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Chapter 2: Problem 19

The total number of passengers riding a certain city bus during the morning shift is 1000 . If the child's fare is \(\$ .50\), the adult fare is \(\$ 1.50\), and the total revenue from the fares in the morning shift is \(\$ 1300\), how many children and how many adults rode the bus during the morning shift?

Short Answer

Expert verified
There were 200 children and 800 adults on the bus during the morning shift.

Step by step solution

01

Define the variables

Let's define x as the number of children and y as the number of adults on the bus.
02

Set up the equations

The total number of passengers is given as 1000, so we have: \(x + y = 1000\) The total revenue from the fares is given as \(1300, and it is also given that the child's fare is \).50, and the adult fare is $1.50. So, the second equation is: \(0.50x + 1.50y = 1300\) Now we have a system of linear equations: \( \begin{cases} x + y = 1000 \\ 0.50x + 1.50y = 1300 \end{cases} \)
03

Solve the system of equations

We can solve this system of linear equations using different methods such as substitution, elimination or matrix method. We will use the elimination method in this case. Multiply the first equation by 0.50 to make the x-coefficients the same in both equations: \(0.50x + 0.50y = 500\) Now, we subtract the new equation from the second equation to eliminate x from the equation: \((0.50x + 1.50y) - (0.50x + 0.50y) = 1300 - 500\) \(1.00y = 800\) Solve for y: \(y = \frac{800}{1} = 800\) Now that we have the number of adults, we can plug the value of y into any of the original equations to find x. We will use the first equation: \(x + 800 = 1000\) \(x = 1000 - 800\) \(x = 200\)
04

Interpret the solution

So, there were 200 children and 800 adults on the bus during the morning shift.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations is a collection of two or more equations with the same set of variables. These systems can help us solve problems where multiple conditions need to be met at the same time. In this exercise, we're dealing with a system of two linear equations:
  • First equation: Represents the total number of passengers, given as 1000. So, it is expressed as: \(x + y = 1000\), where \(x\) is the number of children and \(y\) is the number of adults.
  • Second equation: Represents the total revenue from ticket fares, given as \(1300\). This is expressed as: \(0.50x + 1.50y = 1300\), where \(0.50\) is the child's fare and \(1.50\) is the adult's fare.
With these equations, we can find the exact number of children and adults that were on the bus. This type of problem is common when the situation can be described by interrelated conditions and needs a precise solution.
Revenue Calculation
Revenue calculation involves determining the total earnings from the number of items sold or used, often considering different rates of payment. In this case, solving for the bus fares helps us understand how the total revenue of \(1300\) was earned through specific ticket prices.
  • Children's fare was \(\\(0.50\) per ticket.
  • Adult's fare was \(\\)1.50\) per ticket.
By multiplying the ticket prices by the number of tickets sold (passengers), we can calculate the total revenue. So, the equation \(0.50x + 1.50y = 1300\) helps us see how much money came from children's and adults' tickets combined.This type of calculation is vital in any business or scenario where multiple pricing tiers exist. It helps to find out exactly how different groups contribute to the total earnings.
Passenger Distribution
Passenger distribution refers to understanding how many passengers fall into different categories, such as children and adults in this exercise. By solving the system of equations, specifically the passenger count and revenue, we can break down exactly how many children and adults were on the bus.The steps to find the number of passengers in each category include:
  • Using the first equation \(x + y = 1000\), which tells us the total number of passengers is \(1000\).
  • Using the second equation \(0.50x + 1.50y = 1300\) to get the revenue split.
We solved for \(y\), which represented adults, and found \(y = 800\). Then, by substituting back, we found \(x = 200\), which represented children. This solution highlights how passenger distribution shows different user categories within a total, and it allows for better resource management and planning.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a \(2 \times 4\) matrix and \(B\) is a matrix such that \(A B A\) is defined, then the size of \(B\) must be \(4 \times 2\).

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. A dietitian wishes to plan a meal around three foods. The percent of the daily requirements of proteins, carbohydrates, and iron contained in each ounce of the three foods is summarized in the following table: $$\begin{array}{lccc} \hline & \text { Food I } & \text { Food II } & \text { Food III } \\ \hline \text { Proteins }(\%) & 10 & 6 & 8 \\ \hline \text { Carbohydrates }(\%) & 10 & 12 & 6 \\ \hline \text { Iron }(\%) & 5 & 4 & 12 \\ \hline \end{array}$$ Determine how many ounces of each food the dietitian should include in the meal to meet exactly the daily requirement of proteins, carbohydrates, and iron \((100 \%\) of each).

Rainbow Harbor Cruises charges \$16/adult and \(\$ 8 /\) child for a round-trip ticket. The records show that, on a certain weekend, 1000 people took the cruise on Saturday and 800 people took the cruise on Sunday. The total receipts for Saturday were \(\$ 12,800\) and the total receipts for Sunday were \(\$ 9,600\). Determine how many adults and children took the cruise on Saturday and on Sunday.

Jackson Farms has allotted a certain amount of land for cultivating soybeans, corn, and wheat. Cultivating 1 acre of soybeans requires 2 labor-hours, and cultivating 1 acre of corn or wheat requires 6 labor-hours. The cost of seeds for 1 acre of soybeans is \(\$ 12\), for 1 acre of corn is \(\$ 20\), and for 1 acre of wheat is \(\$ 8\). If all resources are to be used, how many acres of each crop should be cultivated if the following hold? a. 1000 acres of land are allotted, 4400 labor-hours are available, and \(\$ 13,200\) is available for seeds. b. 1200 acres of land are allotted, 5200 labor-hours are available, and \(\$ 16,400\) is available for seeds.

BelAir Publishing publishes a deluxe leather edition and a standard edition of its Daily Organizer. The company's marketing department estimates that \(x\) copies of the deluxe edition and \(y\) copies of the standard edition will be demanded per month when the unit prices are \(p\) dollars and \(q\) dollars, respectively, where \(x, y, p\), and \(q\) are related by the following system of linear equations: $$ \begin{aligned} 5 x+y &=1000(70-p) \\ x+3 y &=1000(40-q) \end{aligned} $$ Find the monthly demand for the deluxe edition and the standard edition when the unit prices are set according to the following schedules: a. \(p=50\) and \(q=25\) b. \(p=45\) and \(q=25\) c. \(p=45\) and \(q=20\)
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