proportional relationships. You may recall that in proportional relationships, the variables in the problem are related by a constant factor or ration.The focus of this lesson is comparing and contrasting proportional and non-proportional relationships. You may recall that in proportional relationships, the variables in the problem are related by a constant factor or ration.
Read the following situation and answer the questions below.
Jenny is comparing two online music download services. Company A offers music downloads for $0.99 per song. Company B offers music downloads for $0.49 per song after a $20.00 membership fee. To help her determine the better deal, Jenny creates a table for each company to compare costs.
Copy the tables below into your notes, and fill in the missing information based on the information given in the problem.
Company A
| Number of Songs | Process | Cost to Download | Cost per Song |
| 5 | 5(.99) | $4.95 | $0.99 |
| 10 | $9.90 | $0.99 | |
| 15 | |||
| 20 | |||
| 25 | |||
| 30 | |||
| 35 | |||
| 40 | |||
| 45 | |||
| 50 |
Company B
| Number of Songs | Process | Cost to Download | Cost per Song |
| 5 | 20 + 5(.49) | $22.45 | $4.49 |
| 10 | $24.90 | $2.49 | |
| 15 | |||
| 20 | |||
| 25 | |||
| 30 | |||
| 35 | |||
| 40 | |||
| 45 | |||
| 50 |
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Number of Songs |
Process | Cost to Download | Cost per Song |
| 5 | 5(.99) | $4.95 | $0.99 |
| 10 | 10(.99) | $9.90 | $0.99 |
| 15 | 15(.99) | $14.85 | $0.99 |
| 20 | 20(.99) | $19.80 | $0.99 |
| 25 | 25(.99) | $24.75 | $0.99 |
| 30 | 30(.99) | $29.70 | $0.99 |
| 35 | 35(.99) | $34.65 | $0.99 |
| 40 | 40(.99) | $39.60 | $0.99 |
| 45 | 45(.99) | $44.55 | $0.99 |
| 50 | 50(.99) | $49.50 | $0.99 |
Number of Songs |
Process | Cost to Download | Cost per Song |
| 5 | 20 + 5(.49) | $22.45 | $4.49 |
| 10 | 20 + 10(.49) | $24.90 | $2.49 |
| 15 | 20 + 15(.49) | $27.35 | $1.82 |
| 20 | 20 + 20(.49) | $29.80 | $1.49 |
| 25 | 20 + 25(.49) | $32.25 | $1.29 |
| 30 | 20 + 30(.49) | $34.70 | $1.16 |
| 35 | 20 + 35(.49) | $37.15 | $1.06 |
| 40 | 20 + 40(.49) | $39.60 | $0.99 |
| 45 | 20 + 45(.49) | $42.05 | $0.94 |
| 50 | 20 + 50(.49) | $44.50 | $0.89 |
What do you notice about the cost per song for company A?
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The cost per song remains constant.
For company A, how can you find the cost of downloading any number of songs?
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Multiply the number of songs by 0.99.
What is the equation that relates y, the cost to download, to x, the number of songs, for company A?
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y = 0.99x
Keeping in mind that proportional relationships are in the form y = kx, is this a proportional relationship or a non-proportional relationship?
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This is a proportional relationship.
What do you notice about the cost per song for company B?
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The cost per song decreases as you download more songs.
What is the equation that relates to y, the cost to download, to x, the number of songs, for company B?
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y = 20 + 0.49x
Keeping in mind that proportional relationships are in the form y = kx, is this a proportional relationship or a non-proportional relationship?
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This is a non-proportional relationship.
Read the following problem, and fill in the missing values in the table.
Your parents are trying to get your little sister to help with the yard work. They offer her a quarter for every 10 weeds she pulls and puts in a bucket.
Weeds |
10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
| Pay |
What is the rate of pay?
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The rate of pay is $0.25 for every 10 weeds.
What is the unit rate for the following equation?
0.25 ÷ 10 = 0.025
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The unit rate is 2.5 cents per weed.
Does the ratio of
pay
number of weeds
change or stay the same in the problem?
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The ratio stays the same.
What equation could you write to calculate your little sister’s pay?
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y = 0.025x
Keeping in mind that proportional relationships are in the form y = kx, is this a proportional relationship or a non-proportional relationship?
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This is a proportional relationship.
How do you know?
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The equation is y = k • x.
The rate of pay stays constant.
What difference do you notice between the equations that represent proportional relationships and the equations that represent non-proportional relationships?
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The equations for proportional relationships had multiplication only; the equation for the non-proportional relationship that we saw had both multiplication and addition.