Work out [tex] e:g [/tex]. Give your answer in its simplest form.

[tex]\[
\begin{array}{l}
e:f = 4:3 \\
f:g = 5:6
\end{array}
\][/tex]

Answer: [tex] e:g = \boxed{} \]



Answer :

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]

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