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Difficulty: MediumRatios, Rates, and Proportions

A construction project requires a specific concrete mixture made by combining cement, sand, and gravel in a ratio of 1:2:41 : 2 : 4 by weight. A builder needs to prepare exactly 350350 pounds of this concrete mixture. If the builder currently has 8080 pounds of sand and an unlimited supply of cement and gravel, how many additional pounds of sand must the builder purchase to make the concrete mixture?

Answer: 20 pounds

Answer

The builder must purchase 20 additional pounds of sand.
The ratio of cement to sand to gravel is 1:2:41 : 2 : 4, giving a total of 1+2+4=71 + 2 + 4 = 7 parts. Therefore, sand represents 27\frac{2}{7} of the total weight of the mixture. To find the total sand required for a 350350-pound mixture, we calculate 27×350=100\frac{2}{7} \times 350 = 100 pounds. Subtracting the 8080 pounds of sand already on hand, we find that the builder must purchase 100−80=20100 - 80 = 20 additional pounds of sand.

Step-by-Step Solution

1
Determine the fraction of the concrete mixture that consists of sand using the given ratio of 1:2:41 : 2 : 4 by weight.
Sand constitutes 27\frac{2}{7} of the total weight.
The sum of the ratio parts is 1+2+4=71 + 2 + 4 = 7 parts, and sand represents 22 parts out of these 77 total parts.
2
Calculate the total weight of sand required for 350350 pounds of the concrete mixture.
100100 pounds of sand.
Multiplying the fraction of sand, 27\frac{2}{7}, by the total desired weight of the mixture, 350350 pounds, gives the required weight of sand.
3
Calculate the additional amount of sand needed by subtracting the available sand from the total required sand.
2020 pounds of sand.
Since the builder already has 8080 pounds of sand, subtracting this from the required 100100 pounds yields the weight of additional sand to purchase.

Key Concept

Part-to-whole ratios and proportion scaling
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Topics

Ratios, Rates, and Proportions

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