Rational Coefficient Equations

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Rational Coefficient Equations · Grade 7 · Worksheet 3

1. Aroha is sewing a quilt and needs a total of 2.5 meters of fabric. She already has 0.75 meters. The rest she will buy in pieces that are each 0.35 meters long. How many pieces does Aroha need to buy? Answer: 5 Solution: Find the remaining fabric needed: 2.5 - 0.75 = 1.75 meters. Each piece is 0.35 meters. Let x be the number of pieces.
   Full step-by-step solution

   Step 1: Find the remaining fabric needed: 2.5 - 0.75 = 1.75 meters. Step 2: Each piece is 0.35 meters. Let x be the number of pieces. The equation is 0.35x = 1.75. Step 3: Divide both sides by 0.35: x = 1.75 / 0.35. Step 4: Calculate: 1.75 / 0.35 = 175 / 35 = 5. Aroha needs to buy 5 pieces of fabric.

2. Olivia is saving money to buy a new bicycle that costs $350. She has already saved $80. She plans to earn the rest by walking dogs in her neighborhood. She charges $12.50 per dog walk. After walking some dogs, she still needs $45 more. How many dogs has Olivia walked so far? Answer: 18 Solution: Find the total amount Olivia has earned from dog walking. She needs $350 total. She already has $80, and she still needs $45.
   Full step-by-step solution

   Step 1: Find the total amount Olivia has earned from dog walking. She needs $350 total. She already has $80, and she still needs $45. So the amount earned from dog walking is: $350 - $80 - $45 = $225. Step 2: Let d represent the number of dog walks. Each walk earns $12.50, so the equation is: 12.5d = 225. Step 3: Solve for d by dividing both sides by 12.5: d = 225 / 12.5 = 18. Therefore, Olivia has walked 18 dogs so far.

3. (3/4)x - 7 = 5 Answer: 16 Solution: (3/4)x - 7 = 5 Isolate the term with x by adding 7 to both sides. We do this because -7 is on the left side, and we want to remove it. (3/4)x - 7 + 7 = 5 + 7 (3/4)x = 12 Solve for x by getting rid of the fraction (3/4).
   Full step-by-step solution

   We start with the equation: (3/4)x - 7 = 5 Step 1: Isolate the term with x by adding 7 to both sides. We do this because -7 is on the left side, and we want to remove it. (3/4)x - 7 + 7 = 5 + 7 This simplifies to: (3/4)x = 12 Step 2: Solve for x by getting rid of the fraction (3/4). Since x is multiplied by 3/4, we can multiply both sides by the reciprocal of 3/4, which is 4/3. So: (3/4)x * (4/3) = 12 * (4/3) Step 3: Simplify both sides. On the left: (3/4)*(4/3) = 1, so we have x. On the right: 12 * (4/3) = (12/1)*(4/3) = (12*4)/(1*3) = 48/3 = 16 Therefore: x = 16 We can check: (3/4)*16 - 7 = (48/4) - 7 = 12 - 7 = 5, which matches the original equation.

4. (7/9)x + 14 = (5/6)x - 3 = ? Answer: 306 Solution: Start with the equation (7/9)x + 14 = (5/6)x - 3 Find the least common denominator of 9 and 6, which is 18.
   Full step-by-step solution

   Step 1: Start with the equation (7/9)x + 14 = (5/6)x - 3 Step 2: Find the least common denominator of 9 and 6, which is 18. Step 3: Multiply every term by 18: 18*(7/9)x + 18*14 = 18*(5/6)x - 18*3 Step 4: Simplify: 14x + 252 = 15x - 54 Step 5: Subtract 14x from both sides: 14x - 14x + 252 = 15x - 14x - 54 → 252 = x - 54 Step 6: Add 54 to both sides: 252 + 54 = x - 54 + 54 → 306 = x The answer is 306.

5. Mason is designing a rectangular garden on a coordinate grid. The garden has corners at (0, 0), (18, 0), (18, 10), and (0, 10). He wants to build a straight path from corner (0, 0) to corner (18, 10). Each unit on the grid represents 1 meter. However, the path must have a constant width of 0.5 meters, so the area of the path is 0.5 times the length of the diagonal. What is the area of the path in square meters? Answer: 10.3 Solution: Find the horizontal distance: 18 - 0 = 18 meters Find the vertical distance: 10 - 0 = 10 meters Apply the Pythagorean theorem: diagonal^2 = 18^2 + 10^2 = 324 + 100 = 424 Diagonal = sqrt(424) = 20.5913...
   Full step-by-step solution

   Step 1: Find the horizontal distance: 18 - 0 = 18 meters Step 2: Find the vertical distance: 10 - 0 = 10 meters Step 3: Apply the Pythagorean theorem: diagonal^2 = 18^2 + 10^2 = 324 + 100 = 424 Step 4: Diagonal = sqrt(424) = 20.5913... meters (approximately) Step 5: Area of path = diagonal length x width = 20.5913 x 0.5 = 10.2956... square meters Step 6: Round to one decimal place: 10.3 square meters The area of the path is 10.3 square meters.

6. 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.

7. 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.

8. (1/6)x + 21 = (1/3)x - 16 = ? Answer: 222 Solution: Start with the equation (1/6)x + 21 = (1/3)x - 16. Find the least common denominator of 6 and 3, which is 6. Multiply every term by 6: 6*(1/6)x + 6*21 = 6*(1/3)x - 6*16.
   Full step-by-step solution

   Step 1: Start with the equation (1/6)x + 21 = (1/3)x - 16. Step 2: Find the least common denominator of 6 and 3, which is 6. Step 3: Multiply every term by 6: 6*(1/6)x + 6*21 = 6*(1/3)x - 6*16. Step 4: Simplify: x + 126 = 2x - 96. Step 5: Subtract x from both sides: x - x + 126 = 2x - x - 96, which gives 126 = x - 96. Step 6: Add 96 to both sides: 126 + 96 = x - 96 + 96. Step 7: Simplify: 222 = x, so x = 222. The answer is 222.