The ratio $$ c: g $$ is $$ 2: 3 $$ and the ratio $$ g: t $$ is $$ 5: 4 $$. Write the ratio $$ c: g: t $$ in its simplest form.

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1. We are given two ratios:
   - $$ c : g = 2 : 3 $$, meaning we can write $$ c = 2x $$ and $$ g = 3x $$ for some number $$ x $$.
   - $$ g : t = 5 : 4 $$, meaning we can write $$ g = 5y $$ and $$ t = 4y $$ for some number $$ y $$.

2. Since $$ g $$ appears in both ratios, set the two expressions for $$ g $$ equal to each other:
   $$
   3x = 5y
   $$
   Solve for $$ x $$ or $$ y $$. For simplicity, let’s solve for $$ x $$ in terms of $$ y $$:
   $$
   x = \frac{5y}{3}
   $$

3. To eliminate fractions, choose a value for $$ y $$ so that $$ x $$ is an integer. We can take $$ y = 3 $$. Then:
   $$
   x = \frac{5(3)}{3} = 5
   $$

4. Substitute back to find $$ c $$, $$ g $$, and $$ t $$:
   - $$ c = 2x = 2(5) = 10 $$
   - $$ g = 3x = 3(5) = 15 $$ (or, equivalently, $$ g = 5y = 5(3) = 15 $$)
   - $$ t = 4y = 4(3) = 12 $$

5. Therefore, the combined ratio is:
   $$
   c : g : t = 10 : 15 : 12
   $$
The ratio $$ c : g : t $$ in its simplest form is $$ 10 : 15 : 12 $$.

The ratio is and the ratio is .
Write the ratio in its simplest form.

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The ratio is and the ratio is . Write the ratio in its simplest form.