Ratio Calculation: Finding a to d

We are given two ratios:

• $a : b = 3 : 4$
• $d : b = 4 : 3$

We need to find the ratio $a : d$. Let's rewrite the given ratios in fractional form:

• From $a : b = 3 : 4$, we have $\frac{a}{b} = \frac{3}{4}$.
• From $d : b = 4 : 3$, we have $\frac{d}{b} = \frac{4}{3}$.

To find the ratio $a : d$ (or $\frac{a}{d}$), we can express it using the common term 'b'. We can write $\frac{a}{d}$ as the product of $\frac{a}{b}$ and $\frac{b}{d}$:

$ \frac{a}{d} = \frac{a}{b} \times \frac{b}{d} $

We know $\frac{a}{b} = \frac{3}{4}$. We need to find $\frac{b}{d}$. We are given $\frac{d}{b} = \frac{4}{3}$. Taking the reciprocal of both sides gives:

$ \frac{b}{d} = \frac{1}{\frac{d}{b}} = \frac{1}{\frac{4}{3}} = \frac{3}{4} $

Now, substitute the values of $\frac{a}{b}$ and $\frac{b}{d}$ into the equation for $\frac{a}{d}$:

$ \frac{a}{d} = \frac{3}{4} \times \frac{3}{4} $

$ \frac{a}{d} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16} $

Therefore, the ratio of a to d is $9 : 16$.