## Section3.7Using Quantity Rate Value Tables to Determine Equations

### Subsection3.7.1Overview

Half the battle in solving a system of equations has to do with finding the equations in the first place. In Section 3.7, you will learn a convenient tool to help you set up equations for problems that involve rates.

### Student Page3.7.2Setting Up Systems of Equations with Quantity Rate Value Tables A4US

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.

##### (a)

Fill in the first two rows of Table 3.7.3.1 using information from the story.

##### (c)

Write two equations that arise directly from the table.

##### (d)

Solve the system of equations. Use two different methods. Show your work. Verify that your solution works.

##### (e)

Interpret your solution based on the problem with gummy and taffy candies.

#### 2.

Gavin wants to start-up a food truck that uses locally grown ingredients. He plans to sell crepes and smoothies; he will offer different types of each item. In this case, you need a variation of the QRV table to find each equation needed to solve the system.

##### (c)

Use the equations from Task 3.7.3.2.a and Task 3.7.3.2.b to determine how many crepes and how many smoothies Gavin should be prepared to sell each day to reach his goals. Solve the problem in at least 2 different ways.

### Student Page3.7.4Chemistry Mixture Problems Using Quantity Rate Value Tables A4US

Quantity Rate Value Tables are helpful in setting up equations to solve mixture problems in chemistry and other fields. As you solve these problems, notice that a Rate is needed in all three rows. Add Quantities and Values to obtain equations to solve these problems.

#### 1.

How much 10% sulfuric acid ($$H_2SO_4$$) must be mixed with how much 30% sulfuric acid to make 200 milliliters of 15% sulfuric acid?

##### (a)

Fill in the first two rows of Table 3.7.4.1 using information from the problem.

##### (c)

Determine equations arising from the table.

##### (d)

Solve the system of equations. Show your work. Show that your solution works.

##### (e)

Interpret your solution based on the problem with two different strengths of acid solution.

#### 2.

Solve these two problems similarly to problem 1. Draw your own table for each problem.

##### (a)

You need 20 liters of 20% acid solution. You have jugs of 10% solution and 25% solution. How many liters of each should you combine to get the needed solution?

##### (b)

A medical technician has 20% alcohol solution and 70% alcohol solution. She needs 20 liters of a 40% alcohol solution. What amount of each type of solution should she combine?

#### 3.

For each problem, add Quantities to get an expression. Add Values to get another expression. Use the rate for the Total row to find a relationship between the expressions.

##### (a)

A chemist has 6 liters of a 25% alcohol solution. How much alcohol must he add so that the resulting solution contains 50% alcohol?

##### (b)

How many liters of a 14 percent alcohol solution must be mixed with 20 liters of a 50 percent alcohol solution to get a 20 percent alcohol solution? (Draw your own table.)

#### 4.

Sterling Silver is 92.5% pure silver. How many grams of pure silver and sterling silver must be mixed to obtain 100g of a 94% Silver alloy?