Question

In the first hour of a bake sale, students sold either chocolate chip cookies, which sold for \(\$\)1.30, or brownies, which sold for \(\$\)1.50. What was the ratio of chocolate chip cookies sold to brownies sold during that hour? 
1. The average price for the items sold during that hour was $1.42 
2. The total price for all items sold during that hour was $14.20 
 

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• B. EACH statement ALONE is sufficient to answer the question asked
• C. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• D. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• E. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

Weighted average problems can often be solved quickly using an alligation or "number line" method. The ratio of the quantities is the inverse of the ratio of the distances of their individual values from the average. Here, the distance of 1.30 from 1.42 is 0.12, and the distance of 1.50 from 1.42 is 0.08. The ratio C:B is therefore 0.08:0.12, which simplifies to 2:3.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept: 
The question asks for the ratio of the number of chocolate chip cookies to the number of brownies sold. Let C be the number of cookies and B be the number of brownies. We need to find the value of the ratio \(\frac{C}{B}\). This is a weighted average and Diophantine equation problem. 
 

Step 2: Key Formula or Approach: 
For statement (1), we will use the weighted average formula: \[ \text{Average Price} = \frac{(\text{Price}_1 \times \text{Number}_1) + (\text{Price}_2 \times \text{Number}_2)}{\text{Total Number}} \] For statement (2), we will set up an equation for the total revenue and analyze its integer solutions. 
 

Step 3: Detailed Explanation: 
Analyzing Statement (1): The average price for the items sold during that hour was \$1.42. 
Using the weighted average formula: \[ 1.42 = \frac{1.30 \times C + 1.50 \times B}{C + B} \] Now, we solve this equation for the ratio \(\frac{C}{B}\). \[ 1.42(C + B) = 1.30C + 1.50B \] \[ 1.42C + 1.42B = 1.30C + 1.50B \] Now, group the C terms and B terms: \[ 1.42C - 1.30C = 1.50B - 1.42B \] \[ 0.12C = 0.08B \] To find the ratio \(\frac{C}{B}\), we can rearrange the equation: \[ \frac{C}{B} = \frac{0.08}{0.12} = \frac{8}{12} = \frac{2}{3} \] We have found a unique value for the ratio. Therefore, statement (1) is sufficient. 
Analyzing Statement (2): The total price for all items sold during that hour was \$14.20. 
This gives us an equation for the total revenue: \[ 1.30C + 1.50B = 14.20 \] To work with integers, we can multiply the entire equation by 100: \[ 130C + 150B = 1420 \] Divide by 10 to simplify: \[ 13C + 15B = 142 \] Here, C and B must be non-negative integers representing the number of items sold. We need to find the non-negative integer solutions to this linear Diophantine equation. We can test values or use properties of numbers. Let's look at the units digit. The term \(15B\) will end in a 0 (if B is even) or a 5 (if B is odd). The term \(142\) ends in a 2. 

- Case 1: B is odd. Then \(15B\) ends in 5. So, \(13C\) must end in a 7 for the sum to end in 2 (\(7+5=12\)). For \(13C\) to end in 7, C must end in 9 (since \(3 \times 9 = 27\)). Let's test C=9: \(13(9) + 15B = 117 + 15B = 142 \implies 15B = 25\). B is not an integer. The next value, C=19, would make \(13C\) too large. 

- Case 2: B is even. Then \(15B\) ends in 0. So, \(13C\) must end in a 2. For \(13C\) to end in 2, C must end in 4 (since \(3 \times 4 = 12\)). Let's test C=4: \(13(4) + 15B = 52 + 15B = 142 \implies 15B = 90 \implies B=6\). This is a valid integer solution. The next value, C=14, would make \(13C = 13(14)=182\), which is already greater than 142. 

So, the only possible non-negative integer solution is \(C=4, B=6\). This gives a unique ratio: \(\frac{C}{B} = \frac{4}{6} = \frac{2}{3}\). Therefore, statement (2) is also sufficient. 
 

Step 4: Final Answer: 
Since both statements independently provide enough information to determine the ratio, each statement alone is sufficient.