Question

A store sells two types of candies: Type A at \$5 per pound and Type B at \$8 per pound. If a mixture of these candies weighs 10 pounds and costs \$6.20 per pound, how many pounds of Type A candy are in the mixture?

• A. 4 pounds
• B. 5 pounds
• C. 6 pounds
• D. 7 pounds
• E. 8 pounds

Hint:

Mixture problems can be solved quickly using the alligation method. Write the component prices (5 and 8) and the mixture price (6.20) in the middle. The difference diagonally gives the ratio of the quantities. \(|8 - 6.20| : |5 - 6.20| \rightarrow 1.80 : 1.20 \rightarrow 18:12 \rightarrow 3:2\). The ratio of pounds of Type A to Type B is 3:2. Since there are 10 pounds total, divide it in the ratio 3:2, which gives 6 pounds and 4 pounds. So, 6 pounds of Type A.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a weighted average problem, also known as a mixture problem. We need to find the quantity of one component given the prices of the components, the total quantity, and the price of the mixture.
Step 2: Key Formula or Approach:
Let \(A\) be the number of pounds of Type A candy and \(B\) be the number of pounds of Type B candy. We can set up a system of two equations: one for the total weight and one for the total cost.
1) \(A + B = \text{Total Weight}\)
2) \((\text{Price of A}) \times A + (\text{Price of B}) \times B = \text{Total Cost}\)
Step 3: Detailed Explanation:
We are given:
Total Weight = 10 pounds
Total Cost = \$6.20/pound \(\times\) 10 pounds = \$62
Price of A = \$5/pound
Price of B = \$8/pound
Set up the equations:
1) \(A + B = 10\)
2) \(5A + 8B = 62\)
From equation (1), we can express \(B\) in terms of \(A\): \(B = 10 - A\).
Now, substitute this into equation (2):
\[ 5A + 8(10 - A) = 62 \]
Distribute the 8:
\[ 5A + 80 - 8A = 62 \]
Combine the terms with \(A\):
\[ -3A + 80 = 62 \]
Subtract 80 from both sides:
\[ -3A = 62 - 80 \]
\[ -3A = -18 \]
Divide by -3:
\[ A = 6 \]
Step 4: Final Answer:
There are 6 pounds of Type A candy in the mixture, which corresponds to option (C).