To work out the ratio \( e : g \) given the ratios \( e : f = 4 : 3 \) and \( f : g = 5 : 6 \), follow these steps:
1. Understanding the Ratios:
- \( e : f = 4 : 3 \) means for every 4 units of \( e \), there are 3 units of \( f \).
- \( f : g = 5 : 6 \) means for every 5 units of \( f \), there are 6 units of \( g \).
2. Finding a Common Term:
- To link the ratios, we need a common term, which is \( f \).
- Let's choose a reference value for \( f \). To facilitate the comparison, we use 1 unit of \( f \).
3. Calculating Equivalent Values for \( e \) and \( g \):
- From the ratio \( e : f = 4 : 3 \), if \( f = 1 \),
[tex]\[
e = \frac{4}{3} \times 1 = \frac{4}{3}
\][/tex]
- From the ratio \( f : g = 5 : 6 \), if \( f = 1 \),
[tex]\[
g = \frac{6}{5} \times 1 = \frac{6}{5}
\][/tex]
4. Finding the Ratio \( e : g \):
- Now, we have \( e = \frac{4}{3} \) and \( g = \frac{6}{5} \).
- To find \( e : g \), we take the ratio of these values:
[tex]\[
e : g = \left( \frac{4}{3} \right) : \left( \frac{6}{5} \right)
\][/tex]
- This can be converted to a single fraction by dividing:
[tex]\[
e : g = \frac{\frac{4}{3}}{\frac{6}{5}} = \frac{4}{3} \times \frac{5}{6} = \frac{4 \times 5}{3 \times 6} = \frac{20}{18} = \frac{10}{9}
\][/tex]
5. Simplifying the Ratio:
- The ratio \( \frac{10}{9} \) is in its simplest form since the greatest common divisor (GCD) of 10 and 9 is 1.
Thus, the ratio \( e : g \) in its simplest form is
[tex]\[
e : g = 10 : 9
\][/tex]
Therefore, the answer is [tex]\( e : g = \boxed{10 : 9} \)[/tex].
