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Difficulty: MediumFractions, Decimals, and Percentages

A coffee roasting company creates a custom espresso blend using three types of coffee beans: Arabica, Robusta, and Liberica. In the blend, 38\frac{3}{8} of the total weight is Arabica beans and 40%40\% of the total weight is Robusta beans. The remaining weight of the blend consists of Liberica beans. If a batch of this custom blend contains 1818 ounces of Liberica beans, what is the total weight, in ounces, of the coffee bean batch?

  1. A
    30
  2. B
    48
  3. C
    62
  4. D
    72
  5. 80Answer

Answer

The total weight of the coffee bean batch is 80 ounces.
To find the total weight of the batch, first express each given portion as a decimal or percentage of the total weight. The Arabica beans make up 38=0.375\frac{3}{8} = 0.375 or 37.5%37.5\% of the total weight, and Robusta beans make up 40%40\% of the total weight. Together, Arabica and Robusta represent 37.5%+40%=77.5%37.5\% + 40\% = 77.5\% of the total weight. Consequently, Liberica beans represent the remaining 100%−77.5%=22.5%100\% - 77.5\% = 22.5\% (or 0.2250.225) of the batch. Since 1818 ounces equals 22.5%22.5\% of the total weight WW, we set up the equation 0.225W=180.225 W = 18 and solve for WW: W=180.225=80W = \frac{18}{0.225} = 80 ounces.

Step-by-Step Solution

1
Convert the fractional share of Arabica beans into a decimal or percentage.
38=0.375=37.5%\frac{3}{8} = 0.375 = 37.5\%
Converting all components to a common format (percentages or decimals) allows easy addition of the component shares.
2
Find the combined percentage share of Arabica and Robusta beans.
37.5%+40%=77.5%37.5\% + 40\% = 77.5\%
Adding the individual percentages determines the total fraction of the batch occupied by Arabica and Robusta.
3
Calculate the percentage share representing Liberica beans.
100%−77.5%=22.5%100\% - 77.5\% = 22.5\% (or 0.2250.225 in decimal form)
The remaining percentage of the batch must equal the Liberica portion.
4
Set up and solve the equation to find the total batch weight WW.
0.225×W=18  ⟹  W=180.225=800.225 \times W = 18 \implies W = \frac{18}{0.225} = 80 ounces
Dividing the given weight of Liberica beans by its fractional representation yields the total weight of the entire coffee blend.

Key Concept

Combining fractions and percentages to solve part-to-whole word problems
Estimated Time:1m 30s
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Topics

Fractions, Decimals, and Percentages

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