William Bought Some Tickets To See His Favorite Singer. He Bought A Total Of 15 Tickets, Consisting Of Adult And Children's Tickets. The Adult Tickets Cost $ 30 \$30 $30 Per Ticket, And The Children's Tickets Cost $ 20 \$20 $20 Per Ticket. If He

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Introduction

William is a huge fan of his favorite singer and has decided to buy tickets to see her live in concert. He purchased a total of 15 tickets, which include both adult and children's tickets. The adult tickets cost $30\$30 per ticket, while the children's tickets cost $20\$20 per ticket. In this article, we will explore the mathematical problem of determining the number of adult and children's tickets William bought.

The Problem

Let's denote the number of adult tickets as AA and the number of children's tickets as CC. We know that the total number of tickets is 15, so we can write the equation:

A+C=15A + C = 15

We also know that the total cost of the tickets is the sum of the cost of the adult tickets and the cost of the children's tickets. Since the adult tickets cost $30\$30 per ticket and the children's tickets cost $20\$20 per ticket, we can write the equation:

30A+20C=Total Cost30A + 20C = \text{Total Cost}

However, we don't know the total cost of the tickets. To solve this problem, we need to find the total cost of the tickets.

Finding the Total Cost

To find the total cost of the tickets, we need to know the number of adult and children's tickets. Let's assume that William bought xx adult tickets and yy children's tickets. Then, we can write the equation:

x+y=15x + y = 15

We also know that the total cost of the tickets is the sum of the cost of the adult tickets and the cost of the children's tickets. Since the adult tickets cost $30\$30 per ticket and the children's tickets cost $20\$20 per ticket, we can write the equation:

30x+20y=Total Cost30x + 20y = \text{Total Cost}

Now, we have two equations and two variables. We can solve this system of equations to find the values of xx and yy.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution. We can solve the first equation for yy:

y=15−xy = 15 - x

Now, we can substitute this expression for yy into the second equation:

30x+20(15−x)=Total Cost30x + 20(15 - x) = \text{Total Cost}

Simplifying this equation, we get:

30x+300−20x=Total Cost30x + 300 - 20x = \text{Total Cost}

Combine like terms:

10x+300=Total Cost10x + 300 = \text{Total Cost}

Now, we can solve for xx:

10x=Total Cost−30010x = \text{Total Cost} - 300

x=Total Cost−30010x = \frac{\text{Total Cost} - 300}{10}

Now, we can substitute this expression for xx into the first equation:

x+y=15x + y = 15

Total Cost−30010+y=15\frac{\text{Total Cost} - 300}{10} + y = 15

Simplifying this equation, we get:

y=15−Total Cost−30010y = 15 - \frac{\text{Total Cost} - 300}{10}

Now, we have expressed yy in terms of the total cost. We can substitute this expression for yy into the second equation:

30x+20y=Total Cost30x + 20y = \text{Total Cost}

30x+20(15−Total Cost−30010)=Total Cost30x + 20\left(15 - \frac{\text{Total Cost} - 300}{10}\right) = \text{Total Cost}

Simplifying this equation, we get:

30x+300−25(Total Cost−300)=Total Cost30x + 300 - \frac{2}{5}(\text{Total Cost} - 300) = \text{Total Cost}

Combine like terms:

30x−25(Total Cost−300)+300=Total Cost30x - \frac{2}{5}(\text{Total Cost} - 300) + 300 = \text{Total Cost}

Now, we can solve for xx:

30x−25(Total Cost−300)=Total Cost−30030x - \frac{2}{5}(\text{Total Cost} - 300) = \text{Total Cost} - 300

30x=Total Cost−300+25(Total Cost−300)30x = \text{Total Cost} - 300 + \frac{2}{5}(\text{Total Cost} - 300)

30x=75(Total Cost−300)30x = \frac{7}{5}(\text{Total Cost} - 300)

x=7150(Total Cost−300)x = \frac{7}{150}(\text{Total Cost} - 300)

Now, we can substitute this expression for xx into the first equation:

x+y=15x + y = 15

7150(Total Cost−300)+y=15\frac{7}{150}(\text{Total Cost} - 300) + y = 15

Simplifying this equation, we get:

y=15−7150(Total Cost−300)y = 15 - \frac{7}{150}(\text{Total Cost} - 300)

Now, we have expressed yy in terms of the total cost. We can substitute this expression for yy into the second equation:

30x+20y=Total Cost30x + 20y = \text{Total Cost}

30(7150(Total Cost−300))+20(15−7150(Total Cost−300))=Total Cost30\left(\frac{7}{150}(\text{Total Cost} - 300)\right) + 20\left(15 - \frac{7}{150}(\text{Total Cost} - 300)\right) = \text{Total Cost}

Simplifying this equation, we get:

75(Total Cost−300)+300−715(Total Cost−300)=Total Cost\frac{7}{5}(\text{Total Cost} - 300) + 300 - \frac{7}{15}(\text{Total Cost} - 300) = \text{Total Cost}

Combine like terms:

75(Total Cost−300)+300−1415(Total Cost−300)=Total Cost\frac{7}{5}(\text{Total Cost} - 300) + 300 - \frac{14}{15}(\text{Total Cost} - 300) = \text{Total Cost}

2115(Total Cost−300)+300−1415(Total Cost−300)=Total Cost\frac{21}{15}(\text{Total Cost} - 300) + 300 - \frac{14}{15}(\text{Total Cost} - 300) = \text{Total Cost}

715(Total Cost−300)+300=Total Cost\frac{7}{15}(\text{Total Cost} - 300) + 300 = \text{Total Cost}

Now, we can solve for the total cost:

715(Total Cost−300)=Total Cost−300\frac{7}{15}(\text{Total Cost} - 300) = \text{Total Cost} - 300

Total Cost−300=157(Total Cost−300)\text{Total Cost} - 300 = \frac{15}{7}(\text{Total Cost} - 300)

Total Cost=156(Total Cost−300)\text{Total Cost} = \frac{15}{6}(\text{Total Cost} - 300)

Total Cost=52(Total Cost−300)\text{Total Cost} = \frac{5}{2}(\text{Total Cost} - 300)

Total Cost=52Total Cost−52(300)\text{Total Cost} = \frac{5}{2}\text{Total Cost} - \frac{5}{2}(300)

Total Cost−52Total Cost=−52(300)\text{Total Cost} - \frac{5}{2}\text{Total Cost} = -\frac{5}{2}(300)

−32Total Cost=−750-\frac{3}{2}\text{Total Cost} = -750

Total Cost=500\text{Total Cost} = 500

Now that we have found the total cost, we can substitute this value into the expression for xx:

x=7150(Total Cost−300)x = \frac{7}{150}(\text{Total Cost} - 300)

x=7150(500−300)x = \frac{7}{150}(500 - 300)

x=7150(200)x = \frac{7}{150}(200)

x=1400150x = \frac{1400}{150}

x=283x = \frac{28}{3}

Since xx represents the number of adult tickets, we can round down to the nearest whole number:

x=9x = 9

Now that we have found the value of xx, we can substitute this value into the expression for yy:

y=15−7150(Total Cost−300)y = 15 - \frac{7}{150}(\text{Total Cost} - 300)

y=15−7150(500−300)y = 15 - \frac{7}{150}(500 - 300)

y=15−7150(200)y = 15 - \frac{7}{150}(200)

y=15−1400150y = 15 - \frac{1400}{150}

y=15−283y = 15 - \frac{28}{3}

y=15−9.33y = 15 - 9.33

y=5.67y = 5.67

Since yy represents the number of children's tickets, we can round down to the nearest whole number:

y=5y = 5

Conclusion

Introduction

In our previous article, we solved a mathematical problem involving the purchase of tickets to see a favorite singer. William bought a total of 15 tickets, consisting of adult and children's tickets, and we found that he bought 9 adult tickets and 5 children's tickets. In this article, we will answer some common questions related to this problem.

Q: What is the total cost of the tickets?

A: The total cost of the tickets is $500\$500.

Q: How many adult tickets did William buy?

A: William bought 9 adult tickets.

Q: How many children's tickets did William buy?

A: William bought 5 children's tickets.

Q: What is the cost of an adult ticket?

A: The cost of an adult ticket is $30\$30.

Q: What is the cost of a children's ticket?

A: The cost of a children's ticket is $20\$20.

Q: How many tickets did William buy in total?

A: William bought a total of 15 tickets.

Q: What is the ratio of adult tickets to children's tickets?

A: The ratio of adult tickets to children's tickets is 9:5.

Q: What is the percentage of adult tickets?

A: The percentage of adult tickets is 915×100%=60%\frac{9}{15} \times 100\% = 60\%.

Q: What is the percentage of children's tickets?

A: The percentage of children's tickets is 515×100%=33.33%\frac{5}{15} \times 100\% = 33.33\%.

Q: What is the average cost of a ticket?

A: The average cost of a ticket is $50015=$33.33\frac{\$500}{15} = \$33.33.

Q: What is the total cost of the adult tickets?

A: The total cost of the adult tickets is 9×$30=$2709 \times \$30 = \$270.

Q: What is the total cost of the children's tickets?

A: The total cost of the children's tickets is 5×$20=$1005 \times \$20 = \$100.

Conclusion

In this article, we have answered some common questions related to the problem of William's ticket purchase. We have found that William bought 9 adult tickets and 5 children's tickets, and the total cost of the tickets was $500\$500. We have also calculated various percentages and ratios related to the tickets.