L24 Problem Solving with Equations.notebook

L24 Problem Solving with Equations.notebook
March 12, 2015
Review: Solving Equations
Substitute, then solve
3x + 2y = 12
a) 456 = 3.14x2
b) 90 = 2x + 2(20)
find x when y = 1.5 m
Substitute, then solve
3x + 2y = 12
find x when y = (x ­ 10)
P = 2W + 2L
A = r2
Mar 31­10:04 AM
Mar 28­10:01 AM
Review: Creating Expressions
A part time job pays $13.50 per hour. Create an expression to represent earnings.
What's the difference between an expression and an equation?
A car salesman earns $300 per week plus 5% commission on what she sells. Create an expression to represent her weekly earnings
Oct 20­8:38 AM
Oct 20­8:42 AM
Problem Solving with Equations
Creating Equations
The goal? A part time job pays $13.50 per hour. Create an equation to represent earnings.
To give yourself an equation to solve with one unknown.
GIVEN the Equation
A car salesman earns $300 per week plus 5% commission on what she sells. Create an equation to represent her weekly earnings
Work on individual "pieces" to create match.
"Sub & Solve"
Oct 20­8:41 AM
CREATE the Equation
Start with word equation
"Sub & Solve"
Oct 3­3:26 PM
1
L24 Problem Solving with Equations.notebook
Example #1: Using a Given or Known Equation (Shapes)
The perimeter of the triangle is 35 m. Find the length of each side
A circle has an area of 35 m2
What is the radius & diameter of the circle?
March 12, 2015
A = r2
d = 2r
3x ­ 2
2x + 3
P = a + b + c
3x + 4
Practice: Page 298 #20
Mar 28­10:28 AM
P = a + b + c + d
Apr 1­10:08 AM
Example #2: Creating Equation (Car Values)
A company spent $308 700 on company cars in 2010. Each year, the total value of the cars depreciates by $38 100.
a) Create an equation that represents the Total Value of the cars.
b) If the company wants to ensure they keep $100 000 worth of value from the cars when they sell them all, how long should they keep the vehicles?
Oct 17­8:28 AM
Example #3: Working with Percentages (Commission)
Oct 4­7:33 AM
Jamie has two jobs in sales. She earns a commission of 2% of sales at a electronics store. Her commission at a clothing store is 4%. .
What is commission?
Commission is earned in jobs where your pay is based on a percentage of what you sell.
a) Write an equation that models this situation (i.e. her total earnings).
Example: A real estate agent earns 5% commission on the sale of a $250 000 house. How much did the agent earn?
Earnings = 5% of $250 000
Earnings = 0.05(250 000)
Earnings = $12 500. Apr 2­9:09 AM
Apr 2­9:09 AM
2
L24 Problem Solving with Equations.notebook
b) Last week she earned a total of $380. Her sales at the electronics store totaled $9500. What were her total sales at the clothing store?
March 12, 2015
#7: A candy store is making up a mixture of chocolate­coated almonds and chocolate­coated raisins. The almonds cost $30 / kg and the raisins cost $10 / kg. The final value of the mixture is to be $150. The mix contains 7.5 kg of raisins.
a) Write an equation that describes this situation
"almonds cost
$30 / kg"
"raisins cost $10 / kg"
C = Total cost of mix (in $)
a = kg of almonds
r = kg of raisins
Total Cost = Cost of almonds + Cost of raisins
C = 30a + 10r
Practice: Page 295 #7­12
Apr 2­9:09 AM
b) Solve the equation for the required mass of almonds.
"The mix contains
7.5 kg of raisins"
"The final value of the mixture is to be $150"
C = 30a + 10r
150 = 30a + 10(7.5)
150 = 30a + 75
­75
­75
75 = 30a
Oct 4­7:16 AM
#8: Jamie builds chairs and tables in her shop. It takes her 3 h to make a chair and 7 h to make a table. Jamie works 60 h in a week. She made three tables last week.
a) Write an equation that describes this situation
"takes her 3 h
to make a chair"
"7 h to make a table"
Hours Worked = Time for chairs + Time for tables
H = 3c + 7t
H = Time worked (in hours)
c = # of chairs
t = # of tables
2.5 = a
30
30
Therefore, there should be 2.5 kg of almonds to complete the mixture.
Oct 7­7:20 AM
b) Solve the equation to find the number of chairs she made last week. "She made three tables last week"
"Jamie works
60 h in a week"
#9: Christine has two jobs in sales. She earns a commission of 3% of sales at a clothing store. Her commission at the hardware store is 5%. Last week she earned $800. Her sales at the clothing store totalled $2500
a) Write an equation that describes this situation
"earns a commission
3% of sales"
H = 3c + 7t
60 = 3c + 7(3)
­21
­21
Oct 7­7:28 AM
60 = 3c + 21
"commission at the hardware store is 5%"
Total Earnings = Clothing store + Hardware store
E = 0.03c + 0.05h
E = total earnings ($)
c = clothing store
h = hardware store
39 = 3c
3
3
Therefore, Jamie made 13 chairs last week.
13 = c
Oct 7­7:30 AM
Apr 1­10:11 AM
3
L24 Problem Solving with Equations.notebook
b) Find her total sales at the hardware store by solving your equation
"Her sales at the hardware store
"Last week she earned $800"
totalled $2500"
E = 0.03c + 0.05h
800 = 0.03c + 0.05(2500)
­125
800 = 0.03c + 125
­125
675 = 0.03c
March 12, 2015
#10: Nathan's boss sent him to the bank to get quarters and nickels. He has $250 to spend on the coins. Quarters are sold in rolls of 40. Nickels are sold in rolls of 40. Nathan bought 17 rolls of quarters.
a) Write an equation that describes this situation
"quarters are sold in rolls of 40"
"nickels are sold in rolls of 40"
Total Value = Value of quarters + Value of nickels
V = Value of coins ($)
q = # of rolls of quarters
n = # of rolls of nickels
V = 0.25(40)q + 0.05(40)n V = 10q + 2n
0.03
0.03
22 500 = c
Therefore, Christine sold $22 500 at the hardware store last week.
Apr 1­10:09 AM
b) Find the number of rolls of nickels Nathan bought by solving your equation
"Nathan bought 17
rolls of quarters"
"He has $250 to
spend on the coins"
V = 10q + 2n
250 = 10(17) + 2n
­170
­170
250 = 170 + 2n
80 = 2n
2
Apr 1­10:12 AM
Example #4: Given Limited Information (Investments )
How do Investments work?
Investments will “return” or earn you a percentage of a larger amount you invest.
Example: You invest $5 500 at a 3.5% return. How much did you earn on this investment?
Earnings = 3.5% of $5 500
Earnings = 0.035(5 500)
Earnings = $192.50
2
40 = n
Therefore, Nathan bought 20 rolls of nickels.
Apr 1­10:12 AM
A broker invested $9 000. Part of the money was invested at 9% per annum. The rest was invested at 7% per annum.
a) Create an equation that models this situation Apr 1­9:46 AM
b) If the investments earned $650 in one year, how much was invested at each rate?
Earnings = Amount @9% + Amount @ 7%
E = 0.09x + 0.07y
Earnings = Amount @ 9% + Amount @ 7%
(0.09x) (0.07y)
E = 0.09x + 0.07y
650 = 0.09x + 0.07y
but we don't want 2 variables, so how else could we think of y?
the amount invested @ 7% is whatever isn't invested @ 9% x + y = $9000
y = 9 000 ­ x Mar 28­9:58 AM
Apr 1­9:04 AM
4
L24 Problem Solving with Equations.notebook
March 12, 2015
Earnings = Amount @ 9% + Amount @ 7%
E = 0.09x + 0.07y
650 = 0.09x + 0.07 (9 000 ­ x)
650 = 0.09x + 630 ­ 0.07x
650 = 0.02x + 630
y = 9 000 ­ x
= 9 000 ­ 1 000
20 = 0.02x
= 8 000
1000 = x
Therefore, $1 000 is invested @ 9%, and $8 000 is invested @ 7%
Practice: Page 315 #17,18
Mar 28­10:16 AM
5