Equation Word Problems

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Equation Word Problems · Grade 7 · Worksheet 3

1. A rectangular prism is drawn with dimensions 12 cm by 8 cm by 5 cm. A smaller rectangular prism measuring 4 cm by 3 cm by 2 cm is cut out from one corner of the larger prism. What is the volume of the remaining solid? Answer: 456 cm³ Solution: Calculate the volume of the large rectangular prism Volume = length × width × height = 12 cm × 8 cm × 5 cm = 480 cm³ Calculate the volume of the smaller rectangular prism that was cut out Volume = length × width × height = 4 cm × 3 cm × 2 cm = 24 cm³ Subtract the smaller volume from the larger…
   Full step-by-step solution

   Step 1: Calculate the volume of the large rectangular prism Volume = length × width × height = 12 cm × 8 cm × 5 cm = 480 cm³ Step 2: Calculate the volume of the smaller rectangular prism that was cut out Volume = length × width × height = 4 cm × 3 cm × 2 cm = 24 cm³ Step 3: Subtract the smaller volume from the larger volume 480 cm³ - 24 cm³ = 456 cm³ The answer is 456 cm³.

2. A factory produces two types of electronic components. Type A components sell for $12,500 each and Type B components sell for $8,750 each. Last month, the factory sold 3 times as many Type A components as Type B components. If the total revenue from both types was $1,156,250, how many Type B components were sold? Answer: 25 Solution: Let’s break this down step by step. Let \( x \) = number of Type B components sold. Then number of Type A components sold = \( 3x \) (since they sold 3 times as many Type A as Type B).
   Full step-by-step solution

   Let’s break this down step by step. --- **Step 1: Define variables** Let \( x \) = number of Type B components sold. Then number of Type A components sold = \( 3x \) (since they sold 3 times as many Type A as Type B). --- **Step 2: Write the revenue equation** Price of Type A = $12,500 Price of Type B = $8,750 Revenue from Type A = \( 12500 \times 3x \) Revenue from Type B = \( 8750 \times x \) Total revenue = \( 12500 \times 3x + 8750 \times x = 1,156,250 \) --- **Step 3: Simplify the equation** First, \( 12500 \times 3x = 37500x \) Then \( 37500x + 8750x = 46250x \) So: \( 46250x = 1,156,250 \) --- **Step 4: Solve for \( x \)** Divide both sides by 46250: \( x = 1,156,250 / 46,250 \) --- **Step 5: Perform the division** First, note that \( 46,250 \times 25 = 1,156,250 \) Let’s check: \( 46,250 \times 20 = 925,000 \) \( 46,250 \times 5 = 231,250 \) Sum: \( 925,000 + 231,250 = 1,156,250 \) ✅ So \( x = 25 \). --- **Step 6: Interpret the result** \( x \) was the number of Type B components sold. So **Type B components sold = 25**. --- **Final answer:** 25

3. Kaia is saving money to buy a new laptop that costs $1,215. She already has $347 saved. She plans to save an equal amount each week for the next 14 weeks. How much money must Kaia save each week to reach her goal? Answer: 62 Solution: Determine the remaining amount Kaia needs to save. Total cost = $1,215, amount already saved = $347. Remaining amount = 1215 - 347 = $868.
   Full step-by-step solution

   Step 1: Determine the remaining amount Kaia needs to save. Total cost = $1,215, amount already saved = $347. Remaining amount = 1215 - 347 = $868. Step 2: Let x be the amount Kaia saves each week. She will save for 14 weeks, so the total saved over 14 weeks is 14x. Step 3: Write the equation: 14x = 868. Step 4: Solve for x by dividing both sides by 14: x = 868 ÷ 14 = 62. Step 5: Check: 14 × 62 = 868, and 868 + 347 = 1,215. The answer is 62.

4. Sophia is saving money to buy a new laptop that costs $1,200. She already has $320 saved. She plans to save an equal amount each week for the next 16 weeks. How much money does she need to save each week to reach her goal? Answer: 55 Solution: Let x = the amount Sophia needs to save each week. The total amount saved after 16 weeks is 320 + 16x. Set up the equation: 320 + 16x = 1200.
   Full step-by-step solution

   Step 1: Let x = the amount Sophia needs to save each week. Step 2: The total amount saved after 16 weeks is 320 + 16x. Step 3: Set up the equation: 320 + 16x = 1200. Step 4: Subtract 320 from both sides: 16x = 880. Step 5: Divide both sides by 16: x = 55. Step 6: Check: 320 + 16(55) = 320 + 880 = 1200. Correct. The answer is 55.

5. Emma is saving money to buy a new bicycle that costs $135. She already has $57. She earns $13 each week walking her neighbor's dog. How many more weeks will she need to save to have exactly enough money to buy the bicycle? Answer: 6 Solution: Determine how much more money Emma needs. She needs $135 total and has $57, so the remaining amount is $135 - $57 = $78. Let w represent the number of weeks she needs to save.
   Full step-by-step solution

   Step 1: Determine how much more money Emma needs. She needs $135 total and has $57, so the remaining amount is $135 - $57 = $78. Step 2: Let w represent the number of weeks she needs to save. Each week she saves $13, so the equation is 13w = 78. Step 3: Solve for w by dividing both sides by 13: w = 78 / 13 = 6. Step 4: Check: After 6 weeks, she saves 6 × $13 = $78. Adding her $57 gives $57 + $78 = $135, which is exactly enough. The answer is 6.

6. Mere is designing a rectangular garden on a coordinate grid. One corner of the garden is at (0, 0), and the opposite corner is at (18, 12). She wants to plant a row of flowers along the diagonal from (0, 0) to (18, 12). What is the length of the flower row? (Round to the nearest whole number.) Answer: 22 Solution: Identify the length and width of the rectangle from the coordinates. The length is the horizontal distance from (0,0) to (18,0), which is 18 units.
   Full step-by-step solution

   Step 1: Identify the length and width of the rectangle from the coordinates. The length is the horizontal distance from (0,0) to (18,0), which is 18 units. The width is the vertical distance from (0,0) to (0,12), which is 12 units. Step 2: The diagonal forms the hypotenuse of a right triangle with legs of length 18 and 12. Use the Pythagorean theorem: c^2 = a^2 + b^2. Step 3: c^2 = 18^2 + 12^2 = 324 + 144 = 468. Step 4: c = sqrt(468). Step 5: sqrt(468) = sqrt(36 * 13) = 6 * sqrt(13) ≈ 6 * 3.6055 = 21.633. Step 6: Round to the nearest whole number: 22. The answer is 22 units.

7. Olivia is designing a rectangular garden. She draws a diagram where the length of the garden is 25 meters more than its width. The perimeter of the garden is 150 meters. What is the area of the garden in square meters? Answer: 1250 Solution: Let the width of the garden be w meters. The length is 25 meters more than the width, so the length is w + 25 meters. The formula for the perimeter of a rectangle is P = 2 * length + 2 * width.
   Full step-by-step solution

   Step 1: Let the width of the garden be w meters. Step 2: The length is 25 meters more than the width, so the length is w + 25 meters. Step 3: The formula for the perimeter of a rectangle is P = 2 * length + 2 * width. Step 4: Substitute the given perimeter and expressions: 150 = 2*(w + 25) + 2*w Step 5: Simplify the equation: 150 = 2w + 50 + 2w Step 6: Combine like terms: 150 = 4w + 50 Step 7: Subtract 50 from both sides: 100 = 4w Step 8: Divide both sides by 4: w = 25 Step 9: The width is 25 meters. The length is w + 25 = 25 + 25 = 50 meters. Step 10: The area of a rectangle is length * width = 50 * 25 = 1250 square meters. The answer is 1250.