To find the ratio [tex]\( e : g \)[/tex] given the following ratios:
[tex]\[ e : f = 4 : 3 \][/tex]
[tex]\[ f : g = 5 : 6 \][/tex]
we need to link these two ratios together by the common term [tex]\( f \)[/tex].
### Step-by-Step Solution:
1. Convert the given ratios into fractional form:
From [tex]\( e : f = 4 : 3 \)[/tex]:
[tex]\[
\frac{e}{f} = \frac{4}{3}
\][/tex]
From [tex]\( f : g = 5 : 6 \)[/tex]:
[tex]\[
\frac{f}{g} = \frac{5}{6}
\][/tex]
2. Combine the two ratios by eliminating the common term [tex]\( f \)[/tex]:
We need [tex]\( \frac{e}{g} \)[/tex]. To combine these, let's first express [tex]\( f \)[/tex] in terms of a common measure:
From [tex]\( e : f = 4 : 3 \)[/tex]:
[tex]\[
e = \frac{4}{3} f
\][/tex]
From [tex]\( f : g = 5 : 6 \)[/tex]:
[tex]\[
f = \frac{5}{6} g
\][/tex]
3. Substitute the expression for [tex]\( f \)[/tex] from the second ratio into the first ratio:
Using [tex]\( f = \frac{5}{6} g \)[/tex] in [tex]\( e = \frac{4}{3} f \)[/tex]:
[tex]\[
e = \frac{4}{3} \left( \frac{5}{6} g \right)
\][/tex]
Simplify the fraction:
[tex]\[
e = \frac{4 \cdot 5}{3 \cdot 6} g
\][/tex]
[tex]\[
e = \frac{20}{18} g
\][/tex]
4. Simplify the ratio [tex]\( \frac{20}{18} \)[/tex]:
The greatest common divisor (GCD) of 20 and 18 is 2. To simplify:
[tex]\[
\frac{e}{g} = \frac{20}{18} = \frac{20 \div 2}{18 \div 2} = \frac{10}{9}
\][/tex]
5. Express the simplified ratio [tex]\( e : g \)[/tex]:
[tex]\[
e : g = 10 : 9
\][/tex]
### Answer:
[tex]\[ e : g = 10 : 9 \][/tex]
