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Answer :
To simplify the fraction [tex]\(\frac{45}{12}\)[/tex] using prime factorization, follow these steps:
1. Prime Factorization:
- Start by finding the prime factors of both the numerator and the denominator.
- The prime factorization of 45 is [tex]\(3 \times 3 \times 5\)[/tex].
- The prime factorization of 12 is [tex]\(2 \times 2 \times 3\)[/tex].
2. Cancel Out Common Factors:
- Identify and cancel out any common factors in the numerator and the denominator.
- Both 45 and 12 share a common factor of 3.
- Cancel the 3 from both the numerator and the denominator:
[tex]\[
\frac{3 \times 3 \times 5}{2 \times 2 \times 3} \longrightarrow \frac{3 \times 5}{2 \times 2}
\][/tex]
3. Resulting Simplified Fraction:
- After canceling the common factor, we multiply the remaining factors:
- The simplified numerator is [tex]\(3 \times 5 = 15\)[/tex].
- The simplified denominator is [tex]\(2 \times 2 = 4\)[/tex].
4. Final Simplified Fraction:
- The fraction [tex]\(\frac{45}{12}\)[/tex] simplifies to [tex]\(\frac{15}{4}\)[/tex].
So, the simplified fraction in its simplest form is [tex]\(\frac{15}{4}\)[/tex].
Referring to the options given in the multiple choice:
- a. [tex]\(\frac{10}{1}\)[/tex]
- b. [tex]\(\frac{\not z \times 3 \times 5}{2 \times 2 \times \not z^{\prime}}\)[/tex]
- c. [tex]\(\frac{15}{4}\)[/tex]
- d. [tex]\(\frac{3 \times 3 \times 5}{2 \times 2 \times 3}\)[/tex]
The correct answer is option c. [tex]\(\frac{15}{4}\)[/tex].
1. Prime Factorization:
- Start by finding the prime factors of both the numerator and the denominator.
- The prime factorization of 45 is [tex]\(3 \times 3 \times 5\)[/tex].
- The prime factorization of 12 is [tex]\(2 \times 2 \times 3\)[/tex].
2. Cancel Out Common Factors:
- Identify and cancel out any common factors in the numerator and the denominator.
- Both 45 and 12 share a common factor of 3.
- Cancel the 3 from both the numerator and the denominator:
[tex]\[
\frac{3 \times 3 \times 5}{2 \times 2 \times 3} \longrightarrow \frac{3 \times 5}{2 \times 2}
\][/tex]
3. Resulting Simplified Fraction:
- After canceling the common factor, we multiply the remaining factors:
- The simplified numerator is [tex]\(3 \times 5 = 15\)[/tex].
- The simplified denominator is [tex]\(2 \times 2 = 4\)[/tex].
4. Final Simplified Fraction:
- The fraction [tex]\(\frac{45}{12}\)[/tex] simplifies to [tex]\(\frac{15}{4}\)[/tex].
So, the simplified fraction in its simplest form is [tex]\(\frac{15}{4}\)[/tex].
Referring to the options given in the multiple choice:
- a. [tex]\(\frac{10}{1}\)[/tex]
- b. [tex]\(\frac{\not z \times 3 \times 5}{2 \times 2 \times \not z^{\prime}}\)[/tex]
- c. [tex]\(\frac{15}{4}\)[/tex]
- d. [tex]\(\frac{3 \times 3 \times 5}{2 \times 2 \times 3}\)[/tex]
The correct answer is option c. [tex]\(\frac{15}{4}\)[/tex].
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