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Test for proportional relationships

Learning goals

  • Tests tables for a constant ratio.
  • Tests graphs for a line through the origin.
  • Tests equations for the form y = kx.
  • Explains why a relationship is or is not proportional.

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Test for proportional relationships 1
Which equation represents a proportional relationship? a) \(y=5x\) b) \(y=x+5\) c) \(y=5\)

Hints

- Recall the standard equation form for a proportional relationship. - Look for an equation where the output is a constant multiple of the input. - Check whether any added constant remains when the input is \(0\).

Solution

1. A proportional relationship can be written in the form \(y=kx\). 2. Choice a), \(y=5x\), has exactly that form. 3. Choices b) and c) do not have the form \(y=kx\).

Answer

a) \(y=5x\)
Test for proportional relationships 2
The graph shows four plotted pairs in a relationship. Is the relationship proportional? Explain using the graph.
Figure for problem 551931

Hints

- Look at the overall pattern made by the plotted points. - Check whether the origin is one of the plotted pairs. - Recall the graph shape associated with a proportional relationship.

Solution

1. The plotted pairs are \((0,0)\), \((1,3)\), \((2,6)\), and \((3,9)\). 2. The points line up on a straight line that passes through the origin. 3. Therefore, the relationship is proportional.

Answer

Yes. The plotted points lie on one straight line through the origin, so the relationship is proportional.
Test for proportional relationships 3
The table shows two pairs. <table><tr><td>\(x\)</td><td>\(2\)</td><td>\(4\)</td></tr><tr><td>\(y\)</td><td>\(3\)</td><td>\(6\)</td></tr></table> Does the table represent a proportional relationship? Give one numerical check that supports your answer.

Hints

- A proportional table must use the same value of \(\frac{y}{x}\) for every pair. - Compare how the second pair scales from the first pair. - One consistent multiplicative scale factor should work for both coordinates.

Solution

1. When \(x\) doubles from \(2\) to \(4\), \(y\) also doubles from \(3\) to \(6\). Equivalently, \(\frac{3}{2}=\frac{6}{4}=1.5\). 2. Therefore, the two-pair table represents a proportional relationship with constant of proportionality \(1.5\).

Answer

Yes. For example, \(\frac{3}{2}=\frac{6}{4}=1.5\), so both pairs use the same constant of proportionality.
Test for proportional relationships 4
Does the graphed line show a proportional relationship? Explain using where the line crosses the vertical axis.
Figure for problem 559482

Hints

- Find the point on the line with an \(x\)-coordinate of zero. - Which point must every proportional graph pass through? - Compare that point with what the line shows.

Solution

1. The line crosses the vertical axis at \((0,2)\), not at the origin. 2. A proportional graph must pass through \((0,0)\), so this relationship is not proportional.

Answer

No. The line crosses the vertical axis at \((0,2)\) instead of \((0,0)\).
Test for proportional relationships 5
Is the graphed relationship proportional? Explain using the graph's shape and where it crosses the axes.
Figure for problem 559483

Hints

- Passing through the origin is one feature to check. - Does the graph rise at the same rate everywhere? - Recall the full shape of a proportional graph.

Solution

1. The graph includes the origin, which is necessary for proportionality. 2. The graph is curved rather than a straight line. 3. Since a proportional relationship has a straight-line graph through the origin, this relationship is not proportional.

Answer

No. The curve passes through \((0,0)\), but it is not a straight line, so the relationship is not proportional.
Test for proportional relationships 6
A farm stand sells apples in different bags: - A \(3\,\text{lb}\) bag costs \(\$4.20\). - A \(6\,\text{lb}\) bag costs \(\$7.80\). - A \(10\,\text{lb}\) bag costs \(\$12.00\). Use calculations to determine whether the relationship weight \(\to\) price is proportional. Justify your answer.

Hints

- What must be true about the price per pound in a proportional relationship? - Find the price for exactly \(1\,\text{lb}\) for each bag. - Compare the unit prices.

Solution

1. Find the price per pound for each bag: \(4.20\div 3=\$1.40\) per pound, \(7.80\div 6=\$1.30\) per pound, and \(12.00\div 10=\$1.20\) per pound. 2. Since the unit prices are not equal, the relationship is not proportional.

Answer

The relationship is not proportional because the unit prices are different: \(\$1.40\), \(\$1.30\), and \(\$1.20\) per pound.
Test for proportional relationships 7
A car-sharing service charges a fixed \(\$5.00\) booking fee plus \(\$0.50\) for each mile driven. a) Find the total cost for a \(10\)-mile trip and for a \(20\)-mile trip. b) Is the relationship between miles driven and total cost proportional? Justify your answer using your results from part a).

Hints

- In a proportional relationship, what happens to the output when the input is doubled? - Calculate both total costs before comparing them. - Separate the fixed part of the cost from the part that changes with distance.

Solution

1. For \(10\) miles, the cost is \(5.00+10\cdot 0.50=\$10.00\). 2. For \(20\) miles, the cost is \(5.00+20\cdot 0.50=\$15.00\). 3. If the relationship were proportional, doubling the miles would double the cost. Doubling \(\$10.00\) gives \(\$20.00\), not \(\$15.00\). 4. Therefore, the relationship is not proportional.

Answer

a) The costs are \(\$10.00\) for \(10\) miles and \(\$15.00\) for \(20\) miles. b) The relationship is not proportional because doubling the distance does not double the total cost.
Test for proportional relationships 8
Determine whether the table represents a proportional relationship by calculating \(\frac{y}{x}\) for every pair. If it is proportional, find the value of \(y\) when \(x=20\). <table> <tr><td>\(x\)</td><td>\(4\)</td><td>\(7\)</td><td>\(12\)</td><td>\(15\)</td></tr> <tr><td>\(y\)</td><td>\(14\)</td><td>\(24.5\)</td><td>\(42\)</td><td>\(52.5\)</td></tr> </table>

Hints

- What must be true about \(\frac{y}{x}\) in a proportional relationship? - Calculate the ratio for every column. - Apply the constant ratio to the new input.

Solution

1. Calculate the ratios: \(14\div 4=3.5\), \(24.5\div 7=3.5\), \(42\div 12=3.5\), and \(52.5\div 15=3.5\). 2. Since all ratios are equal, the relationship is proportional with constant of proportionality \(k=3.5\). 3. For \(x=20\), \(y=3.5\cdot 20=70\).

Answer

The relationship is proportional because \(\frac{y}{x}=3.5\) for every pair. When \(x=20\), \(y=70\).
Test for proportional relationships 9
Two tables are shown. Decide which table represents a proportional relationship and which does not. Justify your decision by comparing \(\frac{y}{x}\) values. Table A: <table> <tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td></tr> <tr><td>\(y\)</td><td>\(1.5\)</td><td>\(3\)</td><td>\(4.5\)</td><td>\(6\)</td></tr> </table> Table B: <table> <tr><td>\(x\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td></tr> <tr><td>\(y\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td><td>\(5\)</td></tr> </table>

Hints

- Calculate the ratio of \(y\) to \(x\) for each pair in both tables. - What condition defines a proportional relationship? - Is checking only the first pair enough?

Solution

1. For Table A, the ratios are \(1.5\div 1=1.5\), \(3\div 2=1.5\), \(4.5\div 3=1.5\), and \(6\div 4=1.5\). Table A is proportional. 2. For Table B, \(2\div 1=2\), while \(3\div 2=1.5\). Since the ratios are not equal, Table B is not proportional.

Answer

Table A is proportional because every ratio \(\frac{y}{x}\) equals \(1.5\). Table B is not proportional because its ratios are not constant.
Test for proportional relationships 10
Analyze the complete pairs in the table. Are they consistent with a proportional relationship? If so, state the constant of proportionality \(k\), where \(y=kx\). Then assume the missing entries are intended to continue that proportional relationship and complete them. <table> <tr><td>\(x\)</td><td>\(2.5\)</td><td>\(4\)</td><td>\(6\)</td><td>\(\square\)</td><td>\(15\)</td></tr> <tr><td>\(y\)</td><td>\(6.25\)</td><td>\(10\)</td><td>\(\square\)</td><td>\(30\)</td><td>\(\square\)</td></tr> </table>

Hints

- Compare the complete input-output pairs to see whether the same multiplicative relationship holds. - Distinguish what the shown data establish from the assumption about the missing entries. - Use the established relationship to check each proposed completion.

Solution

1. The known complete-pair ratios are \(6.25\div2.5=2.5\) and \(10\div4=2.5\), so those pairs are consistent with \(k=2.5\). 2. Under the stated assumption that the missing entries continue that proportional relationship, use \(y=2.5x\). 3. For \(x=6\), \(y=2.5\cdot6=15\). 4. For \(y=30\), \(x=30\div2.5=12\). 5. For \(x=15\), \(y=2.5\cdot15=37.5\).

Answer

The complete pairs are consistent with a proportional relationship with \(k=2.5\). Under the stated continuation assumption, \(y=2.5x\), and the missing values are \(15\), \(12\), and \(37.5\), in table order.
Test for proportional relationships 11
A square has side length \(a\). Consider these two relationships: Relationship 1: side length \(a\) \(\rightarrow\) perimeter \(P\) Relationship 2: side length \(a\) \(\rightarrow\) area \(A\) Use a doubling test to explain why Relationship 1 is proportional but Relationship 2 is not. Use \(a=3\,\text{cm}\) and \(a=6\,\text{cm}\) in your explanation.

Hints

- Recall the formulas for the perimeter and area of a square. - In a proportional relationship, doubling the input doubles the output. - Compare each value at \(a=3\) with the corresponding value at \(a=6\).

Solution

1. When \(a=3\,\text{cm}\), the perimeter is \(P=4\cdot3=12\,\text{cm}\). When the side length doubles to \(6\,\text{cm}\), the perimeter is \(P=4\cdot6=24\,\text{cm}\). The perimeter also doubles, so the relationship is proportional. 2. When \(a=3\,\text{cm}\), the area is \(A=3^2=9\,\text{cm}^2\). When the side length doubles to \(6\,\text{cm}\), the area is \(A=6^2=36\,\text{cm}^2\). The area is multiplied by \(4\), not \(2\), so the relationship is not proportional.

Answer

Relationship 1 is proportional because doubling \(a\) from \(3\,\text{cm}\) to \(6\,\text{cm}\) doubles \(P\) from \(12\,\text{cm}\) to \(24\,\text{cm}\). Relationship 2 is not proportional because doubling \(a\) from \(3\,\text{cm}\) to \(6\,\text{cm}\) multiplies \(A\) by \(4\), from \(9\,\text{cm}^2\) to \(36\,\text{cm}^2\).
Test for proportional relationships 12
Consider two everyday situations. a) Number of equally priced admission tickets with no quantity discount \(\to\) total price. b) Miles traveled in a taxi \(\to\) fare when the taxi charges a fixed starting fee plus a fixed price per mile. For each situation, decide whether the relationship is proportional. Explain how the taxi's starting fee affects proportionality.

Hints

- Check whether doubling the input exactly doubles the output. - Does a taxi charge anything before the first mile is traveled? - How does adding a fixed starting value affect a graph?

Solution

1. The ticket relationship is proportional because the price per ticket is constant. Doubling the number of tickets doubles the total price. 2. The taxi relationship is not proportional because the fixed starting fee is charged even when \(0\) miles are traveled. 3. The starting fee makes the fare-to-distance ratio vary and makes the graph start above the origin.

Answer

a) Proportional, because the price per ticket is constant. b) Not proportional, because the fixed starting fee prevents the graph from passing through the origin.
Test for proportional relationships 13
Decide whether each situation represents a proportional relationship. Briefly justify each decision. a) Number of movie tickets sold and total revenue when every ticket has the same price. b) Age of a tree and its current height. c) Side length of a square and its perimeter. d) Travel time and distance for a train moving at a constant speed of \(60\) miles per hour.

Hints

- If the first quantity doubles, must the second also double? - Check whether the ratio of the two quantities stays constant. - Think about which situations have an unchanging rate and which do not.

Solution

1. Ticket revenue equals a fixed ticket price multiplied by the number sold, so a) is proportional. 2. A tree does not gain the same amount of height every year, so b) is not generally proportional. 3. A square's perimeter is \(P=4s\), so c) is proportional. 4. A train moving at constant speed travels \(d=60t\) miles in \(t\) hours, so d) is proportional.

Answer

a) Proportional: revenue is ticket price multiplied by the number of tickets. b) Not proportional: a tree's height is not a constant multiple of its age. c) Proportional: the perimeter is \(4\) times the side length. d) Proportional: the distance in miles is \(60\) times the time in hours.
Test for proportional relationships 14
During a long trip, a delivery van's cumulative gasoline use is recorded. <table> <tr><td>Distance \(s\), in miles</td><td>\(150\)</td><td>\(280\)</td><td>\(420\)</td><td>\(550\)</td></tr> <tr><td>Gasoline \(G\), in gallons</td><td>\(5.1\)</td><td>\(9.5\)</td><td>\(14.3\)</td><td>\(18.7\)</td></tr> </table> a) Use calculations to determine whether \(s\to G\) is exactly proportional. b) Give one reason real measurements may differ slightly from exact proportionality. What does \(\frac{G}{s}\) mean in this context?

Hints

- Calculate gallons per mile for every pair. - What must be true of all ratios for exact proportionality? - Think of conditions besides distance that affect gasoline use. - Identify the units of gallons divided by miles.

Solution

1. The ratios are \(5.1\div 150=0.034\), \(9.5\div 280\approx 0.03393\), \(14.3\div 420\approx 0.03405\), and \(18.7\div 550=0.034\) gallon per mile. 2. Because the ratios are not exactly equal, the measured relationship is not exactly proportional. 3. Small differences may result from traffic, hills, wind, driving style, or measurement error. 4. The ratio \(G\div s\) is average gasoline use per mile.

Answer

a) No. The ratios are very close but not identical. b) Factors such as traffic, terrain, wind, or driving style can change gasoline use. The ratio \(G\div s\) is gallons used per mile.
Test for proportional relationships 15
Which equations represent proportional relationships? Briefly explain each decision using the form \(y=kx\), where \(k\) is constant. a) \(y=3.5x\) b) \(y=\frac{x}{2}\) c) \(y=2x+1\) d) \(y=x^2\) e) \(y=\frac{4}{x}\)

Hints

- What does \(y\div x\) look like for a proportional relationship? - Can you rewrite each expression as a fixed number multiplied by \(x\)? - Test two different nonzero values of \(x\) if the form is unclear.

Solution

1. In a proportional relationship, the equation can be written as \(y=kx\) with the same constant \(k\) for all inputs. 2. a) \(y=3.5x\) has this form, so it is proportional. 3. b) \(y=\frac{1}{2}x\) has this form, so it is proportional. 4. c) The added \(1\) prevents \(y\) from being a constant multiple of \(x\). 5. d) The ratio \(\frac{y}{x}=x\) changes with nonzero \(x\), so it is not proportional. 6. e) The ratio \(\frac{y}{x}=\frac{4}{x^2}\) is not constant, so it is not proportional.

Answer

a) Proportional: \(y=3.5x\). b) Proportional: \(y=\frac{1}{2}x\). c) Not proportional: there is an added \(1\). d) Not proportional: \(y\div x=x\) changes as \(x\) changes. e) Not proportional: \(y\div x=\frac{4}{x^2}\) is not constant.
Test for proportional relationships 16
A garden tank is empty when a pump starts. The table shows the volume \(V\), in gallons, after \(t\) minutes. <table> <tr><td>Time \(t\), in minutes</td><td>\(4\)</td><td>\(10\)</td><td>\(15\)</td></tr> <tr><td>Volume \(V\), in gallons</td><td>\(30\)</td><td>\(75\)</td><td>\(112.5\)</td></tr> </table> a) Do these measurements fit a proportional relationship? Show how you know. b) For a proportional model that fits the data, find \(k\) and explain its meaning. c) Which point must the graph of this model pass through? Explain using the empty tank.

Hints

- Compare the ratio of water volume to elapsed time in each column. - Think about what one minute's worth of water represents. - What does the statement about the empty tank tell you about the graph?

Solution

1. The measured ratios are \(30\div4=7.5\), \(75\div10=7.5\), and \(112.5\div15=7.5\). They agree, so the measurements fit a proportional model. Three measurements alone do not prove that the rate stays constant at all other times. 2. In this model, \(k=7.5\,\text{gal/min}\) is the amount added each minute. 3. Its graph passes through \((0, 0)\) because the tank is empty when timing starts.

Answer

a) Yes. Each measured ratio \(V\div t\) is \(7.5\); this supports a proportional model, but does not prove the rate is always constant. b) \(k=7.5\,\text{gal/min}\), the modeled filling rate. c) \((0, 0)\), because the tank contains no water when timing starts.
Test for proportional relationships 17
Yuna claims, “When a circle's diameter doubles, its circumference also doubles. Therefore, circumference is proportional to diameter.” Use \(C=\pi d\) to test Yuna's claim. Show how the circumference changes when the diameter changes from \(d\) to \(2d\), and explain whether the relationship is proportional.

Hints

- What happens to a product when one factor doubles? - For a proportional relationship, how should the output change when the input doubles? - Does \(C\div d\) stay constant?

Solution

1. The original circumference is \(C=\pi d\). 2. For double the diameter, the new circumference is \(C_{\text{new}}=\pi(2d)=2\pi d=2C\). 3. The ratio \(\frac{C}{d}=\pi\) is constant for all positive diameters. 4. Therefore, Yuna is correct: circumference is proportional to diameter.

Answer

Yuna is correct. Doubling the diameter gives \(C_{\text{new}}=\pi(2d)=2C\), and the ratio \(\frac{C}{d}=\pi\) is constant.
Test for proportional relationships 18
Graphs a) and b) show two relationships between \(x\) and \(y\). Which graph represents a proportional relationship? Explain using a feature of the graph.
Figure for problem 550208

Hints

- Recall the graph feature that every proportional relationship must have. - Check whether each straight line passes through the origin. - A straight-line pattern alone is not enough to make a relationship proportional.

Solution

1. Graph a) is a straight line that passes through \((0,0)\), so it represents a proportional relationship. 2. Graph b) does not pass through the origin, so it is not proportional.

Answer

Graph a) is proportional because its line passes through \((0,0)\). Graph b) is not proportional because its line does not pass through the origin.
Test for proportional relationships 19
Rosa sees that a relationship contains the point \((3,12)\). She says, “The relationship must be proportional with constant of proportionality \(4\), because \(12\div3=4\).” a) Is Rosa's conclusion justified from that one point alone? Explain. b) A second point from the same relationship is \((6,30)\). Use both points to determine whether the relationship is proportional.

Hints

- Ask what must be true about \(\frac{y}{x}\) for every nonzero point in a proportional relationship. - One data pair can suggest a constant, but consider what information it does not provide. - Compare the ratio from the second point with the ratio from the first.

Solution

1. The point \((3,12)\) gives the candidate ratio \(12\div3=4\), but one point alone does not establish that the same ratio holds for every point in the relationship. 2. For the second point, \(30\div6=5\). 3. Since the ratios \(4\) and \(5\) are different, the two points cannot belong to one proportional relationship of the form \(y=kx\).

Answer

a) No. One point gives a possible value of \(k\), but it does not prove that the ratio is constant for the whole relationship. b) The relationship is not proportional because \(12\div3=4\) while \(30\div6=5\).
Test for proportional relationships 20
Does the graph show one proportional relationship from \(x=0\) to \(x=4\)? Explain using two points with positive \(x\)-values.
Figure for problem 559484

Hints

- How does the graph rise as \(x\) increases? - Choose two points with different positive x-values. - Does the same output-to-input ratio work in both places?

Solution

1. At \((2,4)\), the ratio \(y\div x\) is \(4\div2=2\). 2. At \((4,5)\), the ratio is \(5\div4=1.25\). 3. The ratios differ, so the bent line does not show one proportional relationship.

Answer

No. The graph includes \((2,4)\) and \((4,5)\), giving ratios \(2\) and \(1.25\). One constant rate does not fit the whole graph.
Test for proportional relationships 21
Jordan says, “The plotted points include the origin, so the relationship is proportional.” Is Jordan correct? Check \(y\div x\) at every plotted point where \(x>0\).
Figure for problem 559485

Hints

- Does passing through the origin, by itself, make a relationship proportional? - Find the output-to-input ratio for each point with positive input. - What must be true of every one of those ratios?

Solution

1. The point \((1,2)\) has ratio \(2\div1=2\). 2. The point \((2,4)\) has ratio \(4\div2=2\). 3. The point \((3,8)\) has ratio \(8\div3\), not \(2\). The ratios are not constant, so Jordan is incorrect.

Answer

No. The ratios for \((1,2)\), \((2,4)\), and \((3,8)\) are \(2\), \(2\), and \(\frac{8}{3}\). They do not all match.
Test for proportional relationships 22
A bank charges a fixed \(\$5\) service fee when converting US dollars to Canadian dollars. Only the money left after the fee is converted at a rate of \(1\) US dollar to \(1.35\) Canadian dollars. The bank accepts deposits of at least \(\$5\). a) Make a table showing the Canadian-dollar amounts received for deposits of \(\$20\), \(\$50\), and \(\$100\). b) Write an equation for the Canadian-dollar amount \(D\) received from a deposit of \(x\) US dollars. c) Is the relationship between the deposit and the amount received proportional? Justify your answer mathematically.

Hints

- Decide whether the fee is subtracted before or after the currency conversion. - Compare the output-to-input ratios for at least two deposits. - In a proportional relationship, that ratio must stay constant.

Solution

1. For \(\$20\), \((20-5)\cdot1.35=20.25\,\text{CAD}\); for \(\$50\), \((50-5)\cdot1.35=60.75\,\text{CAD}\); for \(\$100\), \((100-5)\cdot1.35=128.25\,\text{CAD}\). 2. For \(x\ge5\), the rule is \(D=1.35(x-5)\). 3. The relationship is not proportional because \(20.25\div20=1.0125\), while \(60.75\div50=1.215\). The ratios differ.

Answer

a) <table><tr><th>Deposit (US dollars)</th><td>\(\$20\)</td><td>\(\$50\)</td><td>\(\$100\)</td></tr><tr><th>Amount received (CAD)</th><td>\(20.25\,\text{CAD}\)</td><td>\(60.75\,\text{CAD}\)</td><td>\(128.25\,\text{CAD}\)</td></tr></table> b) \(D=1.35(x-5)\) for \(x\ge5\). c) No. The ratios \(20.25\div20\) and \(60.75\div50\) are different, so the relationship is not proportional.
Test for proportional relationships 23
Consider each geometric relationship. Decide whether it is proportional. a) Side length of an equilateral triangle \(\to\) perimeter of the triangle. b) Radius of a circle \(\to\) area of the circle. c) Edge length of a cube \(\to\) total length of all its edges. d) Edge length of a cube \(\to\) surface area of the cube.

Hints

- Write a formula for each relationship. - Check whether multiplying the input by a factor multiplies the output by the same factor. - In particular, compare what happens when the input is doubled.

Solution

1. a) Proportional: \(P=3s\), so the perimeter is always three times the side length. 2. b) Not proportional: \(A=\pi r^2\). Doubling the radius multiplies the area by \(4\), not by \(2\). 3. c) Proportional: a cube has \(12\) edges, so \(L=12s\). 4. d) Not proportional: \(S=6s^2\). Doubling the edge length multiplies the surface area by \(4\), not by \(2\).

Answer

a) Proportional. b) Not proportional. c) Proportional. d) Not proportional.
Test for proportional relationships 24
A bike-rental shop offers two pricing plans: Plan A: no starting fee and \(\$2.50\) per hour. Plan B: a \(\$5.00\) starting fee and \(\$1.00\) per hour. For each plan, determine whether the relationship between rental time, in hours, and total price is proportional. Use specific numerical examples to justify your conclusions.

Hints

- Make a small table for \(1\) hour and \(2\) hours under each plan. - Check whether price divided by hours stays constant. - Consider how a fixed starting fee affects the ratio.

Solution

1. Plan A costs \(C=2.50h\). One hour costs \(\$2.50\) and two hours cost \(\$5.00\), giving the same price per hour; it is proportional. 2. Plan B costs \(\$6.00\) for one hour and \(\$7.00\) for two hours. 3. Plan B's price-to-hours ratios are \(6\div1=6\) and \(7\div2=3.5\). They are not equal, so Plan B is not proportional.

Answer

Plan A is proportional: \(\$2.50\) for one hour and \(\$5.00\) for two hours both give \(\$2.50\) per hour. Plan B is not: its \(\$6.00\) for one hour and \(\$7.00\) for two hours give different price-to-hours ratios, \(6\) and \(3.5\).
Test for proportional relationships 25
A candy shop sells loose candy. A \(5\,\text{oz}\) bag costs \(\$1.50\). a) Find what a \(12\,\text{oz}\) bag would cost if price were proportional to weight. b) The \(12\,\text{oz}\) bag actually costs only \(\$3.30\). Show that the two bag prices are not proportional to weight, and explain one reason a store might charge a lower price per ounce for a larger bag.

Hints

- First find the price for \(1\,\text{oz}\) in each bag. - Compare the two unit prices. - Think about why a store might encourage customers to buy a larger bag.

Solution

1. The unit price for the \(5\,\text{oz}\) bag is \(1.50\div 5=\$0.30\) per ounce. 2. A proportional price for \(12\,\text{oz}\) is \(12\cdot 0.30=\$3.60\). 3. The actual unit price for the \(12\,\text{oz}\) bag is \(3.30\div 12=\$0.275\) per ounce. Since \(\$0.275\) per ounce is not equal to \(\$0.30\) per ounce, the prices are not proportional to weight. 4. A store may offer a quantity discount to encourage larger purchases, or packaging and handling may cost less per ounce for larger bags.

Answer

a) The proportional price would be \(\$3.60\). b) The actual unit prices are \(\$0.30\) per ounce and \(\$0.275\) per ounce, so the prices are not proportional to weight. A larger bag may have a quantity discount or lower packaging and handling costs per ounce.
Test for proportional relationships 26
Orchard manager Marisol claims, “The relationship between the number of apple trees and the total harvest, in pounds, is proportional.” a) Under what idealized conditions would Marisol be correct? b) Give two real-world reasons why this relationship usually is not exactly proportional.

Hints

- When would the number of pounds per tree always be the same? - Are all living plants identical in their production? - What resources might not be equally available to every tree?

Solution

1. The relationship would be proportional if every tree produced exactly the same number of pounds of apples. 2. In reality, trees differ in age, size, health, and exposure to pests. 3. Soil quality, sunlight, water, and available nutrients may also vary from tree to tree. 4. If trees are planted too closely, competition for light or nutrients can reduce the yield per tree.

Answer

a) It would be proportional if every tree had the same yield. b) Possible reasons include differences in tree age or health, pests, soil quality, sunlight, water, and competition between trees.
Test for proportional relationships 27
A copy shop offers two pricing plans: Plan A: Each copy costs \(\$0.10\). Plan B: A monthly fee of \(\$2.50\), plus \(\$0.05\) for each copy. a) Write an equation for the total cost under each plan, using \(x\) for the number of copies and \(C_A\) and \(C_B\) for the costs. b) Which plan represents a proportional relationship? Explain using your equations.

Hints

- In each plan, which cost depends on the number of copies, and which cost stays fixed? - What should the cost be for \(0\) copies in a proportional relationship? - Compare each equation with \(y=kx\).

Solution

1. Plan A has cost \(C_A=0.10x\). Plan B has cost \(C_B=0.05x+2.50\). 2. In Plan A, the total cost is a constant multiplied by the number of copies, so the relationship is proportional. 3. Plan B is not proportional because its fixed fee makes the cost \(\$2.50\) when \(x=0\), instead of \(\$0\).

Answer

a) \(C_A=0.10x\) and \(C_B=0.05x+2.50\). b) Only Plan A is proportional. Plan B includes a fixed fee, so its equation is not of the form \(y=kx\) and it costs \(\$2.50\) even for \(0\) copies.
Test for proportional relationships 28
While a backyard pool is being filled, two measurements are recorded. <table> <tr><td>Time \(t\), in minutes</td><td>\(5\)</td><td>\(12\)</td></tr> <tr><td>Water volume \(V\), in gallons</td><td>\(120\)</td><td>\(288\)</td></tr> </table> a) Use calculations to determine whether the two measurements are consistent with a proportional relationship. b) Find the filling rate in gallons per minute. c) How much water is in the pool after \(25\) minutes if the process continues at the same rate?

Hints

- How can a table show that two quantities are proportional? - Find the gallons per minute in both columns. - Use the unit rate for the new time.

Solution

1. The rates are \(120\div 5=24\) and \(288\div 12=24\), so the measurements are consistent with a proportional relationship. 2. The filling rate is \(24\) gallons per minute. 3. After \(25\) minutes, the volume is \(24\cdot 25=600\) gallons.

Answer

a) Yes. Both ratios \(V\div t\) equal \(24\). b) \(24\) gallons per minute. c) \(600\) gallons.
Test for proportional relationships 29
The table shows the distance a car traveled on a highway and the gasoline it used. <table> <tr><td>Distance \(s\), in miles</td><td>\(100\)</td><td>\(250\)</td><td>\(400\)</td><td>\(550\)</td></tr> <tr><td>Gasoline \(V\), in gallons</td><td>\(4\)</td><td>\(10\)</td><td>\(16\)</td><td>\(22\)</td></tr> </table> Are the measurements consistent with gasoline used being proportional to distance? If so, find the constant of proportionality and explain what it means.

Hints

- For each column, compare gasoline used with distance traveled. - What would have to be the same across all columns for the measurements to be proportional? - Describe the units of the ratio you calculated.

Solution

1. Calculate \(V\div s\) for each pair: \(4\div100=0.04\), \(10\div250=0.04\), \(16\div400=0.04\), and \(22\div550=0.04\). 2. All shown ratios are equal, so the measurements are consistent with a proportional relationship. 3. The constant is \(0.04\) gallon per mile, equivalent to \(4\) gallons per \(100\) miles.

Answer

Yes. Every ratio \(V\div s\) equals \(0.04\). The constant means the car used \(0.04\) gallon per mile, or \(4\) gallons per \(100\) miles.
Test for proportional relationships 30
Kojo wants to know whether he walked at a constant speed during three parts of a trip. - Part 1: \(3\) miles in \(1.5\) hours - Part 2: \(5\) miles in \(2.5\) hours - Part 3: \(7.5\) miles in \(4\) hours Use the constant of proportionality to determine whether distance and time are proportional over all three parts and whether Kojo's speed was constant.

Hints

- Find speed by dividing distance by time. - What must be true about distance divided by time for constant speed? - Check whether Kojo covers the same distance each hour in every part.

Solution

1. Part 1 speed: \(3\div 1.5=2\) miles per hour. 2. Part 2 speed: \(5\div 2.5=2\) miles per hour. 3. Part 3 speed: \(7.5\div 4=1.875\) miles per hour. 4. Since the third rate differs, the speed was not constant and the three pairs do not form one proportional relationship.

Answer

Kojo's speed was not constant. It was \(2\) miles per hour for the first two parts and \(1.875\) miles per hour for the third, so the relationship is not proportional across all three parts.
Test for proportional relationships 31
A laboratory tests three metal cylinders that are claimed to be made of the same material. For this problem, treat the listed measurements as exact and assume that cylinders made of the same material must have exactly the same mass-to-volume ratio. - Cylinder A: volume \(20\,\text{cm}^3\), mass \(178\,\text{g}\) - Cylinder B: volume \(50\,\text{cm}^3\), mass \(445\,\text{g}\) - Cylinder C: volume \(12\,\text{cm}^3\), mass \(105\,\text{g}\) a) Use calculations to determine whether the three measurements are consistent with that assumption. b) What mass would Cylinder C need to have the same constant of proportionality as Cylinders A and B?

Hints

- Compare mass divided by volume for each cylinder. - Under the stated assumption, what must be true about those ratios? - For part b, use the common ratio from Cylinders A and B with Cylinder C's volume.

Solution

1. Cylinder A's density is \(178 \div 20=8.9\,\text{g/cm}^3\). 2. Cylinder B's density is \(445 \div 50=8.9\,\text{g/cm}^3\). 3. Cylinder C's density is \(105 \div 12=8.75\,\text{g/cm}^3\). 4. Since Cylinder C's ratio differs, the three measurements are not consistent with one exact mass-to-volume constant. 5. At \(8.9\,\text{g/cm}^3\), Cylinder C would need mass \(12\cdot8.9=106.8\,\text{g}\).

Answer

a) No. Cylinders A and B have density \(8.9\,\text{g/cm}^3\), while Cylinder C has density \(8.75\,\text{g/cm}^3\), so the three exact measurements do not share one constant ratio. b) \(106.8\,\text{g}\).
Test for proportional relationships 32
Two companies rent electric scooters using different pricing plans. - Company A: no starting fee; each minute costs \(\$0.20\). - Company B: a \(\$1.00\) activation fee for each ride, then \(\$0.10\) per minute. Determine which plan represents a proportional relationship between ride time and total cost. Justify your answer using properties of a line through the origin.

Hints

- Write each cost as an equation. - What is the cost at \(0\) minutes? - Where must the graph begin for a proportional relationship?

Solution

1. Company A has equation \(y=0.20x\). Company B has equation \(y=0.10x+1.00\). 2. A proportional relationship has a graph that is a line through \((0, 0)\). 3. For Company A, \(0\) minutes costs \(\$0\), so the graph passes through the origin. 4. For Company B, \(0\) minutes still costs \(\$1.00\), so its graph does not pass through the origin.

Answer

Only Company A is proportional. Company B's activation fee gives a nonzero cost at \(0\) minutes.
Test for proportional relationships 33
The table shows measured gasoline use for a delivery truck. <table> <tr><td>Distance, in miles</td><td>\(80\)</td><td>\(150\)</td><td>\(240\)</td><td>\(310\)</td></tr> <tr><td>Gasoline used, in gallons</td><td>\(2.6\)</td><td>\(4.9\)</td><td>\(7.75\)</td><td>\(10.1\)</td></tr> </table> a) Calculate the gasoline-per-mile ratio for each measurement. Do all four ratios fall between \(0.032\) and \(0.033\) gallon per mile? Explain what this suggests about using a proportional model for estimation. b) Find a suitable estimated unit rate by calculating the mean of the four gasoline-per-mile ratios. Round to the nearest ten-thousandth. c) Use a proportional model with that estimated unit rate to estimate the gasoline used for \(600\) miles. Round to the nearest tenth of a gallon.

Hints

- Find gasoline used per mile for every pair. - Compare every calculated ratio with the interval stated in part a). - Average the four ratios rather than selecting one of them. - Use the estimated unit rate for the new distance.

Solution

1. The ratios are \(2.6\div 80=0.0325\), \(4.9\div 150\approx 0.0327\), \(7.75\div 240\approx 0.0323\), and \(10.1\div 310\approx 0.0326\) gallon per mile. 2. All four ratios lie between \(0.032\) and \(0.033\) gallon per mile. They are close but not identical, so a proportional model is reasonable for estimation but does not describe the measurements exactly. 3. The mean is \(\frac{0.0325+0.032666\ldots+0.032291\ldots+0.032580\ldots}{4}\approx 0.0325\) gallon per mile. This is an estimated unit rate for the model, not an exact constant shared by all four measurements. 4. The estimate is \(600\cdot 0.0325=19.5\) gallons.

Answer

a) Yes. All four ratios are between \(0.032\) and \(0.033\) gallon per mile. This supports using a proportional model for estimation, even though the measured ratios are not exactly equal. b) The estimated unit rate is approximately \(0.0325\) gallon per mile. c) Approximately \(19.5\) gallons.
Test for proportional relationships 34
A copper-wire manufacturer measures the length \(l\) and mass \(m\) of several rolls. <table> <tr><td>Length \(l\), in meters</td><td>\(10\)</td><td>\(25\)</td><td>\(40\)</td><td>\(60\)</td><td>\(100\)</td></tr> <tr><td>Mass \(m\), in grams</td><td>\(185\)</td><td>\(462.5\)</td><td>\(740\)</td><td>\(1110\)</td><td>\(1850\)</td></tr> </table> a) Determine whether length and mass form a proportional relationship. b) Find the constant of proportionality and explain its physical meaning for the wire.

Hints

- Find mass divided by length for each column. - Check whether this ratio stays constant. - Interpret the units grams per meter.

Solution

1. Calculate \(m\div l\): \(185\div 10=18.5\), \(462.5\div 25=18.5\), \(740\div 40=18.5\), \(1110\div 60=18.5\), and \(1850\div 100=18.5\). 2. Since every ratio is equal, the relationship is proportional. 3. The constant \(18.5\,\text{g/m}\) is the wire's mass per meter, or linear density.

Answer

a) Yes. The ratio \(m\div l\) is constant at \(18.5\). b) \(k=18.5\,\text{g/m}\), meaning each meter of wire has a mass of \(18.5\) grams.
Test for proportional relationships 35
A laboratory measures the mass \(m\) of a liquid for several volumes \(V\). <table> <tr><td>Volume \(V\), in \(\text{cm}^3\)</td><td>\(50\)</td><td>\(150\)</td><td>\(300\)</td></tr> <tr><td>Mass \(m\), in grams</td><td>\(40\)</td><td>\(120\)</td><td>\(240\)</td></tr> </table> a) Are the measurements consistent with mass being proportional to volume? Explain. b) Write an equation of the form \(m=kV\). What does \(k\) represent, and what are its units? c) Find the mass of \(450\,\text{cm}^3\) of the liquid if the same rule continues.

Hints

- Which ratio compares mass with its corresponding volume? - What units does mass divided by volume have? - Use the constant you find for the new volume.

Solution

1. The ratios are \(40\div50=0.8\), \(120\div150=0.8\), and \(240\div300=0.8\), so the measured pairs have a constant mass-to-volume ratio. 2. The equation is \(m=0.8V\). The constant \(0.8\,\text{g/cm}^3\) represents the liquid's mass per unit volume, or density. 3. For \(450\,\text{cm}^3\), the mass is \(0.8\cdot450=360\,\text{g}\).

Answer

a) Yes; each mass-to-volume ratio equals \(0.8\). b) \(m=0.8V\). The constant is the density, \(0.8\,\text{g/cm}^3\). c) \(360\,\text{g}\).
Test for proportional relationships 36
In a proportional relationship between \(x\) and \(y\), the ratio \(\frac{y}{x}\) is the same for every nonzero \(x\). Use this ratio test to decide whether each relationship is proportional. a) \(y=12x\) b) \(y=3x+6\) c) \(y=4(x+3)-12\)

Hints

- Could you rewrite a rule in a simpler form before testing its ratios? - Can two different inputs help you rule out a constant ratio? - If you find a constant ratio in a few examples, how could you show it stays constant for every nonzero input?

Solution

1. a) When \(x\ne0\), \(\frac{y}{x}=\frac{12x}{x}=12\), so the relationship is proportional. 2. b) At \(x=1\), \(y=9\) and \(y\div x=9\). At \(x=2\), \(y=12\) and \(y\div x=6\). The ratios differ, so it is not proportional. 3. c) Distribute and simplify: \(y=4x+12-12=4x\). For every nonzero \(x\), \(\frac{y}{x}=4\). Although the original rule contains added and subtracted numbers, they cancel, so the relationship is proportional.

Answer

a) Proportional; \(\frac{y}{x}=12\). b) Not proportional; for \(x=1\) and \(x=2\), the ratios are \(9\) and \(6\). c) Proportional; \(y=4(x+3)-12=4x\), so \(\frac{y}{x}=4\) for every nonzero \(x\).
Test for proportional relationships 37
A copy shop lists these prices for black-and-white copies: - \(10\) copies cost \(\$0.80\). - \(50\) copies cost \(\$4.00\). - \(100\) copies cost \(\$7.50\). a) Determine whether price is proportional to number of copies. Show your calculations. b) Give one real-world reason a store might use pricing that is not proportional.

Hints

- Find the price of one copy for each offer. - What must stay constant in a proportional relationship? - Think about discounts for buying larger quantities.

Solution

1. The unit prices are \(0.80\div 10=\$0.08\), \(4.00\div 50=\$0.08\), and \(7.50\div 100=\$0.075\) per copy. 2. Since the unit prices are not all equal, the relationship is not proportional. 3. The store may offer a quantity discount for large orders.

Answer

a) No. The first two prices are \(\$0.08\) per copy, but \(100\) copies cost \(\$0.075\) per copy. b) A possible reason is a quantity discount.
Test for proportional relationships 38
Test the pairs in the table for proportionality. <table> <tr><td>\(x\)</td><td>\(3\)</td><td>\(5\)</td><td>\(8\)</td><td>\(10\)</td></tr> <tr><td>\(y\)</td><td>\(10\)</td><td>\(16\)</td><td>\(25\)</td><td>\(31\)</td></tr> </table> Justify your conclusion mathematically. If the relationship is not proportional, describe how \(\frac{y}{x}\) changes as \(x\) increases.

Hints

- Calculate the ratio for every column. - Is matching in only two columns enough? - Read the ratio values from left to right and describe the pattern.

Solution

1. The ratios are \(10\div 3\approx 3.33\), \(16\div 5=3.2\), \(25\div 8=3.125\), and \(31\div 10=3.1\). 2. Since the ratios are not equal, the relationship is not proportional. 3. As \(x\) increases, the ratio \(y\div x\) decreases.

Answer

The relationship is not proportional because the ratios \(y\div x\) are not equal. The ratio decreases as \(x\) increases.
Test for proportional relationships 39
Two runners, Lucas and Simon, are training for a marathon. Their distances were measured at different times: Lucas: <table> <tr><td>Time \(t\) (in \(\text{min}\))</td><td>\(10\)</td><td>\(25\)</td><td>\(40\)</td></tr> <tr><td>Distance \(d\) (in \(\text{m}\))</td><td>\(2200\)</td><td>\(5500\)</td><td>\(8800\)</td></tr> </table> Simon: <table> <tr><td>Time \(t\) (in \(\text{min}\))</td><td>\(12\)</td><td>\(20\)</td><td>\(45\)</td></tr> <tr><td>Distance \(d\) (in \(\text{m}\))</td><td>\(2700\)</td><td>\(4500\)</td><td>\(10{,}125\)</td></tr> </table> a) Show that distance is proportional to time for each runner. b) Who runs faster? Justify your answer by comparing the constants of proportionality. c) How far would the faster runner travel in one hour at the same speed?

Hints

- What does the value of distance divided by time represent? - How is speed related to the constant of proportionality? - Pay attention to the units when finding the distance traveled in one hour.

Solution

1. For Lucas, divide each distance by its time: \(2200 \div 10=220\), \(5500 \div 25=220\), and \(8800 \div 40=220\). The constant ratio is \(220\), so the relationship is proportional. 2. For Simon, \(2700 \div 12=225\), \(4500 \div 20=225\), and \(10{,}125 \div 45=225\). The constant ratio is \(225\), so this relationship is also proportional. 3. The constants of proportionality are \(220\,\text{m/min}\) for Lucas and \(225\,\text{m/min}\) for Simon. Simon runs faster because \(225>220\). 4. In \(60\) minutes, Simon travels \(225 \cdot 60=13{,}500\,\text{m}\), or \(13.5\,\text{km}\).

Answer

a) Both relationships are proportional because the ratio \(\frac{d}{t}\) is constant for each runner: \(220\) for Lucas and \(225\) for Simon. b) Simon runs faster because \(225\,\text{m/min}>220\,\text{m/min}\). c) Simon would run \(13{,}500\,\text{m}\), or \(13.5\,\text{km}\), in one hour.
Test for proportional relationships 40
A circular flower bed is enlarged from radius \(r_1=2\,\text{ft}\) to radius \(r_2=5\,\text{ft}\). 1) For each radius, calculate the ratio of area to radius, \(\frac{A}{r}\). 2) Use the ratios to decide whether the relationship “radius \(\to\) area” is proportional. 3) By what factor does the area increase when the radius is multiplied by \(2.5\)?

Hints

- First calculate each circle's area. - Divide each area by its matching radius and compare the ratios. - A proportional relationship must have a constant output-to-input ratio. - How does a length scale factor affect area?

Solution

1. For \(r_1=2\,\text{ft}\), \(A_1=\pi(2\,\text{ft})^2=4\pi\,\text{ft}^2\), so \(\frac{A_1}{r_1}=2\pi\,\text{ft}\approx6.28\,\text{ft}\). 2. For \(r_2=5\,\text{ft}\), \(A_2=\pi(5\,\text{ft})^2=25\pi\,\text{ft}^2\), so \(\frac{A_2}{r_2}=5\pi\,\text{ft}\approx15.71\,\text{ft}\). 3. The ratios are not equal, so area is not proportional to radius. 4. Since area depends on the square of the radius, a radius factor of \(2.5\) gives an area factor of \(2.5^2=6.25\).

Answer

1) \(\frac{A_1}{r_1}=2\pi\,\text{ft}\approx6.28\,\text{ft}\) and \(\frac{A_2}{r_2}=5\pi\,\text{ft}\approx15.71\,\text{ft}\) 2) The relationship is not proportional. 3) The area is multiplied by \(6.25\).
Test for proportional relationships 41
Darius records several time-distance pairs from a bicycle ride. The table shows distance \(d\) after time \(t\). a) Do the three complete pairs support the claim that distance is proportional to time? Justify your answer using the ratios \(\frac{d}{t}\). b) Complete the missing entries so the table continues that proportional relationship. <table> <tr><td>Time \(t\), in hours</td><td>\(1.5\)</td><td>\(2\)</td><td>\(3.5\)</td><td>\(5\)</td><td></td></tr> <tr><td>Distance \(d\), in miles</td><td>\(27\)</td><td>\(36\)</td><td>\(63\)</td><td></td><td>\(108\)</td></tr> </table>

Hints

- Test the claim from the table rather than from the bicycle context. - Compare the distance-to-time ratios for the complete columns. - Use the same ratio in either direction to fill the missing entries.

Solution

1. The ratios are \(27\div1.5=18\), \(36\div2=18\), and \(63\div3.5=18\). All complete pairs agree with a proportional relationship at \(18\,\text{mi/h}\). 2. At \(5\) hours, \(d=18\cdot5=90\) miles. 3. For \(108\) miles, \(t=108\div18=6\) hours.

Answer

a) Yes. All three complete pairs give \(\frac{d}{t}=18\,\text{mi/h}\). b) The missing distance is \(90\) miles, and the missing time is \(6\) hours.
Test for proportional relationships 42
Two cell phone plans list these prices for mobile data. Assume each plan follows the same pricing rule for other data amounts. Which plan is consistent with proportional pricing? Justify your answer, then find the price of \(120\,\text{GB}\) under that plan. **Plan A:** <table> <tr><td>Data, in \(\text{GB}\)</td><td>\(10\)</td><td>\(50\)</td></tr> <tr><td>Price</td><td>\(\$1.00\)</td><td>\(\$5.00\)</td></tr> </table> **Plan B:** <table> <tr><td>Data, in \(\text{GB}\)</td><td>\(10\)</td><td>\(50\)</td></tr> <tr><td>Price</td><td>\(\$5.50\)</td><td>\(\$7.50\)</td></tr> </table>

Hints

- Compare the price of one gigabyte in each column. - Which plan's two prices follow a single multiplier? - Use the plan's assumed pricing rule for the larger data amount.

Solution

1. Plan A's unit prices are \(1.00\div10=0.10\) and \(5.00\div50=0.10\) dollar per gigabyte. 2. Plan B's unit prices are \(5.50\div10=0.55\) and \(7.50\div50=0.15\), which are different. 3. Plan A is consistent with proportional pricing. At its assumed continuing rate, \(120\,\text{GB}\) costs \(0.10\cdot120=\$12.00\).

Answer

Plan A: both listed prices are \(\$0.10\) per gigabyte. Plan B's listed unit prices differ, \(\$0.55\) and \(\$0.15\). Under the stated continuation assumption, \(120\,\text{GB}\) costs \(\$12.00\) with Plan A.
Test for proportional relationships 43
A copy shop charges the prices shown. <table><tr><td>Number of pages \(n\)</td><td>\(10\)</td><td>\(20\)</td><td>\(50\)</td><td>\(100\)</td></tr><tr><td>Total cost \(C\)</td><td>\(\$0.80\)</td><td>\(\$1.60\)</td><td>\(\$4.00\)</td><td>\(\$8.00\)</td></tr></table> 1) Show that cost is proportional to the number of pages. 2) Write an equation for the cost \(C\) of \(n\) pages. 3) How many pages were printed if the bill was \(\$14.40\)? 4) Suppose the shop also charged a one-time \(\$2.00\) fee per order. Would total cost still be proportional to the number of pages? Explain.

Hints

- How could you check whether the price per page is the same in every column? - Express the total cost as a product of a rate and a page count. - Work backward from the bill to the number of pages. - Does adding a fixed fee preserve the form of the equation?

Solution

1. All the cost-to-page ratios are equal: \(0.80\div10=1.60\div20=4.00\div50=8.00\div100=0.08\). 2. The original price rule is \(C=0.08n\). 3. Solve \(14.40=0.08n\): \(n=14.40\div0.08=180\) pages. 4. With a fixed fee, the rule becomes \(C=0.08n+2\). This is not proportional because the cost is no longer just a constant multiple of the page count.

Answer

1) All ratios \(\frac{C}{n}\) equal \(0.08\). 2) \(C=0.08n\). 3) \(180\) pages. 4) No. The new equation \(C=0.08n+2\) has a fixed fee and is not of the form \(C=kn\).
Test for proportional relationships 44
A rectangular prism has a fixed width of \(2\,\text{inches}\) and height of \(5\,\text{inches}\). Its length \(x\) can vary and is always positive. Determine whether each quantity is proportional to length \(x\). Show an equation to support each decision. a) Volume \(V\) b) Surface area \(S\)

Hints

- Which of the prism's dimensions change, and which stay fixed? - Express the volume as length times width times height. - For surface area, include all six faces, then compare the rule with a constant times \(x\).

Solution

1. The volume is \(V=x\cdot2\cdot5=10x\). It is a constant multiple of the length, so volume is proportional to length. 2. The surface area is \(S=2(2x+5x+2\cdot5)=14x+20\). 3. Surface area is not proportional to length because its equation includes the added constant \(20\). Its output-to-input ratio therefore changes with \(x\).

Answer

a) Proportional: \(V=10x\) has the form \(V=kx\). b) Not proportional: \(S=14x+20\) has an additional constant term.
Test for proportional relationships 45
Nadia and Leo hike away from a trailhead. The table shows Nadia's distance from the trailhead, and the graph shows Leo's distance. Time \(x\) is the number of hours after noon. <table><tr><td>Hours after noon</td><td>\(1\)</td><td>\(2\)</td><td>\(4\)</td></tr><tr><td>Miles from trailhead</td><td>\(3\)</td><td>\(6\)</td><td>\(12\)</td></tr></table> a) Do Nadia's table values fit a proportional relationship? Does Leo's graph show a proportional relationship? Explain both answers. b) At \(2\) p.m., who is farther from the trailhead, and by how many miles?
Figure for problem 559486

Hints

- Check whether Nadia's distance divided by the matching time stays the same across her table. - Where is Leo at noon, according to his graph? Does that matter for proportionality? - To compare their locations, use the same time in both representations.

Solution

1. Nadia's table has the same distance-to-time ratio for every listed pair: \(3\div1=6\div2=12\div4=3\). The values fit a proportional relationship. 2. Leo's graph crosses the vertical axis at \((0,1)\), so it is not proportional to hours after noon. Leo is already \(1\) mile from the trailhead at noon. 3. At \(2\) p.m., Nadia is \(6\) miles and Leo is \(7\) miles from the trailhead, so Leo is \(1\) mile farther away.

Answer

a) Nadia's values fit a proportional relationship: each distance divided by the matching time is \(3\). Leo's graph is not proportional because it passes through \((0,1)\), not the origin. b) Leo is \(1\) mile farther from the trailhead. Nadia is \(6\) miles away and Leo is \(7\) miles away at \(2\) p.m.
Test for proportional relationships 46
Two students write equations for \(y\) in terms of \(x\). Student A: \(y=3(x+2)-6\) Student B: \(y=3(x+2)\) a) Which equation describes a proportional relationship? Show how you know. b) Find the output of each equation when \(x=0\), and explain why this matters.

Hints

- Distribute the multiplier in both equations. - Look for any added amount that remains after simplifying. - Substitute zero for the input to check the starting value.

Solution

1. A simplifies to \(y=3x+6-6=3x\), which is proportional. 2. B simplifies to \(y=3x+6\), which is not of the form \(y=kx\). 3. At \(x=0\), A gives \(0\) and B gives \(6\). A proportional relationship must output zero for an input of zero.

Answer

a) Student A's equation is proportional: it simplifies to \(y=3x\). Student B's simplifies to \(y=3x+6\), so it is not proportional. b) A gives \(0\); B gives \(6\). A proportional relationship must have output \(0\) when \(x=0\).

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