The problem asks us to find the value of the expression $\frac{c+d}{c-d}$ given the relationship $\frac{c}{d} = 1 \div \frac{3}{4}$.

Ratio Calculation

First, let's simplify the given ratio $\frac{c}{d}$: $ \frac{c}{d} = 1 \div \frac{3}{4} $ Dividing by a fraction is the same as multiplying by its reciprocal: $ \frac{c}{d} = 1 \times \frac{4}{3} $ $ \frac{c}{d} = \frac{4}{3} $ So, the ratio of $c$ to $d$ is $4:3$.

Finding the Expression $\frac{c+d}{c-d}$

We need to calculate $\frac{c+d}{c-d}$. We can use the property of ratios known as Componendo and Dividendo. This property states that if $\frac{a}{b} = \frac{p}{q}$, then $\frac{a+b}{a-b} = \frac{p+q}{p-q}$.

Applying Componendo and Dividendo to our ratio $\frac{c}{d} = \frac{4}{3}$: $ \frac{c+d}{c-d} = \frac{4+3}{4-3} $ $ \frac{c+d}{c-d} = \frac{7}{1} $ $ \frac{c+d}{c-d} = 7 $

Alternatively, we can substitute $c = \frac{4}{3}d$ into the expression:

$ \frac{c+d}{c-d} = \frac{(\frac{4}{3}d) + d}{(\frac{4}{3}d) - d} $ Factor out $d$ from the numerator and denominator: $ = \frac{d(\frac{4}{3} + 1)}{d(\frac{4}{3} - 1)} $ Cancel out $d$ (assuming $d \neq 0$): $ = \frac{\frac{4}{3} + \frac{3}{3}}{\frac{4}{3} - \frac{3}{3}} $ $ = \frac{\frac{7}{3}}{\frac{1}{3}} $ Simplify the complex fraction: $ = \frac{7}{3} \times \frac{3}{1} $ $ = 7 $ Both methods yield the same result.