To determine which fractions are equivalent to [tex]\(\frac{4}{12}\)[/tex], we need to simplify [tex]\(\frac{4}{12}\)[/tex] and identify other fractions that simplify to the same form.
First, let’s simplify [tex]\(\frac{4}{12}\)[/tex]:
1. Find the greatest common divisor (GCD) of the numerator (4) and the denominator (12).
2. The GCD of 4 and 12 is 4.
3. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{4 \div 4}{12 \div 4} = \frac{1}{3}
\][/tex]
So, the simplest form of [tex]\(\frac{4}{12}\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
Now, let's simplify each of the given fractions to see if they are equivalent to [tex]\(\frac{1}{3}\)[/tex]:
1. [tex]\(\frac{2}{10}\)[/tex]:
- Find the GCD of 2 and 10, which is 2.
- Simplify: [tex]\(\frac{2 \div 2}{10 \div 2} = \frac{1}{5}\)[/tex]
- [tex]\(\frac{1}{5}\)[/tex] is not equal to [tex]\(\frac{1}{3}\)[/tex].
2. [tex]\(\frac{2}{6}\)[/tex]:
- Find the GCD of 2 and 6, which is 2.
- Simplify: [tex]\(\frac{2 \div 2}{6 \div 2} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{1}{3}\)[/tex] is equal to [tex]\(\frac{1}{3}\)[/tex].
3. [tex]\(\frac{1}{6}\)[/tex]:
- The numerator and denominator are already simplified, and there’s no common divisor other than 1.
- [tex]\(\frac{1}{6}\)[/tex] is already in its simplest form and is not equal to [tex]\(\frac{1}{3}\)[/tex].
4. [tex]\(\frac{1}{9}\)[/tex]:
- The numerator and denominator are already simplified, and there’s no common divisor other than 1.
- [tex]\(\frac{1}{9}\)[/tex] is already in its simplest form and is not equal to [tex]\(\frac{1}{3}\)[/tex].
5. [tex]\(\frac{1}{3}\)[/tex]:
- The numerator and denominator are already simplified, and there’s no common divisor other than 1.
- [tex]\(\frac{1}{3}\)[/tex] is equal to [tex]\(\frac{1}{3}\)[/tex].
After simplifying all given fractions, we find that the fractions equivalent to [tex]\(\frac{4}{12}\)[/tex] are:
[tex]\[
\boxed{\frac{2}{6} \text{ and } \frac{1}{3}}
\][/tex]
