To find out how much lime juice should be added if the new recipe uses 6 cups of sugar, we need to maintain the ratio of sugar to lime juice as in the original recipe.
The original ratio is:
2 cups of sugar : 1/2 cup of lime juice.
Let's call the unknown amount of lime juice that needs to be added for the new recipe as "L" cups.
The new ratio will be:
6 cups of sugar : L cups of lime juice.
We want these two ratios to be equivalent, so we can express this equality as a proportion:
[tex]\[
\frac{2 \text{ cups of sugar}}{1/2 \text{ cup of lime juice}} = \frac{6 \text{ cups of sugar}}{L \text{ cups of lime juice}}.
\][/tex]
We can solve for L by cross-multiplying:
[tex]\[
2 \cdot L = \left(\frac{1}{2}\right) \cdot 6,
\][/tex]
[tex]\[
2L = 3.
\][/tex]
Now, solve for L by dividing both sides of the equation by 2:
[tex]\[
L = \frac{3}{2}.
\][/tex]
So, the amount of lime juice L needed for the new recipe is [tex]\( \frac{3}{2} \)[/tex] cups or 1.5 cups of lime juice. This maintains the same sugar to lime juice ratio as the original recipe.
