Variation – Why Does It Matter?
Scarlett and Thomas both have jobs working at different stores at the mall. If they both work for a 5 hour
shift, they each earn $100.
a) Predict how much you think they will each earn if they work a 10 hour shift.
b) Scarlett earns $20 per hour. Complete the following table showing her earnings.
Hours
Worked 0 1 2 3 4 5 6 7 8 9 10
Money Earned
c) Is this an example of direct or partial variation? DIRECT / PARTIAL
d) Thomas earns $40 each shift, plus $12 per hour. Complete the following table showing his earnings.
Hours
Worked 0 1 2 3 4 5 6 7 8 9 10
Money Earned
e) Is this an example of direct or partial variation? DIRECT / PARTIAL
f) Was your prediction correct for both Scarlett and Thomas? How did the type of variation affect your answer?
g) Determine which of the following are direct variation, and which are partial.
i) Kelly earns $50 when she works 5 hours, and $80 when she works 8 hours.
Calculating the Initial Value
1. Nick earns $90 if he works an 8 hour shift. He earns $123.75 if he works an 11 hour shift.
a) Is this direct or partial variation? DIRECT / PARTIAL
b) How much does Nick earn per hour? c) How much is Nick’s initial pay?
d) Write an equation to show how much money Nick earns (E) when he works for n hours.
e) Calculate how much money Nick would earn for a 5 hour shift.
2. Annie earns $95 when she works for 6 hours. She earns $132.50 if she works for 9 hours.
a) Is this direct or partial variation? DIRECT / PARTIAL
b) How much does Annie earn per hour? c) How much is Annie’s initial pay?
d) Write an equation to show how much money Annie earns (E) when she works for n hours.
3. The cost of hiring an electrician is shown on the following graph.
a) Why doesn’t the line extend all the way to the y-axis?
b) Is this an example of direct or partial variation?
c) How much does the electrician charge per hour?
d) How much is the electrician’s initial fee?
e) Write an equation to show the cost (C) of hiring an electrician for n hours.
f) Calculate the cost of hiring an electrician for a job that takes 3 hours.
500 550 600 650 700 750 800 850 900 950 1000
4 6 8 10 12 14
Hours Worked
Cos
t ($
4. A charity rented a banquet hall to hold a masquerade ball as a fundraiser. The profits earned by selling tickets for the masquerade ball are shown in the following table.
a) Is this direct or partial variation?
b) How much did one ticket cost? c) How much did it cost to rent the hall?
d) Write an equation to show the profit (P) earned by the charity, where n is the number of tickets sold.
e) Complete the missing values in the table.
f) How many tickets will they need to sell in order to break even?
Tickets Sold Profits
40 $4500
75 $10625
90 $13250
150
Variation & Initial Value Practice
1. For each of the following, determine the type of variation, rate of change, initial value, and equation.
a) A machine makes bolts at a constant rate. In 3 hours it can make 3741 bolts. In 7 hours, it can make 8729 bolts.
b) A plant grows at a constant rate. After 8 days, the height of the plant is 17.6 cm. After 17 days, the height of the plant is 20.3 cm.
2. Determine the initial value of each of the following and state whether it is an example of direct or partial variation.
a) Hours
Worked
Money
Earned b)
Temperature (oC)
Height (cm)
2 20 3 140
3 30 9 110
4 40 15 80
Variation
Rate of Change Initial Value
Equation
Variation
Rate of Change Initial Value
3. The following graph shows the cost of hiring a mechanic, where C is the cost and n is the number of hours that the mechanic spends working on your car.
a) Is this an example of direct or partial variation? Justify your answer.
b) What is the initial value of the graph? What does it represent?
c) What is the rate of change of the graph? What does it represent?
d) Write the equation of the cost.
e) Predict the cost if it takes 10 hours to repair your car.
f) How long did the repairs take if it cost $365.00 to repair your car?
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
0 2 4
4. Veronica hired a painter and paid $290 for 3 hours of work. Saul hired the same painter and paid
$420 for 5 hours of work.
a) How much does the painter charge per hour?
b) How much is the painter’s initial fee?
c) Determine and equation to calculate the cost (C) of hiring the painter when he works for n hours.
d) Calculate the cost of hiring the painter for 2 hours.
5. Stella sells cupcakes at a bake sale. She charges $1.25 per cupcake. She sold a total of 150 cupcakes and made a profit of $144.75. How much did it cost her to buy the ingredients for the cupcakes?
ANSWERS:
1a] Direct Variation; IV = 0 bolts; ROC = 1247 bolts/hour; B = 0 + 1247h
1b] Partial Variation; IV = 15.2 cm; ROC = 0.3 cm/day; H = 15.2 + 0.3d
2a] IV = $0; Direct Variation
2b] IV = 155; Partial Variation
3a] Partial Variation; It does not start at 0
3b] $75; This is the initial cost of bringing your car in for repairs or maybe the cost of the parts
3c] $40 per hour; This is how much the mechanic charges per hour to work on the car
3d] C = 75 + 40n
3e] $475
3f] 7.25 hours
4a] $65 per hour
4b] $95
4c] C = 95 + 65n
4d] $225
4e] No. If she used the same painter it would only cost $615.
