Sure! Let's tackle the problem [tex]\( \frac{3}{7} \div 1 \frac{1}{5} \)[/tex] step-by-step.
Firstly, we need to convert the mixed number [tex]\( 1 \frac{1}{5} \)[/tex] into an improper fraction.
1. Convert the mixed number to an improper fraction:
- A mixed number [tex]\( 1 \frac{1}{5} \)[/tex] means 1 whole and [tex]\( \frac{1}{5} \)[/tex].
- To convert this into an improper fraction:
[tex]\[
1 \frac{1}{5} = \frac{1 \cdot 5 + 1}{5} = \frac{5 + 1}{5} = \frac{6}{5}
\][/tex]
Now we have to divide [tex]\( \frac{3}{7} \)[/tex] by [tex]\( \frac{6}{5} \)[/tex].
2. Division by a fraction is equivalent to multiplication by its reciprocal:
- The reciprocal of [tex]\( \frac{6}{5} \)[/tex] is [tex]\( \frac{5}{6} \)[/tex].
- Therefore, we need to multiply [tex]\( \frac{3}{7} \)[/tex] by [tex]\( \frac{5}{6} \)[/tex]:
[tex]\[
\frac{3}{7} \div \frac{6}{5} = \frac{3}{7} \times \frac{5}{6}
\][/tex]
3. Multiply the numerators and the denominators:
- Multiply the numerators: [tex]\( 3 \times 5 = 15 \)[/tex].
- Multiply the denominators: [tex]\( 7 \times 6 = 42 \)[/tex].
[tex]\[
\frac{3}{7} \times \frac{5}{6} = \frac{15}{42}
\][/tex]
4. Simplify the resultant fraction:
- identify the greatest common divisor (GCD) of 15 and 42, which is 3.
- Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{15 \div 3}{42 \div 3} = \frac{5}{14}
\][/tex]
Therefore, the result of [tex]\( \frac{3}{7} \div 1 \frac{1}{5} \)[/tex] is [tex]\( \frac{15}{42} \)[/tex] which simplifies to [tex]\( \frac{5}{14} \)[/tex].
To summarize, the improper fraction result is [tex]\( \frac{15}{42} \)[/tex] and the simplified result is [tex]\( \frac{5}{14} \)[/tex].
