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Rational Coefficient Equations

Grade 7 · Algebra · Worksheet 3

  1. A school is planning a field trip and needs to rent buses. Each bus can hold 42 students. If 315 students are going on the trip, how many buses are needed? Remember that you can't have a fraction of a bus. Answer: ______________
  2. Noah is designing a rectangular garden on a coordinate grid where each unit represents 1 meter. The garden has corners at (0, 0), (18, 0), (18, 14), and (0, 14). Inside the garden, a triangular flower bed is drawn with vertices at (0, 0), (18, 0), and (0, 14). If Noah wants to plant grass in the rectangular garden but NOT in the triangular flower bed, what is the area of the grassy region in square meters? Answer: ______________
  3. Hana is saving money to buy a new laptop that costs $1,200. She has already saved $360. Her grandmother gives her a gift of $150, and then Hana decides to save 2/3 of her remaining weekly allowance until she reaches her goal. If her weekly allowance is $45, how many weeks will it take Hana to save enough money for the laptop after receiving her grandmother's gift? Answer: ______________
  4. Sophia is designing a rectangular garden on a coordinate grid. The garden has corners at (0, 0), (14, 0), (14, 9), and (0, 9). She wants to create a triangular flower bed inside the garden with vertices at (0, 0), (7, 0), and (0, 5). The remaining area of the garden will be covered with grass. What is the area of the grass section? (Each unit on the grid represents 1 meter.) Answer: ______________
  5. (2/7)x + 12 = (3/7)x - 17 = ? Answer: ______________
  6. (7/9)x + 14 = (5/6)x - 3 = ? Answer: ______________
  7. (3/4)x - 7 = 5 Answer: ______________
  8. Isabella is planning a community garden and needs to build a fence around a rectangular plot. The length of the garden is 2.5 times its width. If the perimeter of the garden is 112 meters, what is the width of the garden in meters? Answer: ______________
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Answer Key & Explanations

Rational Coefficient Equations · Grade 7 · Worksheet 3

  1. A school is planning a field trip and needs to rent buses. Each bus can hold 42 students. If 315 students are going on the trip, how many buses are needed? Remember that you can't have a fraction of a bus. Answer: 8 Solution: Divide the total number of students by the capacity of each bus: 315 ÷ 42 = 7.5 Since we cannot have half a bus, we need to round up to the next whole number 7 buses would only hold 7 × 42 = 294 students, which is not enough for all 315 students 8 buses would hold 8 × 42 = 336 students, which is…
    Full step-by-step solution

    Step 1: Divide the total number of students by the capacity of each bus: 315 ÷ 42 = 7.5 Step 2: Since we cannot have half a bus, we need to round up to the next whole number Step 3: 7 buses would only hold 7 × 42 = 294 students, which is not enough for all 315 students Step 4: 8 buses would hold 8 × 42 = 336 students, which is enough for all 315 students Therefore, 8 buses are needed.

  2. Noah is designing a rectangular garden on a coordinate grid where each unit represents 1 meter. The garden has corners at (0, 0), (18, 0), (18, 14), and (0, 14). Inside the garden, a triangular flower bed is drawn with vertices at (0, 0), (18, 0), and (0, 14). If Noah wants to plant grass in the rectangular garden but NOT in the triangular flower bed, what is the area of the grassy region in square meters? Answer: 126 Solution: Find the area of the rectangular garden. Length = 18 meters, Width = 14 meters. Area of rectangle = 18 * 14 = 252 square meters.
    Full step-by-step solution

    Step 1: Find the area of the rectangular garden. Length = 18 meters, Width = 14 meters. Area of rectangle = 18 * 14 = 252 square meters. Step 2: Find the area of the triangular flower bed. The triangle has vertices at (0, 0), (18, 0), and (0, 14). The base is the side from (0, 0) to (18, 0) which is 18 meters. The height is the side from (0, 0) to (0, 14) which is 14 meters. Area of triangle = 1/2 * base * height = 1/2 * 18 * 14 = 1/2 * 252 = 126 square meters. Step 3: Subtract the area of the triangle from the area of the rectangle to find the grassy region. Grassy area = 252 - 126 = 126 square meters. The answer is 126.

  3. Hana is saving money to buy a new laptop that costs $1,200. She has already saved $360. Her grandmother gives her a gift of $150, and then Hana decides to save 2/3 of her remaining weekly allowance until she reaches her goal. If her weekly allowance is $45, how many weeks will it take Hana to save enough money for the laptop after receiving her grandmother's gift? Answer: 23 Solution: Total money Hana has after the gift: $360 + $150 = $510. Amount still needed: $1,200 - $510 = $690. Let w be the number of weeks.
    Full step-by-step solution

    Step 1: Total money Hana has after the gift: $360 + $150 = $510. Step 2: Amount still needed: $1,200 - $510 = $690. Step 3: Let w be the number of weeks. She saves 2/3 of her $45 allowance each week: (2/3) * 45 = $30 per week. Step 4: Equation: 30w = 690. Step 5: Divide both sides by 30: w = 690 / 30 = 23. Hana needs 23 weeks to save enough money.

  4. Sophia is designing a rectangular garden on a coordinate grid. The garden has corners at (0, 0), (14, 0), (14, 9), and (0, 9). She wants to create a triangular flower bed inside the garden with vertices at (0, 0), (7, 0), and (0, 5). The remaining area of the garden will be covered with grass. What is the area of the grass section? (Each unit on the grid represents 1 meter.) Answer: 108.5 Solution: Find the area of the rectangular garden. Length = 14 meters, Width = 9 meters. Area of rectangle = length × width = 14 × 9 = 126 square meters.
    Full step-by-step solution

    Step 1: Find the area of the rectangular garden. Length = 14 meters, Width = 9 meters. Area of rectangle = length × width = 14 × 9 = 126 square meters. Step 2: Find the area of the triangular flower bed. The triangle has vertices at (0, 0), (7, 0), and (0, 5). Base = 7 meters, Height = 5 meters. Area of triangle = (1/2) × base × height = (1/2) × 7 × 5 = (1/2) × 35 = 17.5 square meters. Step 3: Subtract the area of the flower bed from the total garden area: 126 - 17.5 = 108.5 square meters. The grass section has an area of 108.5 square meters.

  5. (2/7)x + 12 = (3/7)x - 17 = ? Answer: 203 Solution: Start with the equation (2/7)x + 12 = (3/7)x - 17. The common denominator is 7. Multiply every term by 7: 7*(2/7)x + 7*12 = 7*(3/7)x - 7*17.
    Full step-by-step solution

    Step 1: Start with the equation (2/7)x + 12 = (3/7)x - 17. Step 2: The common denominator is 7. Multiply every term by 7: 7*(2/7)x + 7*12 = 7*(3/7)x - 7*17. Step 3: Simplify: 2x + 84 = 3x - 119. Step 4: Subtract 2x from both sides: 2x - 2x + 84 = 3x - 2x - 119 → 84 = x - 119. Step 5: Add 119 to both sides: 84 + 119 = x - 119 + 119 → 203 = x. Step 6: The solution is x = 203. The answer is 203.

  6. (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.

  7. (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.

  8. Isabella is planning a community garden and needs to build a fence around a rectangular plot. The length of the garden is 2.5 times its width. If the perimeter of the garden is 112 meters, what is the width of the garden in meters? Answer: 16 Solution: Let w represent the width of the garden in meters. The length is 2.5 times the width, so length = 2.5w. The formula for the perimeter of a rectangle is P = 2(length + width).
    Full step-by-step solution

    Step 1: Let w represent the width of the garden in meters. The length is 2.5 times the width, so length = 2.5w. Step 2: The formula for the perimeter of a rectangle is P = 2(length + width). Substitute the given perimeter and expressions: 112 = 2(2.5w + w). Step 3: Simplify inside the parentheses: 2.5w + w = 3.5w. Step 4: The equation becomes: 112 = 2(3.5w). Step 5: Multiply: 2 * 3.5w = 7w. Step 6: So, 112 = 7w. Step 7: Divide both sides by 7: 112 / 7 = w. Step 8: w = 16. The width of the garden is 16 meters.