Rational Coefficient Equations

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Answer Key & Explanations

Rational Coefficient Equations · Grade 7 · Worksheet 2

1. Mere is designing a rectangular garden on a coordinate grid. The garden has corners at (0, 0), (24, 0), (24, 16), and (0, 16). She plans to plant a row of flowers along a line from the bottom-left corner (0, 0) to a point on the top edge at (12, 16). This line divides the garden into two sections. The equation of the line is y = (4/3)x. If Mere wants to place a decorative stone at the point where x = 9, what is the y-coordinate of that point on the line? Answer: 12 Solution: The equation of the line is y = (4/3)x. We are asked to find the y-coordinate when x = 9. Substitute x = 9 into the equation: y = (4/3) * 9.
   Full step-by-step solution

   Step 1: The equation of the line is y = (4/3)x. We are asked to find the y-coordinate when x = 9. Step 2: Substitute x = 9 into the equation: y = (4/3) * 9. Step 3: Multiply: (4/3) * 9 = (4 * 9) / 3. Step 4: Calculate the numerator: 4 * 9 = 36. Step 5: Divide by the denominator: 36 / 3 = 12. Step 6: So, y = 12. The y-coordinate of the point on the line where x = 9 is 12.

2. (3/7)x - 9 = (1/3)x + 5 = ? Answer: 147/2 Solution: Start with the equation (3/7)x - 9 = (1/3)x + 5. Find the least common denominator of 7 and 3, which is 21. Multiply every term by 21: 21*(3/7)x - 21*9 = 21*(1/3)x + 21*5.
   Full step-by-step solution

   Step 1: Start with the equation (3/7)x - 9 = (1/3)x + 5. Step 2: Find the least common denominator of 7 and 3, which is 21. Step 3: Multiply every term by 21: 21*(3/7)x - 21*9 = 21*(1/3)x + 21*5. Step 4: Simplify: 9x - 189 = 7x + 105. Step 5: Subtract 7x from both sides: 9x - 7x - 189 = 7x - 7x + 105 → 2x - 189 = 105. Step 6: Add 189 to both sides: 2x - 189 + 189 = 105 + 189 → 2x = 294. Step 7: Divide both sides by 2: 2x/2 = 294/2 → x = 147. The answer is 147.

3. Emma is drawing a rectangular garden on a coordinate grid, where each unit represents 1 meter. The garden has corners at (0, 0), (30, 0), (30, 20), and (0, 20). She wants to place a diagonal path from (0, 0) to (30, 20). The length of the path is given by the equation 0.5x + 5 = 20, where x represents the length of the path in meters. What is the length of the diagonal path? Answer: 30 Solution: Start with the equation: 0.5x + 5 = 20 Subtract 5 from both sides: 0.5x = 20 - 5 = 15 Divide both sides by 0.5: x = 15 / 0.5 = 30 The length of the diagonal path is 30 meters.
   Full step-by-step solution

   Step 1: Start with the equation: 0.5x + 5 = 20 Step 2: Subtract 5 from both sides: 0.5x = 20 - 5 = 15 Step 3: Divide both sides by 0.5: x = 15 / 0.5 = 30 Step 4: The length of the diagonal path is 30 meters. The answer is 30.

4. A school is planning a field trip and needs to calculate transportation costs. The total budget for transportation is $12,000. If 3/5 of the budget is allocated for bus rentals and the remaining amount is split equally between train tickets and ferry tickets, how much money is allocated for ferry tickets? Answer: $2400 Solution: Calculate the amount allocated for bus rentals: 3/5 of $12,000 = (3/5) × 12,000 = $7,200 Calculate the remaining budget: $12,000 - $7,200 = $4,800 Since the remaining amount is split equally between train and ferry tickets, divide by 2: $4,800 ÷ 2 = $2,400 Therefore, the amount allocated for…
   Full step-by-step solution

   Step 1: Calculate the amount allocated for bus rentals: 3/5 of $12,000 = (3/5) × 12,000 = $7,200 Step 2: Calculate the remaining budget: $12,000 - $7,200 = $4,800 Step 3: Since the remaining amount is split equally between train and ferry tickets, divide by 2: $4,800 ÷ 2 = $2,400 Therefore, the amount allocated for ferry tickets is $2,400.

5. Mere is planning a rectangular vegetable garden. The length of the garden is 0.75 times its width. If the perimeter of the garden is 84 meters, what is the width of the garden in meters? Answer: 24 Solution: Let w be the width. The length is 0.75w. Perimeter = 2(length + width) = 2(0.75w + w) = 2(1.75w) = 3.5w.
   Full step-by-step solution

   Let w be the width. The length is 0.75w. Perimeter = 2(length + width) = 2(0.75w + w) = 2(1.75w) = 3.5w. Set equal to 84: 3.5w = 84. Divide both sides by 3.5: w = 84 / 3.5 = 24. The width is 24 meters.

6. Emma is planning a road trip from her hometown to visit her grandparents. The total distance is 420 miles. Her car's fuel efficiency is 28 miles per gallon, and the current gas price is $3.75 per gallon. If she also needs to pay a $15 toll for highway usage, how much will Emma spend on gas and tolls for the entire trip? Answer: 71.25 Solution: Calculate the gallons of gas needed for the trip. Distance = 420 miles Fuel efficiency = 28 miles per gallon Gallons needed = 420 ÷ 28 = 15 gallons Calculate the cost of gas.
   Full step-by-step solution

   Step 1: Calculate the gallons of gas needed for the trip. Distance = 420 miles Fuel efficiency = 28 miles per gallon Gallons needed = 420 ÷ 28 = 15 gallons Step 2: Calculate the cost of gas. Gas price = $3.75 per gallon Gas cost = 15 × 3.75 = $56.25 Step 3: Add the toll cost. Toll = $15 Total cost = Gas cost + Toll = 56.25 + 15 = $71.25 Emma will spend $71.25 on gas and tolls for the entire trip.

7. (3/8)x - 14 = (1/4)x + 22 = ? Answer: 288 Solution: Start with the equation (3/8)x - 14 = (1/4)x + 22. The denominators are 8 and 4. The least common denominator is 8.
   Full step-by-step solution

   Step 1: Start with the equation (3/8)x - 14 = (1/4)x + 22. Step 2: The denominators are 8 and 4. The least common denominator is 8. Step 3: Multiply every term by 8: 8*(3/8)x - 8*14 = 8*(1/4)x + 8*22. Step 4: Simplify: 3x - 112 = 2x + 176. Step 5: Subtract 2x from both sides: 3x - 2x - 112 = 2x - 2x + 176 → x - 112 = 176. Step 6: Add 112 to both sides: x - 112 + 112 = 176 + 112 → x = 288. The answer is 288.