Answer :
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].
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].
