Rational Coefficient Equations

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Answer Key & Explanations

Rational Coefficient Equations · Grade 7 · Worksheet 1

1. Liam is designing a rectangular garden with a length that is 3/4 of its width. If the perimeter of the garden is 42 meters, what is the area of the garden in square meters? Answer: 108 Solution: Let the width of the garden be \( w \) meters. The length is \( 3/4 \) of the width, so length \( l = (3/4)w \). The perimeter \( P \) of a rectangle is \( P = 2l + 2w \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is \( 3/4 \) of the width, so length \( l = (3/4)w \). --- **Step 2: Write the perimeter equation** The perimeter \( P \) of a rectangle is \( P = 2l + 2w \). We are told \( P = 42 \). Substitute \( l = (3/4)w \) into the perimeter formula: \[ 2\left( \frac{3}{4}w \right) + 2w = 42 \] --- **Step 3: Simplify and solve for \( w \)** First term: \( 2 \times \frac{3}{4}w = \frac{6}{4}w = \frac{3}{2}w \). So: \[ \frac{3}{2}w + 2w = 42 \] Write \( 2w \) as \( \frac{4}{2}w \) to combine: \[ \frac{3}{2}w + \frac{4}{2}w = \frac{7}{2}w = 42 \] Multiply both sides by \( 2/7 \): \[ w = 42 \times \frac{2}{7} \] \[ w = 6 \times 2 = 12 \] So width \( w = 12 \) meters. --- **Step 4: Find length** \[ l = \frac{3}{4} \times 12 = 9 \ \text{meters} \] --- **Step 5: Find area** Area \( A = l \times w = 9 \times 12 = 108 \) square meters. --- **Final Answer:** 108

2. Noah is saving money to buy a new video game console that costs $248. He already has $61 saved. He plans to mow lawns in his neighborhood for $15.50 per lawn. After mowing a certain number of lawns, he will have exactly enough money to buy the console. How many lawns does Noah need to mow? Answer: 12.06 Solution: Let x be the number of lawns Noah needs to mow. Noah earns 15.50x dollars from mowing x lawns. He already has $61, so total money = 15.50x + 61.
   Full step-by-step solution

   Step 1: Let x be the number of lawns Noah needs to mow. Step 2: Noah earns 15.50x dollars from mowing x lawns. Step 3: He already has $61, so total money = 15.50x + 61. Step 4: This must equal the cost of the console, $248. Step 5: Equation: 15.50x + 61 = 248. Step 6: Subtract 61 from both sides: 15.50x = 187. Step 7: Divide both sides by 15.50: x = 187 / 15.50. Step 8: Calculate: 187 / 15.50 = 12.0645... Step 9: Round to two decimal places: x = 12.06. Noah needs to mow approximately 12.06 lawns. Since he cannot mow a fraction of a lawn, he would need to mow 13 lawns to have enough money.

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. (3/4)x - 7 = (1/2)x + 9 = ? Answer: 64 Solution: Start with the equation (3/4)x - 7 = (1/2)x + 9 Find a common denominator for the fractions (4 and 2), which is 4 Multiply every term by 4 to eliminate denominators: 4*(3/4)x - 4*7 = 4*(1/2)x + 4*9 Simplify: 3x - 28 = 2x + 36 Subtract 2x from both sides: 3x - 2x - 28 = 2x - 2x + 36 Simplify: x -…
   Full step-by-step solution

   Step 1: Start with the equation (3/4)x - 7 = (1/2)x + 9 Step 2: Find a common denominator for the fractions (4 and 2), which is 4 Step 3: Multiply every term by 4 to eliminate denominators: 4*(3/4)x - 4*7 = 4*(1/2)x + 4*9 Step 4: Simplify: 3x - 28 = 2x + 36 Step 5: Subtract 2x from both sides: 3x - 2x - 28 = 2x - 2x + 36 Step 6: Simplify: x - 28 = 36 Step 7: Add 28 to both sides: x - 28 + 28 = 36 + 28 Step 8: Simplify: x = 64 The answer is 64.

5. Matiu is saving money to buy a new surfboard. He has already saved $48.00. Each week he earns money by walking dogs, and he saves 2/3 of his weekly earnings. After 6 weeks, his total savings for the surfboard are $120.00. How much money does Matiu earn each week from walking dogs? Answer: 18 Solution: Let w represent Matiu's weekly earnings in dollars. Each week, he saves 2/3 of his earnings, so weekly savings = (2/3)w. Over 6 weeks, total savings from walking dogs = 6 * (2/3)w = (12/3)w = 4w.
   Full step-by-step solution

   Step 1: Let w represent Matiu's weekly earnings in dollars. Step 2: Each week, he saves 2/3 of his earnings, so weekly savings = (2/3)w. Step 3: Over 6 weeks, total savings from walking dogs = 6 * (2/3)w = (12/3)w = 4w. Step 4: His total savings after 6 weeks = initial $48 + 4w = $120. Step 5: Write the equation: 48 + 4w = 120. Step 6: Subtract 48 from both sides: 4w = 72. Step 7: Divide both sides by 4: w = 18. Step 8: Matiu earns $18 each week from walking dogs. The answer is 18.

6. A triangular garden is drawn on a coordinate plane with vertices at (0, 0), (9, 0), and (0, 7). Each unit on the grid represents 1 meter. A path with a width of 0.5 meters is to be built along the entire base of the triangle from (0, 0) to (9, 0). If the area of the path is given by 0.5 * base, and the remaining garden area must be at least 25 square meters, what is the value of x in the equation 0.5 * (9 - x) * 7 = 25, where x represents the number of meters of base covered by the path? Solve for x. Answer: 1.857142857 Solution: Write the equation from the problem: 0.5 * (9 - x) * 7 = 25. Multiply 0.5 by 7 to simplify: 0.5 * 7 = 3.5, so the equation becomes 3.5 * (9 - x) = 25. Divide both sides by 3.5: (9 - x) = 25 / 3.5.
   Full step-by-step solution

   Step 1: Write the equation from the problem: 0.5 * (9 - x) * 7 = 25. Step 2: Multiply 0.5 by 7 to simplify: 0.5 * 7 = 3.5, so the equation becomes 3.5 * (9 - x) = 25. Step 3: Divide both sides by 3.5: (9 - x) = 25 / 3.5. Step 4: Calculate 25 divided by 3.5: 25 / 3.5 = 250 / 35 = 50 / 7 = 7.14285714286 (approximately). Step 5: So, 9 - x = 50/7. Step 6: Solve for x by subtracting 9 from both sides: -x = 50/7 - 9. Step 7: Write 9 as 63/7: -x = 50/7 - 63/7 = -13/7. Step 8: Multiply both sides by -1: x = 13/7 = 1.857142857. Step 9: The value of x is 1.857142857 meters.

7. (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. Multiply every term by 18: 18*(7/9)x - 18*14 = 18*(5/6)x + 18*3.
   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 each term: 18/9 = 2, so 2*7x = 14x. 18*14 = 252. 18/6 = 3, so 3*5x = 15x. 18*3 = 54. Step 5: The equation becomes 14x - 252 = 15x + 54. Step 6: Subtract 14x from both sides: 14x - 14x - 252 = 15x - 14x + 54, which simplifies to -252 = x + 54. Step 7: Subtract 54 from both sides: -252 - 54 = x + 54 - 54, which simplifies to -306 = x. Step 8: The solution is x = -306. The answer is -306.