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Rational Coefficient Equations

Grade 7 · Algebra · Worksheet 2

  1. Aroha is building a wooden bookshelf. She has a plank that is 7.25 meters long. She cuts off 0.5 meters for waste, then cuts the remaining plank into pieces that are each 3/4 of a meter long to use as shelves. After cutting as many shelves as possible, she has a small leftover piece. How many shelves does Aroha make? Answer: ______________
  2. Mere is saving money to buy a new laptop that costs $1,200. She has already saved $180. Each week, she saves 2/3 of her weekly allowance. Her weekly allowance is $45. How many more weeks will it take Mere to save enough money to buy the laptop? Answer: ______________
  3. A triangular garden is drawn on a coordinate plane with vertices at (0, 0), (9, 0), and (0, 7). Each unit on the grid represents 1 meter. A path with a width of 0.5 meters is to be built along the entire base of the triangle from (0, 0) to (9, 0). If the area of the path is given by 0.5 * base, and the remaining garden area must be at least 25 square meters, what is the value of x in the equation 0.5 * (9 - x) * 7 = 25, where x represents the number of meters of base covered by the path? Solve for x. Answer: ______________
  4. Noah is filling a rectangular swimming pool at his community center. The pool has a total capacity of 15,750 gallons of water. A hose fills the pool at a rate of 0.75 gallons per second. After filling for some time, a second hose is turned on that adds an additional 1.25 gallons per second. Together, the two hoses finish filling the remaining water in 2,100 seconds. How many gallons of water were already in the pool before the second hose was turned on? Answer: ______________
  5. Emma is planning a community fundraiser and needs to mix two types of juice to make a special blend. The first juice is 40% real fruit concentrate and the second is 75% real fruit concentrate. She wants to make 12 liters of a mixture that is 55% real fruit concentrate. How many liters of the 40% concentrate juice should she use? Answer: ______________
  6. Liam is volunteering at a community garden. He needs to prepare a new rectangular plot. The length of the plot is 3/5 of its width. If the total area of the plot must be 135 square meters, what is the width of the plot in meters? Answer: ______________
  7. (3/5)x - 9 = (1/3)x + 7 = ? Answer: ______________
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Answer Key & Explanations

Rational Coefficient Equations · Grade 7 · Worksheet 2

  1. Aroha is building a wooden bookshelf. She has a plank that is 7.25 meters long. She cuts off 0.5 meters for waste, then cuts the remaining plank into pieces that are each 3/4 of a meter long to use as shelves. After cutting as many shelves as possible, she has a small leftover piece. How many shelves does Aroha make? Answer: 9 Solution: Subtract the waste from the total length: 7.25 - 0.5 = 6.75 meters remaining. Each shelf is 3/4 meter long. Write 3/4 as a decimal: 3/4 = 0.75.
    Full step-by-step solution

    Step 1: Subtract the waste from the total length: 7.25 - 0.5 = 6.75 meters remaining. Step 2: Each shelf is 3/4 meter long. Write 3/4 as a decimal: 3/4 = 0.75. Step 3: Divide the remaining length by the length per shelf: 6.75 / 0.75. Step 4: Multiply both numbers by 100 to avoid decimals: 675 / 75. Step 5: Divide: 675 / 75 = 9. Step 6: Since 9 is a whole number, there is no leftover piece. Aroha makes 9 shelves. The answer is 9.

  2. Mere is saving money to buy a new laptop that costs $1,200. She has already saved $180. Each week, she saves 2/3 of her weekly allowance. Her weekly allowance is $45. How many more weeks will it take Mere to save enough money to buy the laptop? Answer: 34 Solution: Calculate Mere's weekly savings: 2/3 of $45 = (2/3) × 45 = 90/3 = $30 per week. Calculate how much more money Mere needs: $1,200 (total cost) - $180 (already saved) = $1,020.
    Full step-by-step solution

    Step 1: Calculate Mere's weekly savings: 2/3 of $45 = (2/3) × 45 = 90/3 = $30 per week. Step 2: Calculate how much more money Mere needs: $1,200 (total cost) - $180 (already saved) = $1,020. Step 3: Divide the remaining amount by weekly savings to find the number of weeks: $1,020 ÷ $30 = 34 weeks. Therefore, Mere needs 34 more weeks to save enough money.

  3. A triangular garden is drawn on a coordinate plane with vertices at (0, 0), (9, 0), and (0, 7). Each unit on the grid represents 1 meter. A path with a width of 0.5 meters is to be built along the entire base of the triangle from (0, 0) to (9, 0). If the area of the path is given by 0.5 * base, and the remaining garden area must be at least 25 square meters, what is the value of x in the equation 0.5 * (9 - x) * 7 = 25, where x represents the number of meters of base covered by the path? Solve for x. Answer: 1.857142857 Solution: Write the equation from the problem: 0.5 * (9 - x) * 7 = 25. Multiply 0.5 by 7 to simplify: 0.5 * 7 = 3.5, so the equation becomes 3.5 * (9 - x) = 25. Divide both sides by 3.5: (9 - x) = 25 / 3.5.
    Full step-by-step solution

    Step 1: Write the equation from the problem: 0.5 * (9 - x) * 7 = 25. Step 2: Multiply 0.5 by 7 to simplify: 0.5 * 7 = 3.5, so the equation becomes 3.5 * (9 - x) = 25. Step 3: Divide both sides by 3.5: (9 - x) = 25 / 3.5. Step 4: Calculate 25 divided by 3.5: 25 / 3.5 = 250 / 35 = 50 / 7 = 7.14285714286 (approximately). Step 5: So, 9 - x = 50/7. Step 6: Solve for x by subtracting 9 from both sides: -x = 50/7 - 9. Step 7: Write 9 as 63/7: -x = 50/7 - 63/7 = -13/7. Step 8: Multiply both sides by -1: x = 13/7 = 1.857142857. Step 9: The value of x is 1.857142857 meters.

  4. Noah is filling a rectangular swimming pool at his community center. The pool has a total capacity of 15,750 gallons of water. A hose fills the pool at a rate of 0.75 gallons per second. After filling for some time, a second hose is turned on that adds an additional 1.25 gallons per second. Together, the two hoses finish filling the remaining water in 2,100 seconds. How many gallons of water were already in the pool before the second hose was turned on? Answer: 11,550 Solution: Find the combined rate of the two hoses: 0.75 + 1.25 = 2.0 gallons per second. Calculate the amount of water added by both hoses in 2,100 seconds: 2.0 × 2,100 = 4,200 gallons.
    Full step-by-step solution

    Step 1: Find the combined rate of the two hoses: 0.75 + 1.25 = 2.0 gallons per second. Step 2: Calculate the amount of water added by both hoses in 2,100 seconds: 2.0 × 2,100 = 4,200 gallons. Step 3: Subtract the added water from the total capacity to find the water already in the pool: 15,750 - 4,200 = 11,550 gallons. The answer is 11,550 gallons.

  5. Emma is planning a community fundraiser and needs to mix two types of juice to make a special blend. The first juice is 40% real fruit concentrate and the second is 75% real fruit concentrate. She wants to make 12 liters of a mixture that is 55% real fruit concentrate. How many liters of the 40% concentrate juice should she use? Answer: 6.86 Solution: Let x be the liters of 40% concentrate juice. Then (12 - x) is the liters of 75% concentrate juice.
    Full step-by-step solution

    Step 1: Let x be the liters of 40% concentrate juice. Then (12 - x) is the liters of 75% concentrate juice. Step 2: Write the equation for the pure fruit concentrate: 0.40x + 0.75(12 - x) = 0.55(12) Step 3: Distribute and simplify: 0.40x + 9 - 0.75x = 6.6 Step 4: Combine like terms: -0.35x + 9 = 6.6 Step 5: Subtract 9 from both sides: -0.35x = -2.4 Step 6: Divide both sides by -0.35: x = 2.4 ÷ 0.35 Step 7: Calculate: x = 6.857... Step 8: Round to two decimal places: x = 6.86 Emma should use 6.86 liters of the 40% concentrate juice.

  6. Liam is volunteering at a community garden. He needs to prepare a new rectangular plot. The length of the plot is 3/5 of its width. If the total area of the plot must be 135 square meters, what is the width of the plot in meters? Answer: 15 Solution: Let w represent the width of the plot in meters. The length is 3/5 of the width, so length = (3/5)w. Area of a rectangle = length x width.
    Full step-by-step solution

    Step 1: Let w represent the width of the plot in meters. Step 2: The length is 3/5 of the width, so length = (3/5)w. Step 3: Area of a rectangle = length x width. So, (3/5)w x w = 135. Step 4: Simplify: (3/5)w^2 = 135. Step 5: Multiply both sides by 5 to eliminate the fraction: 3w^2 = 675. Step 6: Divide both sides by 3: w^2 = 225. Step 7: Take the positive square root (since width is positive): w = 15. The width of the plot is 15 meters.

  7. (3/5)x - 9 = (1/3)x + 7 = ? Answer: 60 Solution: Start with the equation (3/5)x - 9 = (1/3)x + 7. Find a common denominator for 5 and 3, which is 15. Multiply every term by 15: 15*(3/5)x - 15*9 = 15*(1/3)x + 15*7.
    Full step-by-step solution

    Step 1: Start with the equation (3/5)x - 9 = (1/3)x + 7. Step 2: Find a common denominator for 5 and 3, which is 15. Step 3: Multiply every term by 15: 15*(3/5)x - 15*9 = 15*(1/3)x + 15*7. Step 4: Simplify each term: (15/5)*3x - 135 = (15/3)*1x + 105 → 3*3x - 135 = 5*1x + 105 → 9x - 135 = 5x + 105. Step 5: Subtract 5x from both sides: 9x - 5x - 135 = 5x - 5x + 105 → 4x - 135 = 105. Step 6: Add 135 to both sides: 4x - 135 + 135 = 105 + 135 → 4x = 240. Step 7: Divide both sides by 4: 4x/4 = 240/4 → x = 60. The answer is 60.