Certainly! Let's divide the given fractions:
[tex]\[
\frac{x+5}{5} \div \frac{2x+10}{3}
\][/tex]
### Step-by-Step Solution
1. Represent the Division as Multiplication by the Reciprocal:
Division by a fraction is equivalent to multiplying by its reciprocal. Thus,
[tex]\[
\frac{x+5}{5} \div \frac{2x+10}{3} = \frac{x+5}{5} \times \frac{3}{2x+10}
\][/tex]
2. Simplify the Expression:
Now, we'll multiply the numerators together and the denominators together:
[tex]\[
\frac{x+5}{5} \times \frac{3}{2x+10} = \frac{(x+5) \times 3}{5 \times (2x+10)}
\][/tex]
So, the resulting fraction is:
[tex]\[
\frac{3(x+5)}{5(2x+10)}
\][/tex]
3. Simplify the Resulting Fraction:
To simplify the fraction further, let's observe if any common factors can be cancelled in the numerator and the denominator:
Recall that [tex]\(2x+10\)[/tex] can be factored as [tex]\(2(x+5)\)[/tex], so the expression becomes:
[tex]\[
\frac{3(x+5)}{5 \times 2(x+5)} = \frac{3(x+5)}{10(x+5)}
\][/tex]
We notice that [tex]\((x+5)\)[/tex] in the numerator and the denominator are identical and thus can cancel each other out:
[tex]\[
\frac{3(x+5)}{10(x+5)} = \frac{3}{10}
\][/tex]
4. Final Result:
Therefore, after simplifying, the final answer is:
[tex]\[
\frac{3}{10}
\][/tex]
So, the result of the division [tex]\( \frac{x+5}{5} \div \frac{2x+10}{3} \)[/tex] is [tex]\(\frac{3}{10}\)[/tex].
