Answered

How many quarts of water does Paul need to add to keep the consistency of the concrete the same?



Batch 1Batch 2

Dry Ingredients (qt)		515

Water (qt)		2

Verified Answer :

  • Identify the consistency ratio from Batch 1: \(\frac{5 \text{ qt dry}}{2 \text{ qt water}} = 2.5\).
  • Set up a proportion for Batch 2 using the known dry ingredient amount: \(\frac{15 \text{ qt dry}}{w \text{ qt water}} = 2.5\).
  • Solve the equation \(\frac{15}{w} = 2.5\) to find the required amount of water: \(w = 6\).
  • The total amount of water needed for Batch 2 is \(\boxed{6}\) quarts.

Explanation

  1. Problem Analysis and Data Definition

To determine how many quarts of water are needed for Batch 2, we first analyze the requirement for consistency. Maintaining the same consistency implies that the ratio of dry ingredients to water must remain constant between the two batches.

From the provided data:

  • For Batch 1, we have 5 quarts of dry ingredients and 2 quarts of water.
  • For Batch 2, we have 15 quarts of dry ingredients and an unknown amount of water, which we will call \(w\).
  1. Calculating the Consistency Ratio

To find the constant ratio \(R\) of dry ingredients to water, we divide the dry ingredients by the water for Batch 1:
\[R = \frac{\text{Dry}_1}{\text{Water}_1} = \frac{5}{2} = 2.5\]
This ratio means that for every 1 quart of water, Paul must use 2.5 quarts of dry ingredients. Alternatively, for every 1 quart of dry ingredients, he needs \(\frac{2}{5} = 0.4\) quarts of water.

  1. Solving for the Required Water Amount

Since Batch 2 must have the same consistency, we use the same ratio \(R\) for its ingredients. We set up the proportion for Batch 2:
\[\frac{\text{Dry}_2}{w} = \frac{\text{Dry}_1}{\text{Water}_1} = 2.5\]
Substituting the known value of 15 quarts of dry ingredients for Batch 2, we get:
\[\frac{15}{w} = \frac{5}{2}\]
To solve for \(w\), we cross-multiply:
\[5 \times w = 15 \times 2\]
\[5 \times w = 30\]
Dividing both sides by 5:
\[w = \frac{30}{5} = 6\]
Thus, 6 quarts of water are needed for Batch 2 to maintain the same consistency as Batch 1.

Examples

This problem demonstrates the concept of direct proportions, which is essential for scaling recipes or manufacturing mixtures. Whether you are adjusting a concrete mix for a construction project or doubling a recipe in the kitchen, understanding that the ratio between ingredients must stay constant ensures the final product maintains its intended quality and characteristics. By setting up a proportion, you can easily determine the necessary quantities for any size batch.

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