Find the Ratio p:q from Given Equation

The problem asks us to find the ratio \(p : q\) given the equation \(\frac{{p - q}}{q} = \frac{4}{3}\).

To find the ratio \(p : q\), we need to express \(p\) in terms of \(q\) or find the value of the fraction \(\frac{p}{q}\).

Let's start with the given equation:

\(\frac{{p - q}}{q} = \frac{4}{3}\)

We can separate the terms on the left side of the equation:

\(\frac{p}{q} - \frac{q}{q} = \frac{4}{3}\)

Since \(\frac{q}{q} = 1\) (assuming \(q \neq 0\), which must be true if it's in the denominator), the equation becomes:

\(\frac{p}{q} - 1 = \frac{4}{3}\)

Now, we want to isolate \(\frac{p}{q}\). We can do this by adding 1 to both sides of the equation:

\(\frac{p}{q} = \frac{4}{3} + 1\)

To add \(\frac{4}{3}\) and 1, we need a common denominator. We can write 1 as \(\frac{3}{3}\):

\(\frac{p}{q} = \frac{4}{3} + \frac{3}{3}\)

Now, add the fractions on the right side:

\(\frac{p}{q} = \frac{4 + 3}{3}\)

\(\frac{p}{q} = \frac{7}{3}\)

The fraction \(\frac{p}{q}\) represents the ratio \(p : q\). Therefore, the ratio \(p : q\) is 7 : 3.

The steps followed are:

1. Start with the given equation \(\frac{{p - q}}{q} = \frac{4}{3}\).
2. Separate the terms on the left side: \(\frac{p}{q} - \frac{q}{q} = \frac{4}{3}\).
3. Simplify \(\frac{q}{q}\) to 1: \(\frac{p}{q} - 1 = \frac{4}{3}\).
4. Add 1 to both sides: \(\frac{p}{q} = \frac{4}{3} + 1\).
5. Rewrite 1 as \(\frac{3}{3}\) to add fractions: \(\frac{p}{q} = \frac{4}{3} + \frac{3}{3}\).
6. Add the fractions: \(\frac{p}{q} = \frac{7}{3}\).
7. Express the fraction as a ratio: \(p : q = 7 : 3\).

This method correctly derives the ratio \(p : q\) from the given equation.