Solving the Ratio Problem: $\frac{c^2-a^2}{c^2+a^2}$

This problem involves finding the value of an algebraic expression based on given ratios between variables $a$, $b$, and $c$. We are provided with:

• The ratio $\frac{a}{b} = \frac{4}{5}$
• The ratio $\frac{b}{c} = \frac{15}{16}$

Our goal is to calculate the value of $\frac{c^2-a^2}{c^2+a^2}$.

Step 1: Finding the Ratio Between $a$ and $c$

To find the relationship between $a$ and $c$, we can multiply the two given ratios:

$ \frac{a}{b} \times \frac{b}{c} = \frac{4}{5} \times \frac{15}{16} $

Multiplying the fractions on the left side, the $b$ terms cancel out:

$ \frac{a}{c} = \frac{4 \times 15}{5 \times 16} $

Now, simplify the resulting fraction:

$ \frac{a}{c} = \frac{60}{80} $

To simplify $\frac{60}{80}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 20:

$ \frac{a}{c} = \frac{60 \div 20}{80 \div 20} = \frac{3}{4} $

So, we have established that $\frac{a}{c} = \frac{3}{4}$.

Step 2: Evaluating the Expression $\frac{c^2-a^2}{c^2+a^2}$

We need to find the value of $\frac{c^2-a^2}{c^2+a^2}$. We know $\frac{a}{c} = \frac{3}{4}$. From this, we can find the ratio $\frac{c}{a}$:

$ \frac{c}{a} = \frac{4}{3} $

Now, let's manipulate the expression $\frac{c^2-a^2}{c^2+a^2}$. A convenient way is to divide both the numerator and the denominator by $a^2$ (assuming $a \neq 0$):

$ \frac{\frac{c^2-a^2}{a^2}}{\frac{c^2+a^2}{a^2}} = \frac{\frac{c^2}{a^2} - \frac{a^2}{a^2}}{\frac{c^2}{a^2} + \frac{a^2}{a^2}} $

This simplifies to:

$ \frac{(\frac{c}{a})^2 - 1}{(\frac{c}{a})^2 + 1} $

We already found that $\frac{c}{a} = \frac{4}{3}$. Let's substitute this value:

$ (\frac{c}{a})^2 = (\frac{4}{3})^2 = \frac{16}{9} $

Now substitute this back into the expression:

$ \frac{\frac{16}{9} - 1}{\frac{16}{9} + 1} $

To simplify the numerator and the denominator, we find a common denominator:

Numerator: $ \frac{16}{9} - 1 = \frac{16}{9} - \frac{9}{9} = \frac{16-9}{9} = \frac{7}{9} $

Denominator: $ \frac{16}{9} + 1 = \frac{16}{9} + \frac{9}{9} = \frac{16+9}{9} = \frac{25}{9} $

Now, divide the simplified numerator by the simplified denominator:

$ \frac{\frac{7}{9}}{\frac{25}{9}} = \frac{7}{9} \times \frac{9}{25} $

The 9s cancel out:

$ \frac{7}{25} $

Conclusion

The value of the expression $\frac{c^2-a^2}{c^2+a^2}$ is $\frac{7}{25}$.