Question

In a circus company the price of tickets for adult and children were \$50 and \$30 respectively. The company has sold a total of 1000 tickets. The average (arithmetic mean) price per ticket sold was \$42. How many tickets were sold for children?

• A. 200
• B. 300
• C. 400
• D. 600
• E. 800

Hint:

This type of problem can be solved very quickly using the method of alligation. Write the prices of the two components (30 and 50) and the average price (42) in the middle. The ratio of the number of tickets is the inverse of the ratio of the differences.

Difference 1: \( |42 - 30| = 12 \)
Difference 2: \( |42 - 50| = 8 \)
The ratio of Adult tickets to Children tickets is \(12 : 8\), which simplifies to \(3 : 2\). Since the total is 1000 tickets, divide 1000 in the ratio 3:2. The number of children's tickets is \(\frac{2}{3+2} \times 1000 = \frac{2}{5} \times 1000 = 400\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a weighted average or mixture problem. We can solve it by setting up a system of linear equations representing the total number of tickets and the total revenue.
Step 2: Detailed Explanation:
Let \(A\) be the number of adult tickets sold and \(C\) be the number of children's tickets sold.
Equation for the total number of tickets:
\[ A + C = 1000 \]
Equation for the total revenue:
The total revenue is the average price per ticket multiplied by the total number of tickets.
\[ \text{Total Revenue} = \$42 \times 1000 = \$42,000 \]
The total revenue can also be expressed as the sum of the revenue from adult and children tickets:
\[ 50A + 30C = 42,000 \]
Now we have a system of two equations:
1) \( A + C = 1000 \)
2) \( 50A + 30C = 42,000 \)
We need to solve for \(C\). From equation (1), express \(A\) in terms of \(C\):
\[ A = 1000 - C \]
Substitute this expression for \(A\) into equation (2):
\[ 50(1000 - C) + 30C = 42,000 \]
\[ 50,000 - 50C + 30C = 42,000 \]
\[ 50,000 - 20C = 42,000 \]
Subtract 42,000 from both sides:
\[ 8,000 = 20C \]
Divide by 20:
\[ C = \frac{8000}{20} = 400 \]
Step 3: Final Answer:
There were 400 tickets sold for children. This corresponds to option (C).