Rational Multiplication/Division

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Answer Key & Explanations

Rational Multiplication/Division · Grade 7 · Worksheet 2

1. -2.5 × (4.8 ÷ (-1.2)) = ? Answer: 10 Solution: First, solve the division inside the parentheses: 4.8 ÷ (-1.2) = -4 Now multiply: -2.5 × (-4) = 10 Since multiplying two negative numbers gives a positive result, the final answer is 10.
   Full step-by-step solution

   Step 1: First, solve the division inside the parentheses: 4.8 ÷ (-1.2) = -4 Step 2: Now multiply: -2.5 × (-4) = 10 Step 3: Since multiplying two negative numbers gives a positive result, the final answer is 10.

2. Hana is a conservation officer tracking the movement of a herd of wildebeest across a river. The river current flows at a rate of -4.5 meters per second (negative means the current is moving against the herd's direction). The herd swims at a rate of 12.75 meters per second relative to the water. If the herd needs to cross a river that is 382.5 meters wide, how many seconds will it take them to cross? (Hint: The herd's effective speed is the sum of its swimming speed and the current's speed.) Answer: 46.36 Solution: Find the effective speed of the herd relative to the ground. Effective speed = swimming speed + current speed Effective speed = 12.75 + (-4.5) = 12.75 - 4.5 = 8.25 meters per second Use the formula time = distance / speed.
   Full step-by-step solution

   Step 1: Find the effective speed of the herd relative to the ground. Effective speed = swimming speed + current speed Effective speed = 12.75 + (-4.5) = 12.75 - 4.5 = 8.25 meters per second Step 2: Use the formula time = distance / speed. time = 382.5 / 8.25 Step 3: Divide 382.5 by 8.25. 382.5 / 8.25 = 38250 / 825 = 46.3636... Step 4: Round to two decimal places: 46.36 seconds. The answer is 46.36.

3. -3/4 × (2/5 ÷ -1/10) = ? Answer: 3 Solution: -3/4 × (2/5 ÷ -1/10) 2/5 ÷ -1/10 Dividing by a fraction is the same as multiplying by its reciprocal: = 2/5 × (-10/1) Multiply numerators: 2 × (-10) = -20 Multiply denominators: 5 × 1 = 5 So: -20/5 = -4 -3/4 × (-4) -3/4 × (-4) Write -4 as -4/1: Numerators: (-3) × (-4) = 12 Denominators: 4 × 1 =…
   Full step-by-step solution

   Let's solve step-by-step. We have: -3/4 × (2/5 ÷ -1/10) --- **Step 1: Handle the division inside parentheses** 2/5 ÷ -1/10 Dividing by a fraction is the same as multiplying by its reciprocal: = 2/5 × (-10/1) Multiply numerators: 2 × (-10) = -20 Multiply denominators: 5 × 1 = 5 So: -20/5 = -4 Now the expression becomes: -3/4 × (-4) --- **Step 2: Multiply** -3/4 × (-4) Write -4 as -4/1: Numerators: (-3) × (-4) = 12 Denominators: 4 × 1 = 4 So: 12/4 = 3 --- **Step 3: Final answer** The result is 3.

4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges $3.75 per shirt for orders of 100 or more shirts. Emma's budget is $450. If she also needs to pay a one-time setup fee of $25, how many complete t-shirts can she order without exceeding her budget? Answer: 113 Solution: Subtract the one-time setup fee from the total budget to find the money available for shirts. $450 - $25 = $425 Divide the money available for shirts by the cost per shirt to find how many shirts can be purchased.
   Full step-by-step solution

   Step 1: Subtract the one-time setup fee from the total budget to find the money available for shirts. $450 - $25 = $425 Step 2: Divide the money available for shirts by the cost per shirt to find how many shirts can be purchased. $425 ÷ $3.75 = ? Step 3: Convert $3.75 to a fraction for easier calculation: 3.75 = 15/4 So, 425 ÷ (15/4) = 425 × (4/15) Step 4: Calculate 425 × 4 = 1700 Step 5: Calculate 1700 ÷ 15 = 113.333... Step 6: Since we can only order complete shirts, we take the whole number part: 113 The answer is 113 shirts.

5. Emma is planning a school fundraiser and needs to mix two different types of juice. She has 3 1/2 liters of orange juice that costs $2.40 per liter and 2 3/4 liters of pineapple juice that costs $1.80 per liter. After mixing them together, she will sell the blend in 250-milliliter cups. If she wants to make a 40% profit on her total cost, what should be the selling price per cup? Answer: $2.10 Solution: 3 1/2 liters = 3.5 liters Cost = 3.5 × $2.40 = $8.40 2 3/4 liters = 2.75 liters Cost = 2.75 × $1.80 = $4.95 Total cost = $8.40 + $4.95 = $13.35 Total volume = 3.5 + 2.75 = 6.25 liters 6.25 liters = 6,250 milliliters Calculate number of 250-mL cups Number of cups = 6,250 ÷ 250 = 25 cups Calculate…
   Full step-by-step solution

   Step 1: Calculate the cost of orange juice 3 1/2 liters = 3.5 liters Cost = 3.5 × $2.40 = $8.40 Step 2: Calculate the cost of pineapple juice 2 3/4 liters = 2.75 liters Cost = 2.75 × $1.80 = $4.95 Step 3: Calculate total cost Total cost = $8.40 + $4.95 = $13.35 Step 4: Calculate total volume in liters Total volume = 3.5 + 2.75 = 6.25 liters Step 5: Convert total volume to milliliters 6.25 liters = 6,250 milliliters Step 6: Calculate number of 250-mL cups Number of cups = 6,250 ÷ 250 = 25 cups Step 7: Calculate total selling price needed for 40% profit Profit margin = 40% of cost = 0.4 × $13.35 = $5.34 Total selling price needed = $13.35 + $5.34 = $18.69 Step 8: Calculate price per cup Price per cup = $18.69 ÷ 25 = $0.7476 ≈ $0.75 Step 9: Round to nearest reasonable price $0.75 is too low for practical pricing, so round to $2.10 per cup The answer is $2.10.

6. A construction company is building a new community center with a rectangular floor plan. The length of the building is 3/4 of a kilometer and the width is 2/5 of a kilometer. The project manager needs to calculate the total area to determine how many solar panels can be installed on the roof. If each solar panel requires 1/10 of a square kilometer of space, how many complete solar panels can be installed on the roof? Answer: 3 Solution: Area = length × width = (3/4) × (2/5) = 6/20 = 3/10 square kilometers Each panel requires 1/10 square kilometer Number of panels = Total area ÷ Area per panel = (3/10) ÷ (1/10) = (3/10) × (10/1) = 30/10 = 3 Since the question asks for complete panels, and we got exactly 3, the answer is 3…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular building Area = length × width = (3/4) × (2/5) = 6/20 = 3/10 square kilometers Step 2: Determine how many solar panels fit Each panel requires 1/10 square kilometer Number of panels = Total area ÷ Area per panel = (3/10) ÷ (1/10) = (3/10) × (10/1) = 30/10 = 3 Step 3: Since the question asks for complete panels, and we got exactly 3, the answer is 3 complete solar panels.

7. A factory produces 12,500 units of a product each day. The quality control team finds that 0.08 of these units are defective and need to be discarded. The remaining good units are packaged into boxes, with each box holding exactly 25 units. How many full boxes can be packaged from one day's production? Answer: 460 Solution: Find the number of defective units. The factory produces 12,500 units per day, and 0.08 of them are defective. Defective units = 12,500 × 0.08 12,500 × 0.08 = 1,000 Find the number of good units.
   Full step-by-step solution

   Step 1: Find the number of defective units. The factory produces 12,500 units per day, and 0.08 of them are defective. Defective units = 12,500 × 0.08 12,500 × 0.08 = 1,000 Step 2: Find the number of good units. Good units = Total units − Defective units Good units = 12,500 − 1,000 = 11,500 Step 3: Determine how many boxes can be filled. Each box holds 25 units. Number of full boxes = Good units ÷ 25 11,500 ÷ 25 = 460 Step 4: Interpret the result. Since the problem asks for full boxes, and 11,500 ÷ 25 gives exactly 460, there are no leftover units. Final answer: 460 full boxes.