Rational to Decimal

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Answer Key & Explanations

Rational to Decimal · Grade 7 · Worksheet 3

1. Liam is designing a rectangular garden for his school project. The length of the garden is 12.5 meters, and the width is 0.8 times the length. He needs to buy fencing that costs $4.75 per meter to enclose the entire garden. How much money, in dollars, will Liam need to spend on fencing? Answer: 213.75 Solution: Find the width of the garden. The length is given as 12.5 meters. The width is 0.8 times the length.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the width of the garden.** The length is given as 12.5 meters. The width is 0.8 times the length. Width = 0.8 × 12.5 0.8 × 12.5 = 10 So width = 10 meters. --- **Step 2: Find the perimeter of the rectangular garden.** Perimeter of a rectangle = 2 × (length + width) Perimeter = 2 × (12.5 + 10) First, 12.5 + 10 = 22.5 Then, 2 × 22.5 = 45 So perimeter = 45 meters. --- **Step 3: Calculate the total cost of fencing.** Fencing costs $4.75 per meter. Total cost = Perimeter × Cost per meter Total cost = 45 × 4.75 Let's compute 45 × 4.75: First, 45 × 4 = 180 Then, 45 × 0.75 = 45 × (3/4) = (45 × 3) / 4 = 135 / 4 = 33.75 Now add: 180 + 33.75 = 213.75 --- **Step 4: State the final answer.** Liam will need to spend $213.75 on fencing. --- **Final Answer:** 213.75

2. Liam is conducting a science experiment and needs to create a 15% saline solution. He has 1200 milliliters of pure water. How many grams of salt should he add to the water to achieve the correct concentration? Answer: 211.76 Solution: We want a 15% saline solution. Salt mass / Total mass of solution = 15% = 0.15. We have 1200 mL of pure water.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understanding the problem** We want a 15% saline solution. This means: Salt mass / Total mass of solution = 15% = 0.15. We have 1200 mL of pure water. The density of pure water is about 1 g/mL, so mass of water = 1200 g. Let \( x \) = grams of salt to add. --- **Step 2: Setting up the equation** Total mass of solution = mass of water + mass of salt = \( 1200 + x \) grams. Mass of salt in solution = \( x \) grams. Concentration = \( \frac{x}{1200 + x} = 0.15 \). --- **Step 3: Solving for \( x \)** \[ \frac{x}{1200 + x} = 0.15 \] Multiply both sides by \( 1200 + x \): \[ x = 0.15 \times (1200 + x) \] \[ x = 180 + 0.15x \] \[ x - 0.15x = 180 \] \[ 0.85x = 180 \] \[ x = \frac{180}{0.85} \] --- **Step 4: Calculating \( x \)** \[ 180 / 0.85 = 18000 / 85 \] Divide 18000 by 85: 85 × 211 = 17935, remainder 65 85 × 0.76 = 64.6 (since 85 × 0.01 = 0.85, 65/85 ≈ 0.7647) So \( x \approx 211.76 \) grams. --- **Step 5: Conclusion** Liam should add **211.76 grams** of salt to 1200 mL of pure water to make a 15% saline solution by mass. --- **Final answer:** 211.76

3. Mere draws a large square on a coordinate plane with vertices at (0, 0), (30, 0), (30, 30), and (0, 30). Inside this square, she draws a smaller square with vertices at (9, 9), (21, 9), (21, 21), and (9, 21). What fraction of the large square's area is the area of the smaller square? Express your answer as a decimal. Answer: 0.16 Solution: Find the side length of the large square. The vertices go from x=0 to x=30, so side length = 30 units. Area = 30 × 30 = 900 square units.
   Full step-by-step solution

   Step 1: Find the side length of the large square. The vertices go from x=0 to x=30, so side length = 30 units. Area = 30 × 30 = 900 square units. Step 2: Find the side length of the smaller square. The vertices go from x=9 to x=21, so side length = 21 - 9 = 12 units. Area = 12 × 12 = 144 square units. Step 3: Write the fraction of the large square's area that the small square covers: 144/900. Step 4: Simplify the fraction. Divide numerator and denominator by 12: 144 ÷ 12 = 12, 900 ÷ 12 = 75. So 144/900 = 12/75. Divide further by 3: 12/75 = 4/25. Step 5: Convert 4/25 to a decimal. 4 ÷ 25 = 0.16. The answer is 0.16.

4. Emma is conducting a science experiment to test how different concentrations of salt affect plant growth. She needs to create a solution that is 0.15 salt by weight. If she has already measured 2.8 kilograms of water, how many kilograms of salt should she add to achieve the correct concentration? Answer: 0.494 Solution: Let x be the kilograms of salt to add. The total weight of the solution will be water weight + salt weight = 2.8 + x The concentration is salt weight / total weight = x / (2.8 + x) Set this equal to the desired concentration: x / (2.8 + x) = 0.15 Multiply both sides by (2.8 + x): x = 0.15(2.8 +…
   Full step-by-step solution

   Step 1: Let x be the kilograms of salt to add. Step 2: The total weight of the solution will be water weight + salt weight = 2.8 + x Step 3: The concentration is salt weight / total weight = x / (2.8 + x) Step 4: Set this equal to the desired concentration: x / (2.8 + x) = 0.15 Step 5: Multiply both sides by (2.8 + x): x = 0.15(2.8 + x) Step 6: Distribute: x = 0.42 + 0.15x Step 7: Subtract 0.15x from both sides: x - 0.15x = 0.42 Step 8: Simplify: 0.85x = 0.42 Step 9: Divide both sides by 0.85: x = 0.42 / 0.85 Step 10: Calculate: x = 0.494117... Step 11: Round to three decimal places: x = 0.494 Emma should add 0.494 kilograms of salt.

5. A rectangular swimming pool is drawn on a coordinate plane with vertices at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked off inside the pool with vertices at (5, 3), (15, 3), and (10, 9). What is the area of the pool that is available for swimming (not including the diving area)? Answer: 210 Solution: Length = 20 units (from x=0 to x=20) Width = 12 units (from y=0 to y=12) Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (5, 3), (15, 3), and (10, 9) Base = distance between (5, 3) and (15, 3) = 10 units Height = vertical distance from base to point…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool Length = 20 units (from x=0 to x=20) Width = 12 units (from y=0 to y=12) Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Calculate the area of the triangular diving area The triangle has vertices at (5, 3), (15, 3), and (10, 9) Base = distance between (5, 3) and (15, 3) = 10 units Height = vertical distance from base to point (10, 9) = 9 - 3 = 6 units Area of triangle = 1/2 × base × height = 1/2 × 10 × 6 = 30 square units Step 3: Calculate the swimming area Swimming area = total pool area - diving area Swimming area = 240 - 30 = 210 square units The answer is 210.

6. Hana is measuring ingredients for a traditional Māori rewena bread recipe. The recipe calls for 4/9 of a cup of potato starter. Hana's measuring cup only shows decimal markings. What decimal is equivalent to 4/9? (Use a bar to show any repeating digits.) Answer: 0.4̄ Solution: To convert 4/9 to a decimal, divide 4 by 9. Step 2: 9 does not go into 4, so we add a decimal point and zeros: 4.0000 ÷ 9. Step 3: 9 goes into 40 four times (9 × 4 = 36), remainder 4.
   Full step-by-step solution

   Step 1: To convert 4/9 to a decimal, divide 4 by 9. Step 2: 9 does not go into 4, so we add a decimal point and zeros: 4.0000 ÷ 9. Step 3: 9 goes into 40 four times (9 × 4 = 36), remainder 4. Write 4 after the decimal. Step 4: Bring down another 0 to make 40. 9 goes into 40 four times again, remainder 4. Step 5: This pattern repeats forever — each time we get a remainder of 4 and another 4 in the quotient. Step 6: So 4/9 = 0.4444... which we write as 0.4̄. The answer is 0.4̄.

7. 0.75 × 1.2 ÷ 0.5 = ? Answer: 1.8 Solution: Write the problem clearly. 0.75 × 1.2 ÷ 0.5 Multiplication and division have the same priority, so we go left to right. First, multiply 0.75 × 1.2.
   Full step-by-step solution

   Let's solve step by step. Step 1: Write the problem clearly. 0.75 × 1.2 ÷ 0.5 Step 2: Multiplication and division have the same priority, so we go left to right. First, multiply 0.75 × 1.2. 0.75 × 1.2 = 0.75 × (12/10) = (75/100) × (12/10) = (75 × 12) / (100 × 10) = 900 / 1000 = 0.9 So after the first step: 0.9 ÷ 0.5 Step 3: Now divide 0.9 ÷ 0.5. 0.9 ÷ 0.5 = 0.9 / 0.5 = (9/10) / (5/10) = (9/10) × (10/5) = 9/5 = 1.8 Step 4: Final answer. 0.75 × 1.2 ÷ 0.5 = 1.8

8. Emma is conducting a science experiment to test how different salt concentrations affect plant growth. She needs to create a 0.15 saline solution using 2.8 kilograms of water. How many kilograms of salt should she add to achieve the correct concentration? Answer: 0.42 Solution: The concentration formula is: salt / (salt + water) = 0.15 Let x be the kilograms of salt needed.
   Full step-by-step solution

   Step 1: The concentration formula is: salt / (salt + water) = 0.15 Step 2: Let x be the kilograms of salt needed. Then: x / (x + 2.8) = 0.15 Step 3: Multiply both sides by (x + 2.8): x = 0.15(x + 2.8) Step 4: Distribute: x = 0.15x + 0.42 Step 5: Subtract 0.15x from both sides: 0.85x = 0.42 Step 6: Divide both sides by 0.85: x = 0.42 / 0.85 Step 7: Calculate: x = 0.42 Emma needs to add 0.42 kilograms of salt.