Student Page 3.7.2 Setting Up Systems of Equations with Quantity Rate Value Tables
This activity guides you in creating a system of equations based on known information in a context. Contexts well-suited for use of QRV tables generally involve two unknown quantities that are added together to get a total, different rates (price per pound, price per ticket, level of acidity, etc.) for each quantity, and a value obtained by multiplying the rate by the corresponding quantity. Read carefully and make sense of each step as you solve the problems below.
1.
The admission fee at the Grand Rapids Public Museum is $3 for children and $8 for adults. One day 575 people paid to enter the museum. The museum collected $3450 for admissions.
(a)
Fill in the first two rows of Table 3.7.2.1.
Initially, the number of Children and the number of Adults are unknown. Choose variables to indicate each of these.
The rates (ticket prices) are given in the problem.
How can you use each Quantity and Rate to get the Value?
| Quantity | Rate | Value | |
|---|---|---|---|
| Children | |||
| Adults | |||
| Total | X |
(b)
Fill in the empty cells in the last row of Table 3.7.2.1. The Total Quantity and the Total Value are both given in the problem.
(c)
Write an equation showing the relationship between the Quantity of Children, the Quantity of Adults, and the Total Quantity.
(d)
Write an equation showing the relationship between the Value of Children's tickets, the Value of Adults' tickets, and the Total Value.
(e)
The equations you wrote in Task 3.7.2.1.c and Task 3.7.2.1.d provide a system of linear equations.
(i)
Solve the system in two different ways. Verify that your solution works.
(ii)
How many children and how many adults paid for admission to the museum that day?
2.
Melanie's favorite ways to exercise are playing basketball and running. Her goal is to exercise 60 minutes each day, splitting her time between both basketball and running. She also wants to achieve the equivalent of 10,000 steps per day. Playing basketball for one minute is equivalent to taking 130 steps/minute. Running at the rate of 5 miles per hour is equivalent to taking 185 steps per minute.
(a)
Use Table 3.7.2.2 to set up two equations, one showing the relationship between the number of minutes Melanie will play each sport, and one showing the relationship between the equivalent numbers of steps Melanie will take during each activity.
| Quantity | Rate | Value | |
|---|---|---|---|
| Playing Basketball | |||
| Running | |||
| Total | X |
(b)
Solve the system of equations using two different methods.
(c)
How many minutes of basketball, \(B\text{,}\) and minutes of running, \(R\text{,}\) should Melanie do each day to exercise 60 minutes and take the equivalent of 10,000 steps?
3.
Filiz's favorite ways to exercise in the summer are walking and cycling. Her goal is to exercise 90 minutes each day, splitting her time between both types of exercise. She also wants to achieve the equivalent of 12,000 steps per day. Walking for one minute at the rate of 3.5 miles per hour is equivalent to taking 130 steps/minute. Cycling at the rate of 15 miles per hour is equivalent to taking 160 steps per minute.
(a)
Use Table 3.7.2.3 to set up two equations, one showing the relationship between the number of minutes Filiz will engage in each type of exercise, and one showing the relationship between the equivalent numbers of steps Filiz will take during each activity.
| Quantity | Rate | Value | |
|---|---|---|---|
| Walking | |||
| Cycling | |||
| Total | X |
(b)
Solve the system of equations using two different methods.
(c)
How many minutes of walking, \(W\text{,}\) and how many minutes of cycling, \(C\text{,}\) should Filiz do each day to exercise 90 minutes and take the equivalent of 12,000 steps?
What did you learn about using Quantity Rate Value Tables? How can you use what you learned to set-up systems of equations problems?