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Difficulty: MediumPercentages, Percent Change, and Interest

In January, a coffee roasting facility produced two specialty blends: Roast A and Roast B. Roast A sold for $10\$10 per pound and Roast B sold for $30\$30 per pound. A distributor purchased a total of 100100 pounds of coffee consisting of these two blends at an average price of $18\$18 per pound. In February, the price of Roast A increased by 20%20\%, while the price of Roast B decreased by 4%4\%. If the distributor purchased the exact same quantities of Roast A and Roast B in February as in January, by what percent did the total cost of the order increase?

  1. A
    3.85%3.85\%
  2. 4%4\%Answer
  3. C
    8%8\%
  4. D
    10.4%10.4\%
  5. E
    16%16\%

Answer

The total cost of the order increased by 4%4\%.
The choice stating 4%4\% is correct because the total cost in January is $1,800\$1,800 (6060 lbs of Roast A at $10\$10 and 4040 lbs of Roast B at $30\$30). In February, the price of Roast A rises to $12\$12 and Roast B drops to $28.80\$28.80, bringing the total cost to $1,872\$1,872. The change of $72\$72 represents a 4%4\% increase relative to the baseline January cost of $1,800\$1,800.

Step-by-Step Solution

1
Determine the quantities of Roast A and Roast B purchased in January.
Roast A: 6060 lbs, Roast B: 4040 lbs.
Let aa be the pounds of Roast A and bb be the pounds of Roast B. We are given a+b=100a + b = 100 and 10a+30b=18×100=180010a + 30b = 18 \times 100 = 1800. Substituting b=100−ab = 100 - a gives 10a+30(100−a)=1800  ⟹  −20a=−1200  ⟹  a=6010a + 30(100 - a) = 1800 \implies -20a = -1200 \implies a = 60 and b=40b = 40.
2
Calculate the total cost of the order in January.
Total January Cost = $1,800\$1,800.
Total cost = (60 lbs×$10/lb)+(40 lbs×$30/lb)=$600+$1,200=$1,800(60 \text{ lbs} \times \$10/\text{lb}) + (40 \text{ lbs} \times \$30/\text{lb}) = \$600 + \$1,200 = \$1,800.
3
Calculate the new prices and total cost in February.
Total February Cost = $1,872\$1,872.
New price of Roast A = $10×(1+0.20)=$12/lb\$10 \times (1 + 0.20) = \$12/\text{lb}. New price of Roast B = $30×(1−0.04)=$28.80/lb\$30 \times (1 - 0.04) = \$28.80/\text{lb}. Total February Cost = (60×$12)+(40×$28.80)=$720+$1,152=$1,872(60 \times \$12) + (40 \times \$28.80) = \$720 + \$1,152 = \$1,872.
4
Calculate the percentage increase in total cost from January to February.
Percent Increase = 4%4\%.
Percent Change = February Cost−January CostJanuary Cost×100%=$1,872−$1,800$1,800×100%=$72$1,800×100%=4%\frac{\text{February Cost} - \text{January Cost}}{\text{January Cost}} \times 100\% = \frac{\$1,872 - \$1,800}{\$1,800} \times 100\% = \frac{\$72}{\$1,800} \times 100\% = 4\%.

Key Concept

Weighted Percent Change and Systems of Equations
Estimated Time:1m 30s
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Topics

Percentages, Percent Change, and Interest

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