Rational Multiplication/Division

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Rational Multiplication/Division · Grade 7 · Worksheet 3

1. Liam is designing a rectangular garden that is 3/4 of a meter wide. The area of the garden is 2 1/4 square meters. What is the length of the garden? Answer: 3 Solution: We are told the garden is rectangular, with width = 3/4 meter and area = 2 1/4 square meters. We need to find the length. Write the area in improper fraction form.
   Full step-by-step solution

   We are told the garden is rectangular, with width = 3/4 meter and area = 2 1/4 square meters. We need to find the length. Step 1: Write the area in improper fraction form. 2 1/4 = 2 + 1/4 = 8/4 + 1/4 = 9/4 square meters. Step 2: Recall the formula for the area of a rectangle. Area = length × width Step 3: Substitute the known values into the formula. 9/4 = length × (3/4) Step 4: Solve for the length. To isolate the length, divide both sides by 3/4. Dividing by 3/4 is the same as multiplying by 4/3. So, length = (9/4) × (4/3) Step 5: Multiply the fractions. Multiply numerators: 9 × 4 = 36 Multiply denominators: 4 × 3 = 12 So, length = 36/12 Step 6: Simplify the fraction. 36 ÷ 12 = 3 Step 7: State the answer. The length of the garden is 3 meters.

2. Kaia is a marine biologist tracking the movement of a research submarine. The submarine descends at a rate of -12.5 meters per minute. After diving for 8.4 minutes, it then ascends at a rate of 5.25 meters per minute for 12 minutes. What is the submarine's final depth in meters relative to the surface? (Use negative numbers to indicate depth below the surface.) Answer: -42 Solution: Calculate the depth after the descent. The rate is -12.5 meters per minute for 8.4 minutes. Depth = -12.5 × 8.4 = -105 meters.
   Full step-by-step solution

   Step 1: Calculate the depth after the descent. The rate is -12.5 meters per minute for 8.4 minutes. Depth = -12.5 × 8.4 = -105 meters. Step 2: Calculate the ascent. The rate is +5.25 meters per minute for 12 minutes. Ascent = 5.25 × 12 = 63 meters. Step 3: Combine the descent and ascent. Final depth = -105 + 63 = -42 meters. The submarine ends at a depth of 42 meters below the surface, so the answer is -42.

3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A triangular flower bed is created inside the garden by connecting the midpoints of three sides: the midpoint of the left side, the midpoint of the top side, and the midpoint of the right side. What is the area of this triangular flower bed? Answer: 6 Solution: Vertices: (2, 1), (8, 1), (8, 5), (2, 5). Length in x-direction: from 2 to 8 → length = 6 Height in y-direction: from 1 to 5 → height = 4 So rectangle area = 6 × 4 = 24.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the rectangle's vertices and dimensions** Vertices: (2, 1), (8, 1), (8, 5), (2, 5). Length in x-direction: from 2 to 8 → length = 6 Height in y-direction: from 1 to 5 → height = 4 So rectangle area = 6 × 4 = 24. --- **Step 2: Find the midpoints of the three sides mentioned** - Left side: from (2, 1) to (2, 5) Midpoint = (2, (1+5)/2) = (2, 3) → call this A. - Top side: from (2, 5) to (8, 5) Midpoint = ((2+8)/2, 5) = (5, 5) → call this B. - Right side: from (8, 1) to (8, 5) Midpoint = (8, (1+5)/2) = (8, 3) → call this C. --- **Step 3: Plot the triangle ABC** Points: A = (2, 3), B = (5, 5), C = (8, 3). --- **Step 4: Find the area of triangle ABC** We can use the shoelace formula: Shoelace formula: Area = 1/2 × | (x1y2 + x2y3 + x3y1) − (y1x2 + y2x3 + y3x1) | Let’s list coordinates in order A→B→C→A: A = (2, 3) B = (5, 5) C = (8, 3) Sum1 = x1*y2 + x2*y3 + x3*y1 = 2×5 + 5×3 + 8×3 = 10 + 15 + 24 = 49 Sum2 = y1*x2 + y2*x3 + y3*x1 = 3×5 + 5×8 + 3×2 = 15 + 40 + 6 = 61 Area = 1/2 × |49 − 61| = 1/2 × |−12| = 1/2 × 12 = 6. --- **Step 5: Conclusion** The area of the triangular flower bed is 6. --- **Final answer:** 6

4. Isabella is designing a rectangular garden on a coordinate grid. The garden has vertices at (0, 0), (0, 36), (56, 36), and (56, 0). She plants a triangular flower bed with vertices at (0, 0), (56, 0), and (0, 36). The remaining area of the garden (outside the triangle) is divided into 7 equal rectangular plots for vegetables. What is the area of each rectangular vegetable plot? Answer: 144 Solution: Find the area of the entire rectangular garden. Length = 56, Width = 36. Area = 56 × 36 = 2016 square units.
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular garden. Length = 56, Width = 36. Area = 56 × 36 = 2016 square units. Step 2: Find the area of the triangular flower bed. The triangle has base = 56 (along the bottom from (0,0) to (56,0)) and height = 36 (along the left side from (0,0) to (0,36)). Area of triangle = 1/2 × base × height = 1/2 × 56 × 36 = 1/2 × 2016 = 1008 square units. Step 3: Find the remaining area for vegetables. Remaining area = Total area - Triangle area = 2016 - 1008 = 1008 square units. Step 4: Divide the remaining area into 7 equal rectangular plots. Area of each plot = 1008 ÷ 7 = 144 square units. The answer is 144.

5. A factory produces 12,500 units of a product each day. During quality control, they found that 0.08 of the production was defective and had to be discarded. The remaining good units were packed into boxes, with each box holding 25 units. How many boxes were needed to pack all the good units? 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 are needed. Each box holds 25 units. Number of boxes = Good units ÷ Units per box Number of boxes = 11,500 ÷ 25 Step 4: Perform the division. 11,500 ÷ 25 = 460 Step 5: Conclusion. Since boxes must be whole and the division is exact, 460 boxes are needed to pack all the good units. Final answer: 460

6. A triangular sail for a boat is drawn on a coordinate grid. Its vertices are at (-3, 0), (9, 0), and (0, 7). The sail is made of a special fabric that costs -2.5 dollars per square unit (meaning the company receives a subsidy for using eco-friendly material, so the cost is negative). What is the total cost of the fabric for this sail? Answer: -105 Solution: Find the base and height of the triangle. The base is along the x-axis from (-3,0) to (9,0). The length of the base is 9 - (-3) = 12 units.
   Full step-by-step solution

   Step 1: Find the base and height of the triangle. The base is along the x-axis from (-3,0) to (9,0). The length of the base is 9 - (-3) = 12 units. The height is the vertical distance from the base to the top vertex (0,7), which is 7 units. Step 2: Calculate the area of the triangle. Area = (1/2) × base × height = (1/2) × 12 × 7 = 6 × 7 = 42 square units. Step 3: Multiply the area by the cost per square unit. Cost = 42 × (-2.5) = -105 dollars. The answer is -105.

7. Liam is designing a community garden with a rectangular plot that measures 3/4 mile by 2/5 mile. He wants to divide the garden into equal sections, each with an area of 1/10 square mile. How many complete sections can Liam create in the garden? Answer: 3 Solution: Area = length × width = (3/4) × (2/5) = (3×2)/(4×5) = 6/20 = 3/10 square mile Number of sections = total area ÷ area per section = (3/10) ÷ (1/10) = (3/10) × (10/1) = 30/10 = 3 Since we're asked for complete sections and 3 is a whole number, Liam can create exactly 3 complete sections.
   Full step-by-step solution

   Step 1: Calculate the total area of the garden plot Area = length × width = (3/4) × (2/5) = (3×2)/(4×5) = 6/20 = 3/10 square mile Step 2: Determine how many sections can fit Number of sections = total area ÷ area per section = (3/10) ÷ (1/10) = (3/10) × (10/1) = 30/10 = 3 Step 3: Verify the answer makes sense Since we're asked for complete sections and 3 is a whole number, Liam can create exactly 3 complete sections. The answer is 3.