Rational Coefficient Equations

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Rational Coefficient Equations · Grade 7 · Worksheet 1

1. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked inside the pool with vertices at (0, 0), (8, 0), and (0, 6). What is the area of the swimming pool that is NOT part of the diving area? (Each unit on the grid represents 1 meter.) Answer: 216 Solution: Calculate the area of the rectangular swimming pool. The rectangle has length 20 meters and width 12 meters. Area of rectangle = length × width = 20 × 12 = 240 square meters.
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular swimming pool. The rectangle has length 20 meters and width 12 meters. Area of rectangle = length × width = 20 × 12 = 240 square meters. Step 2: Calculate the area of the triangular diving area. The triangle has base 8 meters and height 6 meters. Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = (1/2) × 48 = 24 square meters. Step 3: Subtract the triangle area from the rectangle area to find the remaining pool area. Remaining area = rectangle area - triangle area = 240 - 24 = 216 square meters. The answer is 216.

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 × 16 + 2/3 × 15 = ? Answer: 22 Solution: 3/4 × 16 + 2/3 × 15 Calculate 3/4 × 16 3/4 × 16 means (3 × 16) / 4. 3 × 16 = 48 48 / 4 = 12 So, 3/4 × 16 = 12 Calculate 2/3 × 15 2/3 × 15 means (2 × 15) / 3.
   Full step-by-step solution

   Let's solve step by step. We have: 3/4 × 16 + 2/3 × 15 --- **Step 1: Calculate 3/4 × 16** 3/4 × 16 means (3 × 16) / 4. 3 × 16 = 48 48 / 4 = 12 So, 3/4 × 16 = 12 --- **Step 2: Calculate 2/3 × 15** 2/3 × 15 means (2 × 15) / 3. 2 × 15 = 30 30 / 3 = 10 So, 2/3 × 15 = 10 --- **Step 3: Add the results** 12 + 10 = 22 --- **Final Answer:** 22

4. Emma is saving money to buy a new bicycle that costs $255. She already has $45 saved. If she saves 2/5 of her weekly allowance of $35 each week, how many more weeks will it take her to have enough money to buy the bicycle? Answer: 15 Solution: Calculate how much more money Emma needs. Total cost of bicycle: $255 Money already saved: $45 Amount still needed: $255 - $45 = $210 Calculate how much Emma saves each week.
   Full step-by-step solution

   Step 1: Calculate how much more money Emma needs. Total cost of bicycle: $255 Money already saved: $45 Amount still needed: $255 - $45 = $210 Step 2: Calculate how much Emma saves each week. Weekly allowance: $35 Fraction saved: 2/5 Weekly savings: (2/5) × $35 = (2 × 35)/5 = 70/5 = $14 Step 3: Calculate the number of weeks needed. Amount needed: $210 Weekly savings: $14 Number of weeks: $210 ÷ $14 = 15 Emma will need 15 more weeks to save enough money for the bicycle.

5. Emma is planning a road trip from her home to her grandmother's house, which is 360 miles away. Her car's fuel efficiency is 30 miles per gallon, and gasoline costs $3.75 per gallon. If she also needs to pay a $12.50 toll during the trip, how much will the entire trip cost? Answer: 57.50 Solution: Distance = 360 miles Fuel efficiency = 30 miles per gallon Gallons needed = 360 ÷ 30 = 12 gallons Gasoline price = $3.75 per gallon Fuel cost = 12 × $3.75 = $45.00 Toll = $12.50 Total cost = $45.00 + $12.50 = $57.50 The answer is $57.50.
   Full step-by-step solution

   Step 1: Calculate the gallons of gasoline needed Distance = 360 miles Fuel efficiency = 30 miles per gallon Gallons needed = 360 ÷ 30 = 12 gallons Step 2: Calculate the fuel cost Gasoline price = $3.75 per gallon Fuel cost = 12 × $3.75 = $45.00 Step 3: Add the toll cost Toll = $12.50 Total cost = $45.00 + $12.50 = $57.50 The answer is $57.50.

6. Harper is baking cookies for a school fundraiser. The recipe calls for 1 cup of sugar for each batch. If Harper uses 8 cups of sugar total, how many batches of cookies did Harper make? Answer: 8 Solution: Identify the total sugar used: 8 cups. Identify the sugar per batch: 1 cup. Divide total sugar by sugar per batch: 8 ÷ 1 = 8 batches.
   Full step-by-step solution

   Step 1: Identify the total sugar used: 8 cups. Step 2: Identify the sugar per batch: 1 cup. Step 3: Divide total sugar by sugar per batch: 8 ÷ 1 = 8 batches. The answer is 8 batches.

7. (3/4)x - 7 = 11 Answer: 24 Solution: We are solving the equation: (3/4)x - 7 = 11 Isolate the term with x by adding 7 to both sides. We do this because -7 is being subtracted from (3/4)x, so the opposite operation is to add 7.
   Full step-by-step solution

   We are solving the equation: (3/4)x - 7 = 11 Step 1: Isolate the term with x by adding 7 to both sides. We do this because -7 is being subtracted from (3/4)x, so the opposite operation is to add 7. (3/4)x - 7 + 7 = 11 + 7 This simplifies to: (3/4)x = 18 Step 2: Solve for x by getting rid of the fraction 3/4. Since x is multiplied by 3/4, we do the opposite operation, which is multiplying by the reciprocal of 3/4. The reciprocal of 3/4 is 4/3. Multiply both sides by 4/3: (4/3) * (3/4)x = 18 * (4/3) Step 3: Simplify both sides. On the left: (4/3) * (3/4) = 1, so we have 1*x or just x. On the right: 18 * (4/3) = (18/1) * (4/3) = (18 * 4) / (1 * 3) = 72 / 3 = 24 So: x = 24 Step 4: Check the solution (optional but good practice). Substitute x = 24 into the original equation: (3/4)*24 - 7 = (72/4) - 7 = 18 - 7 = 11 This matches the original equation's right-hand side, so the solution is correct. Final answer: x = 24

8. A rectangular playground is drawn on a coordinate plane with corners at (0, 0), (16, 0), (16, 11), and (0, 11). A diagonal path runs from (0, 0) to (16, 11), and another diagonal path runs from (16, 0) to (0, 11). The paths cross at the center. Noah needs to place a light post at the crossing point. What are the coordinates of the crossing point? (Each unit represents 1 meter.) Answer: (8, 5.5) Solution: The two diagonals of a rectangle always intersect at the center of the rectangle. Step 2: The center point's x-coordinate is the average of the x-coordinates of opposite corners: (0 + 16)/2 = 16/2 = 8.
   Full step-by-step solution

   Step 1: The two diagonals of a rectangle always intersect at the center of the rectangle. Step 2: The center point's x-coordinate is the average of the x-coordinates of opposite corners: (0 + 16)/2 = 16/2 = 8. Step 3: The center point's y-coordinate is the average of the y-coordinates of opposite corners: (0 + 11)/2 = 11/2 = 5.5. Step 4: Therefore, the coordinates of the crossing point are (8, 5.5). The answer is (8, 5.5).