Certainly! Let's solve this step-by-step.
Given:
- The numerator and denominator of a fraction are in the ratio [tex]\( \frac{3}{5} \)[/tex].
- If 9 is added to both the numerator and the denominator, the new ratio becomes [tex]\( \frac{15}{22} \)[/tex].
We need to find the original fraction.
1. Set up the original fraction:
Let the numerator be [tex]\( 3x \)[/tex] and the denominator be [tex]\( 5x \)[/tex]. Thus, the original fraction is:
[tex]\[
\frac{3x}{5x}
\][/tex]
2. Adjust the fraction by adding 9 to both the numerator and the denominator:
If we add 9 to both, the new numerator and denominator will be [tex]\( 3x + 9 \)[/tex] and [tex]\( 5x + 9 \)[/tex], respectively. Therefore, the new fraction is:
[tex]\[
\frac{3x + 9}{5x + 9}
\][/tex]
3. Set up the equation with the new ratio:
We know the new fraction is equal to [tex]\( \frac{15}{22} \)[/tex]:
[tex]\[
\frac{3x + 9}{5x + 9} = \frac{15}{22}
\][/tex]
4. Cross-multiply to eliminate the fractions:
[tex]\[
22(3x + 9) = 15(5x + 9)
\][/tex]
5. Expand and simplify the equation:
[tex]\[
66x + 198 = 75x + 135
\][/tex]
6. Move all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other side:
[tex]\[
66x - 75x = 135 - 198
\][/tex]
[tex]\[
-9x = -63
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{63}{9}
\][/tex]
[tex]\[
x = 7
\][/tex]
8. Calculate the original numerator and denominator:
- Numerator:
[tex]\[
3x = 3 \times 7 = 21
\][/tex]
- Denominator:
[tex]\[
5x = 5 \times 7 = 35
\][/tex]
9. Write the original fraction:
[tex]\[
\frac{21}{35}
\][/tex]
10. Simplify the fraction if possible:
[tex]\[
\frac{21}{35} = \frac{3}{5} \text{ (already in the simplest form)}
\][/tex]
Thus, the original fraction is [tex]\( \frac{21}{35} \)[/tex] which simplifies to [tex]\( \frac{3}{5} \)[/tex] and indeed matches our initial given ratio of 3:5.
