Math2

back to Dynact Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Chapter 6 Word Problems

In this chapter, you’ll learn how to solve problems involving more than one equation, like finding out when two animals that grow at different rates will be the same size.

The costs for parking in two different parking garages are given in the table at the right.

Two hikers are walking along a marked trail. The first hiker starts at a point 6 mi from the beginning of the trail and walks at a speed of 4 mi>h. At the same time, the second hiker starts 1 mi from the beginning and walks at a speed of 3 mi>h.

The number of right-handed students in a mathematics class is nine times the number of left-handed students. The total number of students in the class is 30. How many right-handed students are in the class? How many left-handed students are in the class?

A plant nursery is growing a tree that is 3 ft tall and grows at an average rate of 1 ft per year. Another tree at the nursery is 4 ft tall and grows at an average rate of 0.5 ft per year. After how many years will the trees be the same height?

At a local fitness center, members pay a $20 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same?

Tickets for a concert cost $10 each if you order them online, but you must pay a service charge of $8 per order. The tickets are $12 each if you buy them at the door on the night of the concert.

Scientists studied the weights of two alligators over a period of 12 months. The initial weight and growth rate of each alligator are shown below. After how many months did the alligators weigh the same amount?

One satellite radio service charges $10 per month plus an activation fee of $20. A second service charges $11 per month plus an activation fee of $15. In what month was the cost of the service the same?

Two professional downhill skiers are racing at the speeds shown in the diagram. Skier 1 starts 5 s before Skier 2. The course is 5000 ft long. Will Skier 2 pass Skier 1? How do you know?

You are looking for an after-school job. One job pays $9 per hour. Another pays $12 per hour, but you must buy a uniform that costs $39. After how many hours of work would your net earnings from either job be the same?

A cell phone provider offers a plan that costs $40 per month plus $.20 per text message sent or received. A comparable plan costs $60 per month but offers unlimited text messaging.

A school is planning a field trip for 142 people. The trip will use six drivers and two types of vehicles: buses and vans. A bus can seat 51 passengers. A van can seat 10 passengers. Write and solve a system of equations to find how many buses and how many vans will be needed.

The measure of one acute angle in a right triangle is four times the measure of the other acute angle. Write and solve a system of equations to find the measures of the acute angles.

Adult tickets to a play cost $22. Tickets for children cost $15. Tickets for a group of 11 people cost a total of $228. Write and solve a system of equations to find how many children and how many adults were in the group.

A farmer grows corn, tomatoes, and sunflowers on a 320-acre farm. This year, the farmer wants to plant twice as many acres of tomatoes as acres of sunflowers. The farmer also wants to plant 40 more acres of corn than of tomatoes. How many acres of each crop should the farmer plant?

Michelle and Pam are running a 200-m race. Michelle runs at an average of 7.5 m/s. Pam averages 7.8 m/s, but she starts 1 s after Michelle.

The rectangle shown has a perimeter of 34 cm and the given area. Its length is 5 more than twice its width. Write and solve a system of equations to find the dimensions of the rectangle.

A toy store worker packed two boxes of identical dolls and plush toys for shipping in boxes that weigh 1 oz when empty. One box held 3 dolls and 4 plush toys. The worker marked the weight as 12 oz. The other box held 2 dolls and 3 plush toys. The worker marked the weight as 10 oz. Explain why the worker must have made a mistake.

A hotel offers two activity packages. One costs $192 and includes 3 h of horseback riding and 2 h of parasailing. The second costs $213 and includes 2 h of horseback riding and 3 h of parasailing. What is the cost for 1 h of each activity?

Each of the squares in the figures shown at the right has the same area, and each of the triangles has the same area. The total area of Figure A is 141 cm2. The total area of Figure B is 192 cm2. What is the area of each square and each triangle?

The zoo has two other water tanks that are leaking. One tank contains 10 gal of water and is leaking at a constant rate of 2 gal/h. The second tank contains 6 gal of water and is leaking at a constant rate of 4 gal/h. When will the tanks have the same amount of water? Explain.

break even pt A fashion designer makes and sells hats. The material for each hat costs $5.50. The hats sell for $12.50 each. The designer spends $1400 on advertising. How many hats must the designer sell to break even?

A puzzle expert wrote a new sudoku puzzle book. His initial costs are $864. Binding and packaging each book costs $.80. The price of the book is $2. How many copies must be sold to break even?

identifying constraints The local zoo is filling two water tanks for the elephant exhibit. One water tank contains 50 gal of water and is filled at a constant rate of 10 gal/h. The second water tank contains 29 gal of water and is filled at a constant rate of 3 gal/h. When will the two tanks have the same amount of water? Explain.

You row upstream at a speed of 2 mi>h. You travel the same distance downstream at a speed of 5 mi>h. What would be your rowing speed in still water? What is the speed of the current? b. Reasoning Suppose your rowing speed in still water is 3 mi>h and the speed of the current is 4 mi>h. What happens when you try to row upstream?

Suppose you graph a system of linear equations and the intersection point appears to be (3, 7). Can you be sure that the ordered pair (3, 7) is the solution? What must you do to be sure?

A group of scientists studied the effect of a chemical on various strains of bacteria. Strain A started with 6000 cells and decreased at a constant rate of 2000 cells per hour after the chemical was applied. Strain B started with 2000 cells and decreased at a constant rate of 1000 cells per hour after the chemical was applied. When will the strains have the same number of cells? Explain

A traveler is walking on a moving walkway in an airport. The traveler must walk back on the walkway to get a bag he forgot. The traveler’s groundspeed is 2 ft>s against the walkway and 6 ft>s with the walkway. What is the traveler’s speed off the walkway? What is the speed of the moving walkway?

A kayaker paddles upstream from camp to photograph a waterfall and returns. The kayaker’s speed while traveling upstream and downstream is shown below. What is the kayaker’s speed in still water? What is the speed of the current?

You have a jar of pennies and quarters. You want to choose 15 coins that are worth exactly $4.35. a. Write and solve a system of equations that models the situation. b. Is your solution reasonable in terms of the original problem? Explain.

You can represent the value of any two-digit number with the expression 10a 1 b, where a is the tens’ place digit and b is the ones’ place digit. For example, if a is 5 and b is 7, then the value of the number is 10(5) 1 7, or 57. What two-digit number is described below? • The ones’ place digit is one more than twice the tens’ place digit. • The value of the number is two more than five times the ones’ place digit.

You want to sell 1-lb jars of mixed peanuts and cashews for $5. You pay $3 per pound for peanuts and $6 per pound for cashews. You plan to combine 4 parts peanuts and 1 part cashews to make your mix. You have spent $70 on materials to get started. How many jars must you sell to break even?

Last year, one fourth of the students in your class played an instrument. This year, 6 students joined the class. Four of the new students play an instrument. Now, one third of the students play an instrument. How many students are in your class now?

inequalities You are planning what to do after school. You can spend at most 6 h daily playing basketball and doing homework. You want to spend less than 2 h playing basketball. You must spend at least 1 12 h on homework. What is a graph showing how you can spend your time?

Suppose you have a job mowing lawns that pays $12 per hour. You also have a job at a clothing store that pays $10 per hour. You need to earn at least $350 per week, but you can work no more than 35 h per week. You must work a minimum of 10 h per week at the clothing store. What is a graph showing how many hours per week you can work at each job?

Two friends agree to split the driving on a road trip from Philadelphia, Pennsylvania, to Denver, Colorado. One friend drives at an average speed of 60 mi>h. The other friend drives at an average speed of 55 mi>h. They want to drive at least 500 mi per day. They plan to spend no more than 10 h driving each day. The friend who drives slower wants to drive fewer hours. What is a graph showing how they can split the driving each day?

You are fencing in a rectangular area for a garden. You have only 150 ft of fence. You want the length of the garden to be at least 40 ft. You want the width of the garden to be at least 5 ft. What is a graph showing the possible dimensions your garden could have? • What variables will you use? What will they represent? • How many inequalities do you need to write?

A jeweler plans to produce a ring made of silver and gold. The price of gold is about $25 per gram. The price of silver is approximately $.40 per gram. She considers the following in deciding how much gold and silver to use in the ring. • The total mass must be more than 10 g but less than 20 g. • The ring must contain at least 2 g of gold. • The total cost of the gold and silver must be less than $90. a. Write and graph the inequalities that describe this situation. b. For one solution, find the mass of the ring and the cost of the gold and silver.

A teacher wants to post a row of student artwork on a wall that is 20 ft long. Some pieces are 8.5 in. wide. Other pieces are 11 in. wide. She is going to leave 3 in. of space to the left of each art piece. She wants to post at least 16 pieces of art. Write and graph a system of inequalities that describes how many pieces of each size she can post.