Let's solve the problem step-by-step:
1. Define the variables:
Let [tex]\( \frac{a}{b} \)[/tex] be the fraction.
Given that the ratio of the numerator to the denominator is [tex]\( 3:5 \)[/tex], we can write:
[tex]\[
\frac{a}{b} = \frac{3}{5}
\][/tex]
2. Set up the equation based on the given ratio:
From the above ratio, we can express the numerator [tex]\(a\)[/tex] in terms of the denominator [tex]\(b\)[/tex]:
[tex]\[
a = \frac{3}{5} b
\][/tex]
3. Establish the second condition:
It is given that if 9 is added to both the numerator and the denominator, the new fraction is in the ratio [tex]\( 15:22 \)[/tex]. Thus, we can write:
[tex]\[
\frac{a + 9}{b + 9} = \frac{15}{22}
\][/tex]
4. Express the second equation in terms of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a + 9 = \frac{15}{22} (b + 9)
\][/tex]
5. Substitute [tex]\(a = \frac{3}{5} b \)[/tex] into the second equation:
[tex]\[
\frac{3}{5} b + 9 = \frac{15}{22} (b + 9)
\][/tex]
6. Clear the fraction by finding a common denominator and multiplying through:
Multiply both sides by [tex]\( 110 \)[/tex] (which is the least common multiple of 5 and 22):
[tex]\[
110 \left(\frac{3}{5} b + 9\right) = 110 \left( \frac{15}{22} (b + 9) \right)
\][/tex]
Simplify each side:
[tex]\[
66b + 990 = 75 (b + 9)
\][/tex]
7. Distribute and simplify the equation:
[tex]\[
66b + 990 = 75b + 675
\][/tex]
Rearrange to isolate [tex]\(b\)[/tex]:
[tex]\[
66b + 990 - 75b = 675
\][/tex]
[tex]\[
-9b + 990 = 675
\][/tex]
Isolate [tex]\(b\)[/tex]:
[tex]\[
-9b = 675 - 990
\][/tex]
[tex]\[
-9b = -315
\][/tex]
[tex]\[
b = \frac{-315}{-9}
\][/tex]
[tex]\[
b = 35
\][/tex]
8. Find the value of [tex]\(a\)[/tex]:
Substitute [tex]\( b = 35 \)[/tex] back into [tex]\(a = \frac{3}{5} b\)[/tex]:
[tex]\[
a = \frac{3}{5} \times 35
\][/tex]
[tex]\[
a = 21
\][/tex]
9. Determine the fraction:
Therefore, the fraction is:
[tex]\[
\frac{a}{b} = \frac{21}{35}
\][/tex]
We have successfully determined that the fraction is [tex]\( \frac{21}{35} \)[/tex].
