To determine by what fraction \(\frac{9}{35}\) should be multiplied to get \(\frac{7}{15}\), we can follow these steps:
1. Identify the given fractions:
[tex]\[
\text{Fraction 1} = \frac{9}{35}
\][/tex]
[tex]\[
\text{Fraction 2} = \frac{7}{15}
\][/tex]
2. Set up the equation:
We want to find the unknown fraction \( x \) such that:
[tex]\[
\frac{9}{35} \times x = \frac{7}{15}
\][/tex]
3. Isolate \( x \):
To solve for \( x \), divide both sides of the equation by \(\frac{9}{35}\). This is equivalent to multiplying by the reciprocal of \(\frac{9}{35}\):
[tex]\[
x = \frac{\frac{7}{15}}{\frac{9}{35}}
\][/tex]
4. Simplify the complex fraction:
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
x = \frac{7}{15} \times \frac{35}{9}
\][/tex]
5. Perform the multiplication:
Multiply the numerators together and the denominators together:
[tex]\[
x = \frac{7 \times 35}{15 \times 9}
\][/tex]
[tex]\[
x = \frac{245}{135}
\][/tex]
6. Simplify the fraction:
Determine the greatest common divisor (GCD) of 245 and 135 and divide the numerator and denominator by this value. The GCD of 245 and 135 is 5:
[tex]\[
x = \frac{245 \div 5}{135 \div 5}
\][/tex]
[tex]\[
x = \frac{49}{27}
\][/tex]
So, the fraction by which \(\frac{9}{35}\) should be multiplied to obtain \(\frac{7}{15}\) is:
[tex]\[
\boxed{\frac{49}{27}}
\][/tex]
In decimal form, this fraction is approximately:
[tex]\[
1.814814814814815
\][/tex]
