Q1: A shirt originally costs $40. It is on sale for 25% off. What is the sale price of the shirt? (a) $10 (b) $25 (c) $30 (d) $35
Solution:
Ans: (c) Explanation: To find the sale price, first calculate the discount amount: \(25\% \text{ of } 40 = 0.25 \times 40 = 10\). Then subtract the discount from the original price: \(40 - 10 = 30\). The sale price is $30. Option (a) is just the discount amount, not the final price. Option (b) represents an incorrect calculation. Option (d) is the result of subtracting only 5 instead of 10.
Q2: A car travels 150 miles in 3 hours. What is the car's average speed in miles per hour? (a) 30 mph (b) 45 mph (c) 50 mph (d) 60 mph
Solution:
Ans: (c) Explanation: The formula for average speed is \(\text{speed} = \frac{\text{distance}}{\text{time}}\). Substituting the values: \(\text{speed} = \frac{150}{3} = 50\) mph. Option (a) results from dividing 90 by 3. Option (b) comes from incorrect division. Option (d) is the result of dividing 180 by 3.
Q3: If a quantity increases from 50 to 75, what is the percent increase? (a) 25% (b) 33% (c) 50% (d) 75%
Solution:
Ans: (c) Explanation: The percent increase formula is \(\frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%\). Calculating: \(\frac{75 - 50}{50} \times 100\% = \frac{25}{50} \times 100\% = 0.5 \times 100\% = 50\%\). Option (a) is just the absolute difference, not the percent. Option (b) would result from dividing by 75 instead of 50. Option (d) is the new value, not the percent increase.
Q4: A recipe calls for 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) 6 cups (b) 10 cups (c) 12 cups (d) 16 cups
Solution:
Ans: (c) Explanation: The ratio \(2:3\) means for every 2 cups of flour, you need 3 cups of sugar. Set up a proportion: \(\frac{2}{3} = \frac{8}{x}\). Cross-multiply: \(2x = 24\), so \(x = 12\) cups of sugar. Option (a) would result from adding 8 - 2 to 3. Option (b) is from incorrect proportion setup. Option (d) doubles the flour amount instead of applying the ratio.
Q5: What is 35% expressed as a decimal? (a) 0.035 (b) 0.35 (c) 3.5 (d) 35.0
Solution:
Ans: (b) Explanation: To convert a percent to a decimal, divide by 100 or move the decimal point two places to the left. \(35\% = \frac{35}{100} = 0.35\). Option (a) results from moving the decimal point three places. Option (c) moves it one place. Option (d) doesn't convert at all.
Sub-section A2: Fill in the Blank
Q6: A unit rate is a rate in which the second quantity is equal to __________.
Solution:
Ans: 1 Explanation: A unit rate compares a quantity to one unit of another quantity. For example, miles per 1 hour or cost per 1 item. The denominator of a unit rate is always 1.
Q7: The formula to find the percentage of a number is: \(\text{Percentage} =\) __________ \(\times \text{Number}\).
Solution:
Ans: Rate (or Decimal equivalent) Explanation: To find a percentage of a number, multiply the number by the percentage expressed as a decimal. For example, to find 20% of 50, calculate \(0.20 \times 50 = 10\). The rate or decimal equivalent is the percentage divided by 100.
Q8: If the original price of an item is $80 and it is marked up by 15%, the markup amount is $__________.
Solution:
Ans: 12 Explanation: The markup amount is calculated by finding 15% of the original price: \(0.15 \times 80 = 12\). The markup is $12, which would be added to the original price to get the new selling price.
Q9: Two quantities are in proportion if their __________ are equal.
Solution:
Ans: ratios Explanation: A proportion is an equation stating that two ratios are equal. For example, \(\frac{2}{3} = \frac{4}{6}\) is a proportion because both ratios simplify to the same value.
Q10: When a quantity decreases from 100 to 80, the percent decrease is __________%.
Solution:
Ans: 20 Explanation: The percent decrease is calculated using the formula \(\frac{\text{original} - \text{new}}{\text{original}} \times 100\%\). Substituting: \(\frac{100 - 80}{100} \times 100\% = \frac{20}{100} \times 100\% = 20\%\).
Section B: Apply Your Learning
Q11: Maria is buying a bicycle that costs $250. The store is offering a discount of 30% on all bicycles. After the discount, there is a sales tax of 8% applied to the discounted price. What is the total amount Maria will pay for the bicycle?
Solution:
Ans: Step 1: Calculate the discount amount. Discount = \(30\% \text{ of } 250 = 0.30 \times 250 = 75\) Step 2: Subtract the discount from the original price to find the discounted price. Discounted price = \(250 - 75 = 175\) Step 3: Calculate the sales tax on the discounted price. Sales tax = \(8\% \text{ of } 175 = 0.08 \times 175 = 14\) Step 4: Add the sales tax to the discounted price to find the total amount. Total amount = \(175 + 14 = 189\) Final Answer: $189
Q12: A train travels at a constant speed and covers 180 miles in 2.5 hours. At this same rate, how far will the train travel in 4 hours?
Solution:
Ans: Step 1: Find the unit rate (speed in miles per hour). Speed = \(\frac{180 \text{ miles}}{2.5 \text{ hours}} = 72\) mph Step 2: Use the speed to calculate the distance traveled in 4 hours. Distance = \(\text{speed} \times \text{time} = 72 \times 4 = 288\) miles Final Answer: 288 miles
Q13: A store sells oranges at a rate of 5 oranges for $3.00. How much would it cost to buy 13 oranges at this rate?
Solution:
Ans: Step 1: Find the unit rate (cost per orange). Cost per orange = \(\frac{\$3.00}{5} = \$0.60\) per orange Step 2: Multiply the unit rate by the number of oranges. Total cost = \(0.60 \times 13 = 7.80\) Final Answer: $7.80
Q14: The population of a town was 8,000 last year. This year, the population increased to 9,200. What is the percent increase in the population?
Solution:
Ans: Step 1: Calculate the absolute increase in population. Increase = \(9200 - 8000 = 1200\) Step 2: Use the percent increase formula. Percent increase = \(\frac{\text{increase}}{\text{original}} \times 100\% = \frac{1200}{8000} \times 100\%\) Step 3: Simplify the fraction and calculate. \(\frac{1200}{8000} = 0.15\) \(0.15 \times 100\% = 15\%\) Final Answer: 15%
Q15: A recipe for 4 servings requires 3 cups of milk. If you want to make enough for 10 servings, how many cups of milk will you need?
Solution:
Ans: Step 1: Set up a proportion relating servings to cups of milk. \(\frac{3 \text{ cups}}{4 \text{ servings}} = \frac{x \text{ cups}}{10 \text{ servings}}\) Step 2: Cross-multiply to solve for \(x\). \(4x = 3 \times 10\) \(4x = 30\) Step 3: Divide both sides by 4. \(x = \frac{30}{4} = 7.5\) Final Answer: 7.5 cups
Q16: A store bought a jacket for $60 and wants to sell it at a 40% markup. What should the selling price be?
Solution:
Ans: Step 1: Calculate the markup amount. Markup = \(40\% \text{ of } 60 = 0.40 \times 60 = 24\) Step 2: Add the markup to the original cost to find the selling price. Selling price = \(60 + 24 = 84\) Final Answer: $84
Section C: Evidence-Based Reasoning (CER)
Q17: Explain why understanding unit rates is important when comparing prices in a grocery store. Use specific examples to support your explanation.
Solution:
Ans: Claim: Understanding unit rates is essential for making informed purchasing decisions because it allows shoppers to accurately compare prices of items sold in different quantities. Evidence: For example, one store might sell a 12-ounce box of cereal for $3.60, while another store sells an 18-ounce box for $4.86. Without calculating the unit rate (price per ounce), it's difficult to determine which is the better deal. The first box has a unit rate of \(\frac{\$3.60}{12} = \$0.30\) per ounce, while the second box has a unit rate of \(\frac{\$4.86}{18} = \$0.27\) per ounce. Many grocery stores also display unit prices on shelf labels to help consumers make these comparisons quickly. Reasoning: By converting different package sizes to a common unit of measurement, shoppers can directly compare the cost-effectiveness of products regardless of their packaging size. This mathematical skill helps consumers save money and make rational economic choices. Without understanding unit rates, shoppers might mistakenly choose a more expensive option simply because the total price appears lower, when in reality they are paying more per unit of the product.
Q18: A clothing store advertises "25% off everything" during a sale. Explain whether a customer who buys a $80 jacket during the sale and then returns it for store credit (receiving the sale price back) should have to pay $20 to repurchase the same jacket at full price after the sale ends. Support your answer with mathematical reasoning.
Solution:
Ans: Claim: Yes, the customer would need to pay an additional $20 because the sale price and the full price represent different values, and the discount does not work in reverse. Evidence: During the sale, the customer paid \(80 - (0.25 \times 80) = 80 - 20 = \$60\) for the jacket. If they receive store credit of $60 and the jacket now costs the full price of $80, they would need to pay \(80 - 60 = \$20\) more. The percent discount of 25% off $80 equals $20 off, but to go from $60 back up to $80 requires adding $20, which represents a \(\frac{20}{60} \times 100\% = 33.3\%\) increase from the sale price, not a 25% increase. Reasoning: This situation illustrates an important concept about percentages: a percent decrease and the corresponding percent increase are calculated from different base values. The 25% discount is calculated from the original $80, while the increase needed to return to $80 is calculated from the reduced amount of $60. This demonstrates why percentage changes are not reversible and why understanding the reference value in percentage calculations is critical for making accurate financial decisions.
Section D: Extended Thinking
Q19: A store uses a two-stage markup system. First, they mark up an item by 50% from the wholesale cost. Then, during a special sale, they offer 30% off the marked-up price. A customer claims this is equivalent to a 20% markup from the wholesale cost (since 50% - 30% = 20%). Is the customer correct? Prove your answer mathematically using a wholesale cost of $100, and explain the general principle.
Solution:
Ans: Step 1: Calculate the marked-up price after a 50% markup from wholesale cost. Let wholesale cost = $100 Markup amount = \(0.50 \times 100 = 50\) Marked-up price = \(100 + 50 = 150\) Step 2: Calculate the sale price after 30% off the marked-up price. Discount amount = \(0.30 \times 150 = 45\) Sale price = \(150 - 45 = 105\) Step 3: Calculate the actual percent increase from wholesale to sale price. Increase from wholesale = \(105 - 100 = 5\) Percent increase = \(\frac{5}{100} \times 100\% = 5\%\) Step 4: Compare with the customer's claim of 20%. The customer claimed a 20% markup, which would result in a price of \(100 + (0.20 \times 100) = 100 + 20 = \$120\). Since \(105 \neq 120\), the customer is incorrect. Step 5: Explain the general principle. The customer's error is assuming that successive percentage changes can be simply added or subtracted. However, the 30% discount is applied to the marked-up price of $150, not the original wholesale cost of $100. This is why \(50\% - 30\% \neq 20\%\) in this context. The base value for each percentage calculation is different, so the percentage changes cannot be combined arithmetically. This principle applies to all successive percentage changes: they must be calculated sequentially, not by simple addition or subtraction of the percentages. Final Answer: No, the customer is incorrect. The actual markup from wholesale to final sale price is only 5%, not 20%. Successive percentage changes cannot be added or subtracted because they are calculated from different base values.
Q20: A water tank is being filled and drained simultaneously. Water flows in at a rate of 15 gallons per minute, and water flows out at a rate of 8 gallons per minute. The tank starts with 50 gallons of water. (a) Write an expression for the amount of water in the tank after \(t\) minutes. (b) How long will it take for the tank to contain 120 gallons? (c) If the tank has a maximum capacity of 200 gallons, how long will it take to fill completely?
Solution:
Ans: Part (a): Find the net rate of water change. Water flows in at 15 gallons/min and out at 8 gallons/min. Net rate = \(15 - 8 = 7\) gallons per minute (increase) Let \(W(t)\) represent the amount of water after \(t\) minutes. Since the tank starts with 50 gallons and gains 7 gallons each minute: \[W(t) = 50 + 7t\] Part (b): Find when the tank contains 120 gallons. Set \(W(t) = 120\) and solve for \(t\): \(50 + 7t = 120\) \(7t = 120 - 50\) \(7t = 70\) \(t = \frac{70}{7} = 10\) minutes Part (c): Find when the tank reaches 200 gallons capacity. Set \(W(t) = 200\) and solve for \(t\): \(50 + 7t = 200\) \(7t = 200 - 50\) \(7t = 150\) \(t = \frac{150}{7} = 21\frac{3}{7}\) minutes, or approximately 21.43 minutes Verification: Check the answer for part (c). After \(\frac{150}{7}\) minutes, water added = \(7 \times \frac{150}{7} = 150\) gallons Total water = \(50 + 150 = 200\) gallons ✓ Final Answer: (a) \(W(t) = 50 + 7t\) gallons; (b) 10 minutes; (c) \(21\frac{3}{7}\) minutes or approximately 21.43 minutes
The document Mixed Questions Set: Rates And Percentages is a part of the Grade 7 Course Math Grade 7.
FAQs on Mixed Questions Set: Rates And Percentages
1. What is the significance of Section A: Quick Check in the exam?
Ans. Section A: Quick Check is designed to assess students' understanding of key concepts and foundational knowledge. It typically includes straightforward questions that test recall and comprehension of the material covered in the associated article or curriculum.
2. How does Section B: Apply Your Learning enhance student understanding?
Ans. Section B: Apply Your Learning encourages students to apply the knowledge they have gained to new situations or problems. This section often includes practical scenarios or case studies that require critical thinking and the application of concepts learned, thereby deepening understanding.
3. What is the purpose of Section C: Evidence-Based Reasoning (CER)?
Ans. Section C: Evidence-Based Reasoning (CER) aims to develop students' ability to formulate scientific arguments based on evidence. Students are required to construct claims, provide reasoning, and support their claims with relevant evidence, which fosters analytical thinking and scientific literacy.
4. In what way does Section D: Extended Thinking challenge students?
Ans. Section D: Extended Thinking presents complex problems or questions that require in-depth analysis and extended responses. It challenges students to synthesise information, draw connections between concepts, and engage in higher-order thinking, thereby enhancing their critical thinking skills.
5. How can students effectively prepare for each section of the exam?
Ans. Students can prepare for each section by reviewing the material thoroughly, engaging in active learning techniques such as summarising and questioning, and practising past exam questions. For Section A, focus on key facts; for Section B, practice application scenarios; for Section C, work on crafting evidence-based arguments; and for Section D, develop skills in critical analysis and synthesis of information.
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