Answered

Oliver wants to make lemonade, and he found two different recipes online. Simple Lemonade calls for 1.5 cups of lemon juice for every 8 cups of water.



This equation shows how many cups of lemon juice, $y$, are needed for every $x$ cups of water in the recipe for Classic Lemonade:

$y = 0.2x$



Oliver wants to use the recipe with the greatest amount of lemon juice per cup of water. Which recipe should he choose?

  • Simple Lemonade
  • Classic Lemonade



How much lemon juice would that recipe use with 10 cups of water?

[ ] cups of lemon juice

Verified Answer :

  • Calculate the ratio for Simple Lemonade: \(\frac{1.5}{8} = 0.1875\).
  • Identify the ratio for Classic Lemonade from the equation \(y = 0.2x\): \(\frac{y}{x} = 0.2\).
  • Compare the ratios: \(0.2 > 0.1875\), so Classic Lemonade has the greater ratio.
  • Calculate the lemon juice for 10 cups of water in the Classic recipe: \(0.2 \times 10 = 2\). Final answer: \(\boxed{2}\).

Explanation

  1. Calculating and Comparing Ratios

To determine which recipe has the greater lemon juice per cup of water, we must first calculate and compare the ratios of lemon juice (y) to water (x) for both the Simple Lemonade and the Classic Lemonade recipes. The ratio is defined as \(\frac{y}{x}\). For the Simple Lemonade recipe, Oliver needs \(1.5\) cups of lemon juice for every \(8\) cups of water. Thus, the ratio is \(\frac{1.5}{8} = 0.1875\). For the Classic Lemonade recipe, the relationship is given by the equation \(y = 0.2x\), which means the ratio \(\frac{y}{x} = 0.2\). Comparing these two values, we see that \(0.2 > 0.1875\), so the Classic Lemonade recipe has the greater ratio of lemon juice per cup of water.

  1. Calculating Lemon Juice for 10 Cups of Water

Since the Classic Lemonade recipe has the higher ratio, we use its formula to calculate the required amount of lemon juice when the amount of water is \(10\) cups. By substituting \(x = 10\) into the equation \(y = 0.2x\), we get \(y = 0.2 \times 10 = 2\). Therefore, for \(10\) cups of water, the Classic Lemonade recipe requires \(2\) cups of lemon juice.

  1. Conclusion

The comparison shows that the Classic Lemonade recipe has the higher concentration of lemon juice. For \(10\) cups of water, this recipe will use \(2\) cups of lemon juice. The final result for the amount of lemon juice is \(\boxed{2}\).

Examples

Understanding how to compare ratios is essential in many real-life situations, such as cooking and baking, where you might want to find the most flavorful recipe. By expressing recipes as mathematical ratios, you can easily determine which mixture is stronger or better suited to your preferences, just like comparing the concentration of lemon juice in different lemonade recipes to make the perfect glass.

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