Direct and Partial
Variation
When you get a part­time job,you may be paid by the hour.
Suppose you start working 8 h per week.
What happens to your pay cheque if you increase your time to 16 h per week?
The amount you are paid varies directly with the number of hours you work.
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Example
Nicole works part­time at a bookstore. Her earnings for the past three weeks are shown.
Hours Worked (h) Pay ($) Hourly Rate ($/h)
6 45 12 90
8 60
a) Plot the points and create a graph to model this relationship.
Pay ($)
b) Describe the relationship between pay and hours worked.
120
100
80
60
40
20
0
2 4 6 8 10 12 14
Hours Worked
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c) Interpolate to find Nicole's earnings for 9 h of work.
d) Extrapolate to find how many hours Nicole must work to earn $120.
e) What is the slope of the line? Explain
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We can describe the relation using an
algebraic model.
Nicole's pay, P, varies directly with the number of hours, h, worked.
The constant of variation, k, is the rate at which the two variables change.
Nicole's rate of pay is $7.50/h, so k = 7.50.
P = kh
P = 7.50h
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Example
The Alvari family travels 225 km to their cottage.
Jamie records their progress.
Time, t, (h) Distance, d, (km)
0.5 43
1.5 129
2.0 172
a) Determine the constant of variation. Explain what it means.
b) Write an equation that models this direct variation.
c) Use the equation to determine how far the family has travelled after 2.5 h.
d) How long will the trip take?
e) What assumptions must you make to answer parts
c) and d)?
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Some employees are paid in two ways: a fixed amount plus an amount that varies depending on performance. Here are some examples:
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•
•
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car salesperson
waiter/waitress
professional athlete
computer salesperson
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Example
Eva is a computer salesperson. Her earnings depend on sales performance: the more she sells, the more money she earns. At the end of each month, Eva is paid $2000 plus 10% commission on sales for the month.
Complete the chart.
Sales, S ($) Commission ($) Earnings, E, ($) (S,E)
0
5 000
10 000
15 000
20 000
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a) Plot the ordered pairs (S,E). Draw a line through the points. Does the line pass through the origin? Explain.
E
Earnings ($)
5000
4000
3000
2000
1000
0
5
10
15
20
25 S
Sales ($ 1000s)
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b) Write an equation to model the relationship between total earnings and sales.
c) Use the equation to determine Eva's earnings if she has monthly sales of $17 500.
d) Repeat c) using the graph. Do the results match?
e) Describe how the graph would change if Eva were paid: $3 000 per month plus 10% of sales.
$2 000 per month plus 15% of sales.
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Eva's earnings vary in part with the amount of sales. This is an example of a partial variation relation.
E = 0.1S + 2000
Variable Part
Fixed Part
The fixed part does not change.
The variable part changes depending on how much she sells.
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Example
A taxicab driver charges $2.50 plus $0.50/km for each trip. a) Set up a table of values and graph the relation, using C to represent total cost in dollars, and d to represent distance in kilometres.
b) Use the graph to predict the cost of a 26­km trip.
c) Use the graph to predict how far you could travel for $8.00.
d) Write the equation that models this partial variation relation.
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Key Concepts
Direct Variation
Relation
Partial Variation
Relation
One variable is a constant
multiple of the other.
One variable is a constant
multiple of the other, plus
a constant value.
The fixed multiple is called
the constant of variation, k.
There are two parts:
one is fixed and one is variable.
The graph is a straight line passing through the origin.
The graph is a straight line
that does not pass through
the origin.
The general equation is
y = kx.
The general equation is
y = kx + c.
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From our examples:
Direct Variation
We can describe the relation using an
algebraic model.
Nicole's pay, P, varies directly with the number of hours, h, worked.
The constant of variation, k, is the rate at which the two variables change.
Nicole's rate of pay is $7.50/h, so k = 7.50.
Partial Variation
Eva's earnings vary in part with the amount of sales. This is an example of a partial variation relation.
E = 0.1S + 2000
Variable Part
Fixed Part
The fixed part does not change.
The variable part changes depending on how much she sells.
P = kh
P = 7.50h
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