Question

1. \frac{4}{9} = \frac{r - 3}{6}

Explanation:

Step1: Cross - multiply the equation

Given the equation \(\frac{4}{9}=\frac{r - 3}{6}\), cross - multiplying (using the property that if \(\frac{a}{b}=\frac{c}{d}\), then \(a\times d=b\times c\)) gives us \(4\times6 = 9\times(r - 3)\).
\[4\times6=9\times(r - 3)\]
\[24 = 9r-27\]

Step2: Add 27 to both sides

To isolate the term with \(r\), we add 27 to both sides of the equation.
\[24 + 27=9r-27 + 27\]
\[51 = 9r\]

Step3: Divide both sides by 9

Now, we divide both sides of the equation by 9 to solve for \(r\).
\[r=\frac{51}{9}\]
Simplify the fraction \(\frac{51}{9}\) by dividing both the numerator and the denominator by 3, we get \(r = \frac{17}{3}\) or \(r = 5\frac{2}{3}\).

Answer:

\(r=\frac{17}{3}\) (or \(r = 5\frac{2}{3}\))