Example 1
Completion requirements
| Example 1 |
Simplify the expression \(\frac{315}{840}\).
Method 1: GCF
Step 1: Determine the GCF.
Because it is not obvious what the GCF is, break down both numbers into their prime factors.
\(\begin{align}
315 &= {\color{red}3} \times 3 \times {\color{red}5} \times {\color{red}7} \\
840 &= 2 \times 2 \times 2 \times {\color{red}3} \times {\color{red}5} \times {\color{red}7} \\
{\rm{GCF}} &= 3 \times 5 \times 7 = 105 \\
\end{align}\)
Step 2: Divide the numerator and denominator by the GCF.
Method 2: Prime Factorization
Step 1: Write out the prime factors of each number.
This method starts basically the same way as the previous method, but in this method, write out the prime factors as a fraction.
Step 2: Reduce the numerator and denominator. Then, simplify.
Step 1: Determine the GCF.
Because it is not obvious what the GCF is, break down both numbers into their prime factors.
\(\begin{align}
315 &= {\color{red}3} \times 3 \times {\color{red}5} \times {\color{red}7} \\
840 &= 2 \times 2 \times 2 \times {\color{red}3} \times {\color{red}5} \times {\color{red}7} \\
{\rm{GCF}} &= 3 \times 5 \times 7 = 105 \\
\end{align}\)
Step 2: Divide the numerator and denominator by the GCF.
\[\frac{{315 \div 105}}{{840 \div 105}} = \frac{3}{8}\]
Method 2: Prime Factorization
Step 1: Write out the prime factors of each number.
This method starts basically the same way as the previous method, but in this method, write out the prime factors as a fraction.
\[\frac{{315}}{{840}} = \frac{{3 \times 3 \times 5 \times 7}}{{2 \times 2 \times 2 \times 3 \times 5 \times 7}}\]
Step 2: Reduce the numerator and denominator. Then, simplify.
\[\frac{{{\color{red}\cancel {\color{#444}3}}} \times 3 \times
{{\color{red}\cancel {\color{#444}5}}} \times {{\color{red}\cancel {\color{#444}7}}}}{{2 \times 2
\times 2 \times {{\color{red}\cancel {\color{#444}3}}} \times {{\color{red}\cancel {\color{#444}5}}}
\times {{\color{red}\cancel {\color{#444}7}}}}} = \frac{3}{8}\]