[Math HelperKeymaster]

Start with the files at the bottom of these directions. First read the one entitled “READ THIS FIRST” and then open the file called “Systems of Equations Problems with Answers”.

It is difficult to learn how to do story problems because there are so many different types. If you want to do well on this week’s test, FOLLOW THESE INSTRUCTIONS!

1) Go through ALL the story problems provided and try to solve them. Pretend it’s a Practice Test. Check your answers with the key provided.

2) Pick ONE of the problems that you got right (that has not already been solved by a classmate), and demonstrate its solution for the rest of us.

3) Study how your classmates solved the problems that you missed. Remember that these may be on the test!

To demonstrate your problem, select Start a New Conversation and make BOTH the problem number and topic (#10 Jarod and the Bunnies) the subject of your post.

Begin your post with a statement of the problem so that we can understand what you are doing.

The answers are at the end of the file, so don’t just give an answer—we can already see what the answers are. Don’t post an explanation unless your answer matches the correct one!

Your goal should be to explain this problem so well that a classmate who “just doesn’t get it” will be able to understand it completely!

This is a moderated forum. Your posting will say PENDING and will not be visible to the rest of the class until I approve it. Occasionally, more than one person will tackle a problem before they can see the work of others. In that case, credit will be given to all posters. Once the solution to a problem has become visible, that problem is off limits and you will need to choose a different problem in order to get credit.

I will indicate in the grading comments if corrections need to be made. If you haven’t received credit, first double-check for my comments in the gradebook. If everything looks OK, then message me asking me to check on it.

You must make the necessary corrections and have your work posted in order to receive credit.

For this particular Forum, no responses are required – your initial post is worth the full 10 points. Should you choose to respond to a classmate, a request for clarification on the procedure used, a suggestion for an alternate method of solving the problem or a general comment about the technique would all be appropriate. I’m sure that a “thank you” for an exceptionally clear explanation would also be welcome!

Please sign ALL your Forum posts with the name that you like to be called – it makes it so much easier for the rest of us to address you by your preferred name when we respond.

Initial Post Due: Sunday, by 11:55 p.m., ET

 

Systems of Equations

1) A vendor sells hot dogs and bags of potato chips. A customer buys 5 hot

dogs and 4 bags of potato chips for $17.00. Another customer buys 4 hot

dogs and 2 bags of potato chips for $11.50. Find the cost of each item.

1)

2) The Globe Theater sold 556 tickets for a play. Tickets cost $22 per adult

and $10 per senior citizen. If total receipts were $7924, how many senior

citizen tickets were sold?

2)

3) A tour group split into two groups when waiting in line for food at a fast

food counter. The first group bought 6 slices of pizza and 5 soft drinks

for $31.12. The second group bought 5 slices of pizza and 4 soft drinks

for $25.64. How much does one slice of pizza cost?

3)

4) Beverly scored 22 points in a recent basketball game without making

any 3-point shots. She scored 16 times, making several free throws

worth 1 point each and several field goals worth two points each. How

many free throws did she make? How many 2-point field goals did she

make?

4)

5) Ray has found that his new car gets 31 miles per gallon on the highway

and 26 miles per gallon in the city. He recently drove 285 miles on 10

gallons of gasoline. How many miles did he drive on the highway? How

many miles did he drive in the city?

5)

6) A textile company has specific dyeing and drying times for its different

cloths. A roll of Cloth A requires 60 minutes of dyeing time and 50

minutes of drying time. A roll of Cloth B requires 55 minutes of dyeing

time and 35 minutes of drying time. The production division allocates

2060 minutes of dyeing time and 1500 minutes of drying time for the

week. How many rolls of each cloth can be dyed and dried?

6)

7) A bank teller has 46 $5 and $20 bills in her cash drawer. The value of the

bills is $695. How many $5 bills are there?

7)

8) Jimmy always throws loose change into a pencil holder on his desk and

takes it out every two weeks. This time it is all nickels and dimes. There

are 2 times as many dimes as nickels, and the value of the dimes is $0.60

more than the value of the nickels. How many nickels and dimes does

Jimmy have?

8)

9) A flat rectangular piece of aluminum has a perimeter of 58 inches. The

length is 13 inches longer than the width. Find the width.

9)

1

10) John is having a problem with rabbits getting into his vegetable garden,

so he decides to fence it in. The length of the garden is 12 feet more than

5 times the width. He needs 72 feet of fencing to do the job. Find the

length and width of the garden.

10)

11) Two angles are complementary if the sum of their measures is 90°. The

measure of the first angle is 54° more than two times the second angle.

Find the measure of each angle.

11)

12) The three angles in a triangle always add up to 180°. If one angle in a

triangle is 36° and the second is 2 times the third, what are the three

angles?

12)

13) An isosceles triangle is one in which the measures of two of the sides are

equal. The perimeter of an isosceles triangle is 39 mm. If the length of

the equal sides is 6 times the length of the third side, find the dimensions

of the triangle.

13)

14) A chemist needs 50 milliliters of a 42% solution but has only 32% and

57% solutions available. Find how many milliliters of each that should be

mixed to get the desired solution.

14)

15) Two lines that are not parallel are shown. Suppose that the measure of

angle 1 is (4x + 4y)°, the measure of angle 2 is 6y°, and the measure of

angle 3 is (x + y)°. Find x and y.

15)

16) The manager of a bulk foods establishment sells a trail mix for $6 per

pound and premium cashews for $12 per pound. The manager wishes to

make a 108-pound trail mix-cashew mixture that will sell for $11 per

pound. How many pounds of each should be used?

16)

17) A college student earned $8900 during summer vacation working as a

waiter in a popular restaurant. The student invested part of the money at

9% and the rest at 8%. If the student received a total of $752 in interest at

the end of the year, how much was invested at 9%?

17)

2

18) A retired couple has $180,000 to invest to obtain annual income. They

want some of it invested in safe Certificates of Deposit yielding 6%. The

rest they want to invest in AA bonds yielding 10% per year. How much

should they invest in each to realize exactly $16,400 per year?

18)

19) A certain aircraft can fly 798 miles with the wind in 3 hours and travel

the same distance against the wind in 7 hours. What is the speed of the

wind?

19)

20) Jane and Ed row their boat (at a constant speed) 63 miles downstream

for 7 hours, helped by the current. Rowing at the same rate, the trip back

against the current takes 9 hours. Find the rate of the current.

20)

21) Chris and Hal live 72 miles apart in southeastern Illinois. They decide to

bicycle towards each other and meet somewhere in between. Hal’s rate

of speed is 60% of Chris’s. They start out at the same time and meet 5

hours later. Find Hal’s rate of speed.

21)

22) Richard purchased tickets to an air show for 5 adults and 2 children. The

total cost was $167. The cost of a child’s ticket was $4 less than the cost of

an adult’s ticket. Find the price of an adult’s ticket and a child’s ticket.

22)

23) On a buying trip in Los Angeles, Rhoda ordered 120 pieces of jewelry: a

number of bracelets at $4 each and a number of necklaces at $12 each.

She wrote a check for $720 to pay for the order. How many bracelets and

how many necklaces did she purchase?

23)

24) Natalie rides her bike (at a constant speed) for 3 hours, helped by a wind

of 3 miles per hour. Pedaling at the same rate, the trip back against the

wind takes 9 hours. Find find the total round trip distance she traveled.

24)

25) A barge takes 4 hours to move (at a constant rate) downstream for 32

miles, helped by a current of 2 miles per hour. If the barge’s engines are

set at the same pace, find the time of its return trip against the current.

25)

26) Debbie and Isabelle plan to leave their houses at the same time, roller

blade towards each other, and meet for lunch after 3 hours on the road.

Debbie can maintain a speed of 9.9 miles per hour, which is 90% of

Isabelle’s speed. If they meet exactly as planned, what is the distance

between their houses?

26)

3

27) David needs 7 liters of a 27% solution of sulfuric acid for a research

project in molecular biology. He has two supplies of sulfuric acid

solution: one is an unlimited supply of the 21% solution and the other

an unlimited supply of the 42% solution. How many liters of each

solution should he use?

27)

28) Chloe has 4 liters of a 45% solution of sodium hydroxide in a container.

What is the amount and concentration of sodium hydroxide solution she

must add to this in order to end up with 9 liters of 40% solution?

28)

29) During the 1998-1999 season, the Panthers played 40 games. They lost 14

more games than they won. How many games did they win that season?

29)

30) The perimeter of a rectangle is 58 m. If the width were doubled and the

length were increased by 19 m, the perimeter would be 114 m. What are

the length and width of the rectangle?

30)

31) The perimeter of a triangle is 53 cm. The triangle is isosceles now, but if

its base were lengthened by 7 cm and each leg were shortened by 3 cm, it

would be equilateral. Find the length of the base of the original triangle.

31)

32) The side of an equilateral triangle is 6 inches shorter than the side of a

square. The perimeter of the square is 35 inches more than the perimeter

of the triangle. Find the length of a side of the square.

32)

33) The side of an equilateral triangle is 6 inches shorter than the side of a

square. The perimeter of the square is 48 inches more than the perimeter

of the triangle. Find the length of a side of the triangle.

33)

34) A cashier has 50 $20 and $5 bills in her cash drawer. The value of the

bills is $535. How many $20 bills are there?

34)

35) A flat rectangular piece of aluminum has a perimeter of 70 inches. The

length is 11 inches longer than the width. Find the width.

35)

36) A chemist needs 130 milliliters of a 51% solution but has only 37% and

63% solutions available. Find how many milliliters of each that should be

mixed to get the desired solution.

 

Forum: If Only I Had a System…

Applications of Systems of Linear Equalities

The Problem:

 

When students are surveyed about what makes a good math Forum, at least half of the

responses involve

• “discussing how to work problems”
• “seeing how this math applies to real-life situations”

This Forum on applications of systems of equations addresses both of these concerns.

Unfortunately, the typical postings are far from ideal.

This is an attempt to rectify the situation. Please read this in its entirety before you

post your answer!

Pick-up games in the park vs. the NBA:

Shooting hoops in the park may be lots of fun, but it scarcely qualifies as the precision

play of a well-coached team. On the one hand, you have individuals with different

approaches and different skill levels, “doing their own thing” within the general rules of

the game. On the other hand you have trained individuals, using proven strategies and

basing their moves on fundamentals that have been practiced until they are second

nature.

The purpose of learning algebra is to change a natural, undisciplined approach to

individual problem solving into an organized, well-rehearsed system that will work on

many different problems. Just like early morning practice, this might not always be

pleasant; just like Michael Jordan, if you put in the time learning how to do it correctly,

you will score big-time in the end.

But my brain just doesn’t work that way. . .

Nonsense! This has nothing to do with how your brain works. This is a matter of

learning to read carefully, to extract data from the given situation and to apply a

mathematical system to the data in order to obtain a desired answer. Anyone can learn

to do this. It is just a matter of following the system; much like making cookies is a

matter of following a recipe.

“Pick-up Game” Math

It is appalling how many responses involve plugging in numbers until it works.

• “My birthday is the eleventh, so I always start with 11 and work from there.”
• “The story involved both cats and dogs so I took one of the numbers, divided by

2 and then I experimented.”

• “First I fire up Excel…”
• “I know in real-life that hot dogs cost more than Coke, so I crossed my fingers

and started with $0.50 for the Coke…”

The reason these “problem-solving” boards are moderated is so that these creative

souls don’t get everyone else confused!

NBA Math

In more involved problems, where the answer might come out to be something

irrational, like the square root of three, you are not likely to just randomly guess the

correct answer to plug it in. To find that kind of answer by an iterative process (plugging

and adjusting; plugging and adjusting; …) would take lots of tedious work or a computer.

Algebra gives you a relative painless way of achieving your objective without wearing

your pencil to the nub.

The reason that all of the homework has involved x’s and y’s and two equations, is that

we are going to solve these problems that way. Each of these problems is a story about

two things, so every one of these is going to have an x and a y.

In some problems, it’s helpful to use different letters, to help keep straight what the

variables stand for. For example, let L = the length of the rectangle and W = the width.

The biggest advantage to this method is that when you have found that w = 3 you are

more likely to notice that you still haven’t answered the question, “What is the length of

the rectangle?”

Here are the steps to the solution process:

• Figure out from the story what those two things are.

o one of these will be x

o the other will be y

• The first sentence of your solution will be “Let x = ” (or “Let L = ” )

o Unless it is your express purpose to drive your instructor right over the

edge, make sure that your very first word is “Let”

• The second sentence of your solution will be “Let y = ” (or “Let W = ” )
• Each story gives two different relationships between the two things.

o Use one of those relationships to write your first equation.

o Use the second relationship to write the second equation.

• Now demonstrate how to solve the system of two equations. You will be using

either

o substitution

o or elimination – just like in the homework.

More examples…

For this problem, I’d use substitution to solve the system of equations:

The length of a rectangle blah, blah, blah… Let L = the length of the rectangle

… blah, blah, blah twice the width Let W = the width of the rectangle

The length is 6 inches less than twice the width L = 2W – 6

The perimeter of the rectangle is …

Perimeter is 2L + 2W

The perimeter of the rectangle is 56 2L + 2W = 56

For this one, I’d use elimination to solve the system of equations:

Blah, blah, blah bought 2 cokes…

Let x = the price of a coke

.. blah, blah, blah 4 hot dogs Let y = the price of a hot dog

2 cokes plus 4 hot dogs cost 8.00 2x + 4y = 8.00

3 cokes plus 2 hot dogs cost 8.00 3x + 2y = 8.00

For this one, I’d use substitution to solve the system of equations:

One number is blah, blah, blah… Let x = the first number

…blah, blah, blah triple the second number Let y = the second number

The first number is triple the second x = 3y

The sum of the numbers is 24 x + y = 24

Checking your answers vs. Solving the problem

The problem: Two numbers add to give 4 and subtract to give 2. Find the numbers.

Solving the problem:

Let x = the first number

Let y = the second number

Two numbers add to give 4: x + y = 4

Two numbers subtract to give 2: x – y = 2

Our two equations are: x + y = 4

x – y = 2 Adding the equations we get

2x = 6

x = 3 The first number is 3.

x + y = 4 Substituting that answer into equation 1

3 + y = 4

y = 1 The second number is 1.

Checking the answers:

Two numbers add to give 4: 3 + 1 = 4

The two numbers subtract to give 2: 3 – 1 = 2

Do NOT demonstrate how to check the answers that are provided and call that

demonstrating how to solve the problem!

Formulas vs. Solving equations

Formulas express standard relationships between measurements of things in the real

world and are probably the mathematical tools that are used most frequently in real-life

situations.

Solving equations involves getting an answer to a specific problem, sometimes based

on real-world data, and sometimes not. In the process of solving a problem, you may

need to apply a formula. As a member of modern society, it is assumed that you know

certain common formulas such as the area of a square or the perimeter of a rectangle. If

you are unsure about a formula, just Google it. Chances are excellent it will be in one of

the first few hits.

If you are still baffled:

 

• Check out all the examples worked out in the PowerPoints in the Other

Resources section of the Handy Helpers for Section 4.3.

• Message me if you are still confused.

Tips for studying for this test:

 

• Pretend that the list of questions for this Forum is a pre-test.
• Take the pre-test and grade yourself with the answers provided.
• If you missed any, check to see how your classmate demonstrated the problem.
• If it was not one of the problems demonstrated Message me for help!

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