Q1: A car travels 150 miles in 3 hours. What is the constant of proportionality (unit rate) in miles per hour? (a) 50 miles per hour (b) 45 miles per hour (c) 150 miles per hour (d) 3 miles per hour
Solution:
Ans: (a) Explanation: The constant of proportionality is found by dividing the total distance by the total time: \(150 \div 3 = 50\) miles per hour. This represents the unit rate. Option (b) is incorrect because it results from an arithmetic error. Option (c) is the total distance, not the rate. Option (d) is the total time, not the rate.
Q2: Which equation represents a proportional relationship between \(x\) and \(y\)? (a) \(y = 3x + 2\) (b) \(y = 5x\) (c) \(y = x^2\) (d) \(y = 2x - 1\)
Solution:
Ans: (b) Explanation: A proportional relationship has the form \(y = kx\) where \(k\) is the constant of proportionality and the line passes through the origin. Option (b) \(y = 5x\) fits this form. Options (a) and (d) have constant terms added or subtracted, so they do not pass through the origin. Option (c) is a quadratic relationship, not a proportional one.
Q3: The graph of a proportional relationship is always: (a) A curve (b) A straight line through the origin (c) A horizontal line (d) A vertical line
Solution:
Ans: (b) Explanation: A proportional relationship always graphs as a straight line through the origin (0, 0) because when \(x = 0\), \(y\) must equal 0. Option (a) is incorrect because proportional relationships are linear, not curved. Options (c) and (d) do not represent proportional relationships.
Q4: If 4 pounds of apples cost $6, how much do 10 pounds cost assuming a proportional relationship? (a) $12 (b) $15 (c) $10 (d) $24
Solution:
Ans: (b) Explanation: First, find the unit rate: \(6 \div 4 = 1.5\) dollars per pound. Then multiply by 10 pounds: \(1.5 \times 10 = 15\) dollars. Option (a) results from doubling the original cost incorrectly. Option (c) assumes $1 per pound. Option (d) results from multiplying 6 by 4 instead of finding the unit rate.
Q5: The table shows values of \(x\) and \(y\). Is this a proportional relationship? \(x\): 2, 4, 6 \(y\): 8, 16, 24 (a) Yes, because \(y\) increases as \(x\) increases (b) No, because the values are too large (c) Yes, because the ratio \(\frac{y}{x}\) is constant (d) No, because \(x\) and \(y\) are different
Solution:
Ans: (c) Explanation: To determine if a relationship is proportional, check if the ratio \(\frac{y}{x}\) is constant. Here: \(\frac{8}{2} = 4\), \(\frac{16}{4} = 4\), \(\frac{24}{6} = 4\). The ratio is constant at 4, so this is proportional. Option (a) is not sufficient because many non-proportional relationships also increase. Options (b) and (d) are not valid criteria for proportional relationships.
Q6: A recipe uses a ratio of 2 cups of flour to 3 cups of sugar. If you use 8 cups of flour, how many cups of sugar do you need? (a) 10 cups (b) 12 cups (c) 6 cups (d) 16 cups
Solution:
Ans: (b) Explanation: Set up a proportion: \(\frac{2}{3} = \frac{8}{x}\). Cross-multiply: \(2x = 24\), so \(x = 12\) cups of sugar. Option (a) results from adding 2 to 8 instead of using proportions. Option (c) is incorrect scaling. Option (d) results from doubling 8 instead of using the correct ratio.
Q7: Which of the following points lies on the graph of \(y = 7x\)? (a) (2, 12) (b) (3, 21) (c) (4, 30) (d) (5, 40)
Solution:
Ans: (b) Explanation: Substitute each point into the equation \(y = 7x\). For option (b): \(7 \times 3 = 21\), which matches the \(y\)-value. For (a): \(7 \times 2 = 14 \neq 12\). For (c): \(7 \times 4 = 28 \neq 30\). For (d): \(7 \times 5 = 35 \neq 40\).
Q8: A proportional relationship has a constant of proportionality of \(\frac{2}{5}\). If \(x = 15\), what is \(y\)? (a) 5 (b) 6 (c) 7.5 (d) 37.5
Solution:
Ans: (b) Explanation: Use the equation \(y = kx\) where \(k = \frac{2}{5}\). Substitute \(x = 15\): \(y = \frac{2}{5} \times 15 = \frac{30}{5} = 6\). Option (a) results from dividing 15 by 3. Option (c) results from dividing 15 by 2. Option (d) results from multiplying instead of using the correct proportion.
Section B: Fill in the Blanks
Q9: In a proportional relationship represented by \(y = kx\), the letter \(k\) represents the __________.
Solution:
Ans: constant of proportionality (or unit rate) Explanation: The value \(k\) in the equation \(y = kx\) is called the constant of proportionality and represents the unit rate of the relationship.
Q10: The graph of every proportional relationship passes through the point __________.
Solution:
Ans: (0, 0) or the origin Explanation: In a proportional relationship, when \(x = 0\), \(y = 0\), so the graph always passes through the origin at point (0, 0).
Q11: If \(y\) is proportional to \(x\) and \(y = 20\) when \(x = 4\), then the constant of proportionality is __________.
Solution:
Ans: 5 Explanation: The constant of proportionality \(k\) is found using \(k = \frac{y}{x} = \frac{20}{4} = 5\).
Q12: To determine if a table represents a proportional relationship, check if all ratios of \(\frac{y}{x}\) are __________.
Solution:
Ans: equal (or constant or the same) Explanation: A relationship is proportional if the ratio \(\frac{y}{x}\) is constant for all pairs of values.
Q13: The equation of a proportional relationship has the form \(y = kx\) where there is no __________ term.
Solution:
Ans: constant (or added constant or y-intercept) Explanation: Proportional relationships have the form \(y = kx\) with no constant term added or subtracted. The y-intercept must be 0.
Q14: If 5 gallons of paint cover 200 square feet, then 1 gallon covers __________ square feet.
Solution:
Ans: 40 Explanation: The unit rate is found by dividing: \(200 \div 5 = 40\) square feet per gallon.
Section C: Word Problems
Q15: Emma earns money babysitting at a constant rate. She earns $45 for 5 hours of work. How much will she earn for 8 hours of work?
Solution:
Ans: First, find the unit rate (constant of proportionality): \(k = \frac{45}{5} = 9\) dollars per hour
Now use the equation \(y = kx\) where \(k = 9\) and \(x = 8\): \(y = 9 \times 8 = 72\) dollars
Final Answer: Emma will earn $72 for 8 hours of work.
Q16: A machine produces 240 bottles in 6 minutes. At this rate, how many bottles will it produce in 15 minutes?
Solution:
Ans: Find the constant of proportionality (bottles per minute): \(k = \frac{240}{6} = 40\) bottles per minute
Calculate bottles produced in 15 minutes using \(y = kx\): \(y = 40 \times 15 = 600\) bottles
Final Answer: The machine will produce 600 bottles in 15 minutes.
Q17: The cost of bananas is proportional to their weight. If 3 pounds cost $2.25, what is the cost of 7 pounds?
Solution:
Ans: Find the unit rate (cost per pound): \(k = \frac{2.25}{3} = 0.75\) dollars per pound
Calculate the cost of 7 pounds: \(y = 0.75 \times 7 = 5.25\) dollars
Final Answer: The cost of 7 pounds of bananas is $5.25.
Q18: A water tank is being filled at a constant rate. After 4 minutes, the tank contains 60 gallons of water. Write an equation that represents the proportional relationship between time \(t\) (in minutes) and the amount of water \(w\) (in gallons). How much water will be in the tank after 10 minutes?
Solution:
Ans: Find the constant of proportionality: \(k = \frac{60}{4} = 15\) gallons per minute
The equation is: \(w = 15t\)
Calculate water after 10 minutes: \(w = 15 \times 10 = 150\) gallons
Final Answer: The equation is \(w = 15t\) and the tank will contain 150 gallons after 10 minutes.
Q19: The distance a cyclist travels is proportional to time. The cyclist travels 36 miles in 2 hours. If the cyclist maintains this pace, how far will she travel in 5.5 hours?
Solution:
Ans: Find the constant of proportionality (speed): \(k = \frac{36}{2} = 18\) miles per hour
Calculate distance traveled in 5.5 hours: \(d = 18 \times 5.5 = 99\) miles
Final Answer: The cyclist will travel 99 miles in 5.5 hours.
Q20: A store sells juice in bottles. The relationship between the number of bottles purchased and the total cost is proportional. If 4 bottles cost $10, find the constant of proportionality and determine how many bottles can be purchased with $35.
Solution:
Ans: Find the constant of proportionality (cost per bottle): \(k = \frac{10}{4} = 2.5\) dollars per bottle
To find the number of bottles for $35, use \(y = kx\) where \(y = 35\) and \(k = 2.5\): \(35 = 2.5x\) \(x = \frac{35}{2.5} = 14\) bottles
Final Answer: The constant of proportionality is $2.50 per bottle, and 14 bottles can be purchased with $35.
The document Worksheet (with Solutions): Proportional Relationships is a part of the Grade 7 Course Math Grade 7.
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