Chapter 4 Section 1 Solving a Simple Equation pptx

Chapter 4

The Problem Solving Process

■

Ryan’s Roofing charges $50 plus $30 per hour

for emergency roof repair. A homeowner’s bill

was $860 after the last storm. How long did

Ryan’s Roofing spend working on the job?

Strategy

Fixed Charge + Hourly Charge = Bill

Guess:

20 hours

_$50

Let x = # of hours worked

50 + 30(

x

) = 860

x

=

27 hours

Example

■

Evelyn paid $89.25, including 5% tax, for her

Mechanics and Materials

textbook. How

much did the book cost (before taxes)?

Selling Price + Sales Tax = Total Price

Guess: $80

$80 + .05($80) = __________

$84

x + .05x = 89.25

Example

■

Jesse’s test scores in DC and AC Fundamentals

so far are; 73, 75, 89,and 91.

■

What is the lowest score he can get on the next

Example

■

The second angle of an architect’s triangle is

three times as large as the first. The third

angle is 30° more than the first.

❑

Find the measure of each angle.

Angle 1 + Angle 2 + Angle 3 = 180°

A1 A2

A3

Since the second and third angles are

described in terms of first angle, let’s use the variable a to represent the first angle…

a

₊

3

a

₊

a

_{+ 30 = 180}

5

a +

30 = 180

a = 30°

Angle 1 = _____

Angle 2 = _____

Angle 3 = _____

Practice

■

Complete worksheet:

■

#5 – 11

■

Write a “hybrid” equation first.

Diagram if applicable.

■

Make a guess to help write the final

Notes/Worksheet

■ 5.) A computer repair and consulting business charges in-home

visits based on the following rate schedule: $35 plus $28.50 per hour. If a customer has several issues/problems to be solved and they only have $200 to spend, what is the most amount of time (in hours) the computer technician can stay at the customer’s house?

Fixed Amount + Hourly Amount = Customer’s Bill

$35 + Hourly Amount = $200

Notes/Worksheet

■ 6.) The world’s oldest groom was 19 years older than his bride.

Together their ages totaled 187 years. How old were the bride and groom? (Reference Guinness World Records, Millennium Edition)

Groom + Bride = 187

b

Observation: The groom is described in terms of the

bride.

So let the variable be “b” for bride.

+

Notes/Worksheet

■ 7.) Jake paid $63.75 for multi-meter during a 15% off sale. What

was the regular price of the multi-meter?

Meter – Discount = 63.75

$70 originally.

Replace 70 with a variable.

70

–

70(.15)

₌

_59.50

Notes/Worksheet

■ 8.) David did a bike ride around the outer edge of the state of

Wyoming (which is shaped like a rectangle) and accumulated a

distance of 1280 miles. The width of Wyoming is 90 miles less than the length. Determine the width and length of Wyoming.

2(Length) + 2(Width) = 1280

2L

+

2(L – 90)

= 1280

2L + 2L – 180 = 1280

L

Notes/Worksheet

■

9.) Nate’s investment in K-mart stock fell 38% to

$25,560. How much money did Nate lose?

Investment – Loss = 25,560

was $30,000

Replace $30,000 with a variable (

x

30,000

–

30,000(.38)

₌

_18,600

Notes/Worksheet

■

10.) The Iditarod sled-dog race extends for 1049 miles

from Anchorage to Nome. If John is twice as far from

Anchorage as from Nome, how far does John have left to

go?

Dist. to Anch. + Dist. to Nome = 1049

2

n

+

n

_{= 1049}

Nome ∙

Anchorage ∙ Position ∙

n

Notes/Worksheet

■

11.) Ray has an old tractor that he will put in a

neighbor’s upcoming auction. If he wants to pocket

$3200 minimum for the tractor, how much will it have to

sell for at auction so that he gets his price? The

auctioneer receives a 7% commission on all items sold.

Selling Price – Commission = 3200

Guess: Selling price of $3,500.

Replace $3,500 with a variable (x).

3500

–

3500(.07)

=

3255

HOMEWORK

Homework Problems

■

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

1.) Homework

■ When three resistors are connected in series, their resistances are

added to produce a total resistance of 970 Ω. One of them has a resistance of 530 Ω, and the others have resistance levels equal to each other. Find the resistance levels of the other two resistors.

Resistor 1 + Resistor 2 + Resistor 3 = 970

530

Menu

R

2.) Homework

■ A power supply has 2 printed circuit boards that contain a combined

total of 222 components. One board has 6 more than twice the

number of components on the other board. How many components are in each board?

Board 1 + Board 2 = 222

B

Since the board with more

components is described based upon the board with fewer

components, let B = board with the fewer amt. of components.

Menu

2 + 6

3.) Homework

■ An architect determines that if she reduces the dimensions of a

square room by 2 ft. on each side, the perimeter will be 56 ft. What is the length of the original room, before the reduction?

4(length of side) = 56

4(s – 2) = 56

Reduced Floor Plan

s – 2

4.) Homework

■ The sum of two currents is 200 mA, and the larger current is 30 mA

more than the smaller current. Determine the value of the smaller current.

Current 1 + Current 2 = 200

(C + 30) + C = 200

Since the larger current is

described based upon the smaller current, let c = smaller current.

5.) Homework

■ Some resistors cost $0.12 each, while others cost $1.08 each.

Sixty-five resistors cost a total of $22.20. How many of each resistor was purchased?

x + y =

65

x =

65 –

y

.12(65 –

y

) + 1.08

y

= 22.20

Menu

Number of Cheaper Resistors = x

Number of More Expensive Resistors = y

Quantity: # of Cheap + # of Expensive = 65 Quantity: x + y = 65

5.) Homework

■ Some resistors cost $0.12 each, while others cost $1.08 each.

Sixty-five resistors cost a total of $22.20. How many of each resistor was purchased?

Quantity Cost Total Cost

Cheaper Resistor More Expensive Resistor Total

x

y

x + y =

65

$0.12

$1.08

-.12

x

1.08

y

$22.20

x =

65 –

y

.12(65 –

y

)

.12(65 –

.12

x

+ 1.08

y

) + 1.08

y

= 22.20

y

= 22.20

Two variables…cannot solve.

6.) Homework

■ A company plans to issue 24,500 shares of two different kinds of

stocks, which will have a combined value of $800,000. One of the stocks is worth $100 per share and the other stock is worth $25 per share. How many of each stock will be issued?

x + y =

24,500

x =

24,500 –

y

25(24,500 –

y

) + 100

y

= 800,000

Menu

Amount of $25 stocks issued = x

Amount of $100 stocks issued = y

Quantity: Amt. $25 stock + Amt. of $100 stock = 24,500 Quantity: x + y = 24,500

6.) Homework

■ A company plans to issue 24,500 shares of two different kinds of

stocks, which will have a combined value of $800,000. One of the stocks is worth $100 per share and the other stock is worth $25 per share. How many of each stock will be issued?

Quantity Cost Total Cost

Cheaper Stock More Expensive Stock Total

x

y

x + y =

24500

$25

$100

-25

x

100

y

$800,000

x =

24500 –

y

25(24500 –

y

)

25(24500 –

y

) + 100

y

= 800,000

7.) Homework

■ Three different oil storage tanks have a combined capacity of 4,400

gallons. The largest tank holds three times as much as the smallest tank, and twice as much as the other tank. What is the capacity of each tank?

Tank 1+ Tank 2 + Tank 3 = 4,400

Since the smaller tanks are described with reference to the largest tank, let t = size of largest tank.

8.) Homework

■ According to Kirchhoff’s current law, the sum of the currents into a

node equals the current out of the node. The current out of a node is 650 mA and three currents go into it. The largest current is twice the smallest and 100 mA more than the other current. Determine all three currents.

Current 1 + Current 2 + Current 3 = 650

C

Since the two of the currents are described in reference to the largest current, let c = largest current.

C

C - 100

+

+

= 650

9.) Homework

■ A person pays $4800 in state and federal income taxes in a year.

The federal income tax is five times as great as the state income tax. How much does the person pay on each of these income taxes?

State Tax + Federal Tax = 4800

S

Since the federal tax is described in reference to the state tax, let s = state tax paid.

S

5

+

= 4400

10.) Homework

■ A walkway 3 meters wide is constructed along the outside of a

square courtyard. If the perimeter of the courtyard is 320 meters, what is the perimeter of the square formed by the outer edge of the walkway?

4(length of side) = 320

4s = 320

Courtyard

s

s = 80 m

80

+ 3 + 3

80

86

4(86) = Perimeter of outer edge

344 m = Perimeter of outer edge

11.) Homework

■ Two stock investments totaled $15,000. One stock led to a 40%

gain, but the other stock resulted in a 10% loss. If the net result is a profit of $2,000, how much was invested in each stock.

x + y =

15,000

x =

15,000 –

y

.40(15,000 –

y

) – .10

y

= 2,000

Menu

Amount invested in gaining stock = x

Amount invested in losing stock = y

Investment: x + y = $15,000

11.) Homework

■ Two stock investments totaled $15,000. One stock led to a 40%

gain, but the other stock resulted in a 10% loss. If the net result is a profit of $2000, how much was invested in each stock.

Amt. Invested Gain/Loss % Gain/Loss $

Stock that Gained Stock that Lost Total

x

y

x + y =

15000

40%

-10%

-.40

x

–.10

y

$2,200

x =

15000 –

y

.40(15000 –

y

)

.40(15000 –

y

) – .10

y

= 2,200

Example 1

■

Several 6-volt and 12-volt batteries are arranged so that

their individual voltages combine to provide a power

supply of 84-volts.

■

How many of each type are present if the total number of

batteries is 10?

Qty_6v x 6-volts + Qty_12v x 12-volts = 84-volts

Total Voltage from the 6v batteries + Total Voltage from the 12v batteries = 84 volts

Example 2

■

A machinist made 132 items, some of which were

hubs with the rest being threaded rods.

❑

He made 12 more hubs than threaded rods.

How many of each kind of item did he make?

Distance, Rate, Time

■

Formula relating these components:

❑

Distance = Rate x Time

Example 3

■

A car travels at 40 mi/h for 2 hours along a road.

■

A second car starts on the same route 2 hours later,

traveling at 60 mi/h.

■

How many hours will it take for the faster car to

overtake the slower one?

Example 3

Distance

= Distance

D = RT

Example 4

■

A space shuttle is sent to “capture” an

orbiting satellite 6,000 km ahead of it’s

current position.

■

The satellite travels at 27,000 km/h and the

shuttle travels at 29,500 km/h.

Example 4

27,000 km/h 29,500 km/h

6,000 km

X

D = RT

Mixture Problems

Cement: 20% sand

Cement: 30% sand

Cement: 27% sand

+

=

200 lbs of cement _{467 lbs of cement}

667 lbs of cement

Mixture Problem

■

On hand is 100 g of solder that is 50% tin.

■

How many grams of 10% tin solder must be mixed to

end up with solder that is 25% tin?

Solder: 50% tin

Solder: 10% tin

Solder: 25% tin

+

=

Practice

■

Worksheet: Mixture Problems

Example 2

■

The houses on the south side of Elm Street

are consecutive even numbers. Wanda and

Larry are next door neighbors and the sum of

their house numbers is 794. Determine their

house numbers.

Wanda’s House # + Larry’s House # = 794

Example 3

■

The top of the John Hancock Building is a rectangle

whose length is 60 ft more than the width. The

perimeter is 520 ft.

❑

Find the width and length.

❑

Find the area of the rectangle.

width + length + width + length = perimeter

Bird’s eye view from above

2(width) + 2(length) = perimeter

width, let the width be the variable (unknown).

Example 5

■

A taxi ride costs $1.90 plus $1.60 for each

mile traveled. If $18 is budgeted for a taxi

ride, how far can you travel?

Fixed Amount + Amt. Based on Miles Driven = Total Charge

Guess: 10 miles

Example 6

■

Sarah’s investment in Jet Blue stock grew

28% to $448. How much did she invest?

Original Investment + Gain in Value = Current Price

Guess: $400

Example 7

■

Lincoln’s 1863 Gettysburg Address refers to

the year 1776 as “four score and seven years

ago”. Determine what length of time is a

“score”.

Example 8

■

Pam scored 78 on a test that had 4 “fill-ins”

worth 7 points each and 24 multiple-choice

questions worth 3 points each. She had one

fill-in wrong. How many multiple-choice

questions did she get correct?