Equation Word Problems

lessonbunny.com

Answer Key & Explanations

Equation Word Problems · Grade 7 · Worksheet 1

1. Emma is planning a road trip from Chicago to Denver, which is 1,000 miles. Her car's fuel efficiency is 25 miles per gallon, and gasoline costs $3.20 per gallon. She also needs to budget for tolls, which total $18.75 for the entire trip. How much will Emma spend on fuel and tolls combined for her road trip? Answer: $146.75 Solution: Distance = 1,000 miles Fuel efficiency = 25 miles per gallon Gallons needed = 1,000 ÷ 25 = 40 gallons Gas price = $3.20 per gallon Fuel cost = 40 × $3.20 = $128.00 Tolls = $18.75 Total cost = $128.00 + $18.75 = $146.75 The answer is $146.75.
   Full step-by-step solution

   Step 1: Calculate the gallons of fuel needed Distance = 1,000 miles Fuel efficiency = 25 miles per gallon Gallons needed = 1,000 ÷ 25 = 40 gallons Step 2: Calculate the fuel cost Gas price = $3.20 per gallon Fuel cost = 40 × $3.20 = $128.00 Step 3: Add the toll cost Tolls = $18.75 Total cost = $128.00 + $18.75 = $146.75 The answer is $146.75.

2. Emma is designing a rectangular garden on a coordinate grid. She plots the four corners of the garden at points A(0, 0), B(0, y), C(25, y), and D(25, 0). She then draws a straight path from corner A to corner C, dividing the garden into two triangular sections. If the area of the entire garden is 375 square meters, what is the value of y, the width of the garden? Answer: 15 Solution: Identify the length and width from the coordinates. The length is the horizontal distance from (0,0) to (25,0), which is 25 meters. The width is the vertical distance from (0,0) to (0,y), which is y meters.
   Full step-by-step solution

   Step 1: Identify the length and width from the coordinates. The length is the horizontal distance from (0,0) to (25,0), which is 25 meters. The width is the vertical distance from (0,0) to (0,y), which is y meters. Step 2: Write the area formula for a rectangle: Area = length × width. Step 3: Substitute the known area and length: 375 = 25 × y. Step 4: Solve for y by dividing both sides by 25: y = 375 ÷ 25. Step 5: Calculate: 375 ÷ 25 = 15. The answer is 15 meters.

3. Liam is designing a rectangular garden with a length-to-width ratio of 5:3. He wants the garden to have an area of exactly 240 square meters. What should be the dimensions of Liam's garden in meters? Answer: 20 meters by 12 meters Solution: The ratio of length to width is 5:3.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Define variables for length and width** The ratio of length to width is 5:3. Let length = 5x meters Let width = 3x meters --- **Step 2: Write the equation for area** Area of a rectangle = length × width Area = (5x) × (3x) = 15x² We are told the area is 240 square meters, so: 15x² = 240 --- **Step 3: Solve for x** Divide both sides by 15: x² = 240 / 15 x² = 16 Take the positive square root (since dimensions are positive): x = 4 --- **Step 4: Find length and width** Length = 5x = 5 × 4 = 20 meters Width = 3x = 3 × 4 = 12 meters --- **Step 5: Verify** Area = 20 × 12 = 240 square meters ✓ Ratio = 20 : 12 = 5 : 3 ✓ --- **Final answer:** Length = 20 meters, Width = 12 meters

4. Noah is drawing a rectangular garden on a coordinate grid. The garden has vertices at (0, 0), (15, 0), (15, 11), and (0, 11). Inside the garden, Noah draws a circular flower bed with its center at (7, 5) and a radius of 3 units. What is the area of the garden that is NOT covered by the flower bed? (Use π = 3.14) Answer: 136.74 Solution: Find the area of the rectangular garden. Length = 15 units, Width = 11 units. Area of rectangle = length × width = 15 × 11 = 165 square units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. Length = 15 units, Width = 11 units. Area of rectangle = length × width = 15 × 11 = 165 square units. Step 2: Find the area of the circular flower bed. Radius = 3 units. Area of circle = π × radius² = 3.14 × 3² = 3.14 × 9 = 28.26 square units. Step 3: Subtract the area of the flower bed from the garden area. Remaining area = 165 - 28.26 = 136.74 square units. The answer is 136.74 square units.

5. Olivia is saving money to buy a new bicycle that costs $255. She already has $75 saved. She plans to save $15 each week from her allowance. How many weeks will it take Olivia to save enough money to buy the bicycle? Answer: 12 Solution: Determine the amount Olivia still needs to save. Total cost = $255, amount saved = $75. Amount needed = 255 - 75 = $180.
   Full step-by-step solution

   Step 1: Determine the amount Olivia still needs to save. Total cost = $255, amount saved = $75. Amount needed = 255 - 75 = $180. Step 2: Let x represent the number of weeks Olivia needs to save. She saves $15 each week, so the equation is: 15x = 180. Step 3: Solve for x by dividing both sides by 15: x = 180 / 15 = 12. Step 4: It will take Olivia 12 weeks to save enough money to buy the bicycle. Final answer: 12.

6. Liam is designing a rectangular garden with a length-to-width ratio of 5:3. If the total area of the garden needs to be 240 square meters, what will be the actual length of the garden in meters? Answer: 20 Solution: Let the length be 5x and the width be 3x, because the ratio of length to width is 5:3. Write the area formula. Area = length × width Area = (5x) × (3x) = 15x^2 Set the area equal to 240.
   Full step-by-step solution

   Let the length be 5x and the width be 3x, because the ratio of length to width is 5:3. Step 1: Write the area formula. Area = length × width Area = (5x) × (3x) = 15x^2 Step 2: Set the area equal to 240. 15x^2 = 240 Step 3: Solve for x^2. x^2 = 240 / 15 x^2 = 16 Step 4: Solve for x. x = sqrt(16) x = 4 (We take the positive root since length cannot be negative.) Step 5: Find the actual length. Length = 5x = 5 × 4 = 20 meters. Therefore, the actual length of the garden is 20 meters.

7. Mere is designing a rectangular garden on a coordinate grid. The garden has vertices at (2, 3), (2, 19), (14, 19), and (14, 3). She wants to build a square pond in one corner of the garden, with its vertices at (2, 3), (9, 3), (9, 10), and (2, 10). What is the area of the garden that is not covered by the pond? Answer: 143 Solution: Find the dimensions of the rectangular garden. The length is the difference in x-coordinates: 14 - 2 = 12 units. The width is the difference in y-coordinates: 19 - 3 = 16 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangular garden. The length is the difference in x-coordinates: 14 - 2 = 12 units. The width is the difference in y-coordinates: 19 - 3 = 16 units. Area of garden = 12 * 16 = 192 square units. Step 2: Find the side length of the square pond. The side length is the difference in x-coordinates: 9 - 2 = 7 units. (You can also use y-coordinates: 10 - 3 = 7 units.) Area of pond = 7 * 7 = 49 square units. Step 3: Subtract the area of the pond from the area of the garden: 192 - 49 = 143 square units. The answer is 143.

8. Nikau and a friend together have 84 trading cards. Nikau has 3 times as many cards as the friend. How many cards does the friend have? Answer: 21 Solution: Let f = number of cards the friend has. Then Nikau has 3 × f cards. Together: f + 3f = 84 f(3 + 1) = 84 f = 84 ÷ (3 + 1) = 21 cards.
   Full step-by-step solution

   Step 1: Let f = number of cards the friend has. Step 2: Then Nikau has 3 × f cards. Step 3: Together: f + 3f = 84 Step 4: f(3 + 1) = 84 Step 5: f = 84 ÷ (3 + 1) = 21 cards.