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Tarin organized this table to determine the number of adults, \( a \), and children attending a fundraiser based on the total number of people attending and the total amount of money raised.

\begin{tabular}{|c|c|c|c|}
\cline {2-4}
\multicolumn{1}{c|}{} & Number & Cost & Total \\
\hline
Adult & \( a \) & \$20 & \\
\hline
Child & \( 319 - a \) & \$10 & \\
\hline
Total & & & \$5,860 \\
\hline
\end{tabular}

If \$5,860 was raised, which statements are true? Check all that apply.

A. 319 people attended.

B. 52 children attended.

C. 201 adults attended.

D. The total amount of money raised from adult tickets is double the total amount of money raised from children's tickets.

E. The total amount of money raised from adults attending is more than 10 times the total amount of money raised from children attending.



Answer :

GinnyAnswer
Let's delve into the problem step-by-step to determine the accurate answers regarding the number of adults and children attending the fundraiser:

1. Total Number of People and Money Raised:
- We are given that 319 people attended the fundraiser.
- The total amount of money raised is $5,860.

2. Cost per Ticket:
- Each adult ticket costs $20.
- Each child ticket costs $10.

3. Variable Definitions:
- Let \( a \) represent the number of adults who attended.
- Consequently, the number of children who attended will be \( 319 - a \).

4. Formulate the Equation:
- The total money raised can be expressed as:
[tex]\[ 20a + 10(319 - a) = 5860 \][/tex]
- Simplify the equation:
[tex]\[ 20a + 3190 - 10a = 5860 \][/tex]
[tex]\[ 10a + 3190 = 5860 \][/tex]
[tex]\[ 10a = 5860 - 3190 \][/tex]
[tex]\[ 10a = 2670 \][/tex]
[tex]\[ a = 267 \][/tex]
- Therefore, the number of adults \( a \) is 267.

5. Calculate the Number of Children:
- The number of children is:
[tex]\[ 319 - a = 319 - 267 = 52 \][/tex]
- Hence, there are 52 children.

6. Verification of Statements:
- 319 people attended:
- Yes, this statement is true.
- 52 children attended:
- Yes, this statement is true.
- 201 adults attended:
- No, this statement is false (267 adults attended).
- The total amount of money raised from adult tickets is double the total amount of money raised from children's tickets.
- Calculate money from adults: \( 267 \times 20 = 5340 \)
- Calculate money from children: \( 52 \times 10 = 520 \)
- Check if \( 5340 = 2 \times 520 \):
- \( 5340 \neq 1040 \)
- No, this statement is false.
- The total amount of money raised from adults attending is more than 10 times the total amount of money raised from children attending.
- Compare \( 5340 \) with \( 10 \times 520 = 5200 \):
- \( 5340 > 5200 \)
- Yes, this statement is true.

Given these calculations, we find the following statements are true:
- 319 people attended.
- 52 children attended.
- The total amount of money raised from adults attending is more than 10 times the total amount of money raised from children attending.

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