To solve the problem of simplifying the fraction [tex]\(\frac{4}{16}\)[/tex] to its lowest terms and then finding an equivalent fraction with a denominator of 32, follow these steps:
### Step 1: Simplify the Fraction [tex]\(\frac{4}{16}\)[/tex] to Lowest Terms
1. Identify the numerator and the denominator:
[tex]\[
\text{Numerator} = 4, \quad \text{Denominator} = 16
\][/tex]
2. Find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD of 4 and 16 is 4.
3. Divide both the numerator and the denominator by their GCD to simplify the fraction:
[tex]\[
\frac{4 \div 4}{16 \div 4} = \frac{1}{4}
\][/tex]
So, the fraction [tex]\(\frac{4}{16}\)[/tex] simplifies to [tex]\(\frac{1}{4}\)[/tex].
### Step 2: Find an Equivalent Fraction with a Denominator of 32
1. Identify the simplified fraction and the new required denominator:
[tex]\[
\text{Simplified fraction} = \frac{1}{4}, \quad \text{New denominator} = 32
\][/tex]
2. Determine the factor by which to multiply the denominator of the simplified fraction to achieve the new denominator. Since [tex]\(4\)[/tex] needs to be multiplied by [tex]\(8\)[/tex] to get [tex]\(32\)[/tex]:
[tex]\[
4 \times 8 = 32
\][/tex]
3. Multiply both the numerator and the denominator of the simplified fraction by this factor:
[tex]\[
\frac{1 \times 8}{4 \times 8} = \frac{8}{32}
\][/tex]
Thus, the equivalent fraction of [tex]\(\frac{1}{4}\)[/tex] with a denominator of 32 is [tex]\(\frac{8}{32}\)[/tex].
### Summary:
1. The fraction [tex]\(\frac{4}{16}\)[/tex] simplifies to [tex]\(\frac{1}{4}\)[/tex].
2. An equivalent fraction with a denominator of 32 is [tex]\(\frac{8}{32}\)[/tex].
[tex]\[
\boxed{\frac{4}{16} = \frac{1}{4} = \frac{8}{32}}
\][/tex]
