex]\(\frac{4}{7}\)[/tex] on the left matches with [tex]\(\frac{12}{21}\)[/tex] on the right.
- [tex]\(\frac{18}{20}\)[/tex] on the left matches with [tex]\(\frac{9}{10}\)[/tex] on the right.
- [tex]\(\frac{6}{5}\)[/tex] on the left matches with [tex]\(\frac{30}{25}\)[/tex] on the right.
- [tex]\(\frac{81}{90}\)[/tex] on the left matches again with [tex]\(\frac{9}{10}\)[/tex] on the right.

5. Conclusion:
- The matching of fractions is as follows:
- [tex]\(\frac{4}{7}\)[/tex] matches with [tex]\(\frac{12}{21}\)[/tex]
- [tex]\(\frac{18}{20}\)[/tex] matches with [tex]\(\frac{9}{10}\)[/tex]
- [tex]\(\frac{6}{5}\)[/tex] matches with [tex]\(\frac{30}{25}\)[/tex]
- [tex]\(\frac{81}{90}\)[/tex] matches with [tex]\(\frac{9}{10}\)[/tex]

These matches help confirm that the fractions on the left are equivalent to the ones listed on the right, with some options used more than once.

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Rewritten by : Barada