Rational Expressions

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Rational Expressions · Grade 9 · Worksheet 3

1. Nikau is baking cookies for a bake sale. The recipe calls for 6 cups of flour. Nikau decides to make 5 batches of the recipe. How many total cups of flour will Nikau use? Answer: 30 Solution: The recipe calls for 6 cups of flour per batch. Making 5 batches means multiplying the flour per batch by the number of batches: 6 * 5 = 30.
   Full step-by-step solution

   Step 1: The recipe calls for 6 cups of flour per batch. Step 2: Making 5 batches means multiplying the flour per batch by the number of batches: 6 * 5 = 30. Step 3: Therefore, Nikau will use 30 cups of flour in total.

2. (8/(x+7)) + (12/(x-6)) = ? Answer: (20x+36)/((x+7)(x-6)) Solution: Identify the denominators: (x+7) and (x-6) Find the common denominator: (x+7)(x-6) Rewrite each fraction with the common denominator: First fraction: (8/(x+7)) × ((x-6)/(x-6)) = 8(x-6)/((x+7)(x-6)) Second fraction: (12/(x-6)) × ((x+7)/(x+7)) = 12(x+7)/((x+7)(x-6)) Add the numerators: 8(x-6) +…
   Full step-by-step solution

   Step 1: Identify the denominators: (x+7) and (x-6) Step 2: Find the common denominator: (x+7)(x-6) Step 3: Rewrite each fraction with the common denominator: First fraction: (8/(x+7)) × ((x-6)/(x-6)) = 8(x-6)/((x+7)(x-6)) Second fraction: (12/(x-6)) × ((x+7)/(x+7)) = 12(x+7)/((x+7)(x-6)) Step 4: Add the numerators: 8(x-6) + 12(x+7) = 8x - 48 + 12x + 84 = 20x + 36 Step 5: Write the result: (20x + 36)/((x+7)(x-6)) The answer is (20x+36)/((x+7)(x-6)).

3. Tane is designing a triangular garden plot with sides of length 13 meters, 14 meters, and 15 meters. He needs to calculate the area of the triangle to determine how much soil to purchase. Using Heron's formula, what is the area of the triangular garden in square meters? Answer: 84 Solution: Calculate the semi-perimeter (s) of the triangle. s = (13 + 14 + 15) / 2 = 42 / 2 = 21 meters. Apply Heron's formula: Area = sqrt(s(s - a)(s - b)(s - c)), where a = 13, b = 14, c = 15.
   Full step-by-step solution

   Step 1: Calculate the semi-perimeter (s) of the triangle. s = (13 + 14 + 15) / 2 = 42 / 2 = 21 meters. Step 2: Apply Heron's formula: Area = sqrt(s(s - a)(s - b)(s - c)), where a = 13, b = 14, c = 15. Step 3: Calculate the terms inside the square root: s - a = 21 - 13 = 8; s - b = 21 - 14 = 7; s - c = 21 - 15 = 6. Step 4: Multiply these values: s * (s - a) * (s - b) * (s - c) = 21 * 8 * 7 * 6. Step 5: Compute the product: 21 * 8 = 168; 168 * 7 = 1176; 1176 * 6 = 7056. Step 6: Take the square root: sqrt(7056) = 84. The area of the triangular garden is 84 square meters.

4. (4x² - 16)/(x² - 4) ÷ (2x + 4)/(x + 2) = ? Answer: 2 Solution: Write the division as multiplication by the reciprocal: (4x² - 16)/(x² - 4) × (x + 2)/(2x + 4) Factor all polynomials: 4x² - 16 = 4(x² - 4) = 4(x - 2)(x + 2), x² - 4 = (x - 2)(x + 2), 2x + 4 = 2(x + 2) Rewrite the expression: [4(x - 2)(x + 2)]/[(x - 2)(x + 2)] × [(x + 2)]/[2(x + 2)] Cancel…
   Full step-by-step solution

   Step 1: Write the division as multiplication by the reciprocal: (4x² - 16)/(x² - 4) × (x + 2)/(2x + 4) Step 2: Factor all polynomials: 4x² - 16 = 4(x² - 4) = 4(x - 2)(x + 2), x² - 4 = (x - 2)(x + 2), 2x + 4 = 2(x + 2) Step 3: Rewrite the expression: [4(x - 2)(x + 2)]/[(x - 2)(x + 2)] × [(x + 2)]/[2(x + 2)] Step 4: Cancel common factors: (x - 2) cancels, (x + 2) cancels completely Step 5: Simplify: 4/2 = 2 The answer is 2.

5. (x² - 16)/(x + 4) = ? Answer: x - 4 Solution: Recognize that the numerator x² - 16 is a difference of squares.
   Full step-by-step solution

   Step 1: Recognize that the numerator x² - 16 is a difference of squares. Step 2: Factor the numerator: x² - 16 = (x + 4)(x - 4) Step 3: Rewrite the expression: [(x + 4)(x - 4)]/(x + 4) Step 4: Cancel the common factor (x + 4) from numerator and denominator Step 5: The simplified expression is x - 4 The answer is x - 4.

6. Emma is baking cookies and needs to combine two different sugar mixtures. The first mixture has (x² - 9)/(x + 3) cups of sugar, and the second mixture has (x + 3)/(x - 3) cups of sugar. She multiplies these two amounts together to determine the total sugar content. What is the simplified expression for the total number of cups of sugar? Answer: 1 Solution: Write the multiplication problem: [(x² - 9)/(x + 3)] × [(x + 3)/(x - 3)] Factor x² - 9 as (x + 3)(x - 3) Rewrite the expression: [(x + 3)(x - 3)/(x + 3)] × [(x + 3)/(x - 3)] Cancel the common (x + 3) factors: (x - 3)/(1) × (1)/(x - 3) Cancel the common (x - 3) factors: 1/1 The simplified…
   Full step-by-step solution

   Step 1: Write the multiplication problem: [(x² - 9)/(x + 3)] × [(x + 3)/(x - 3)] Step 2: Factor x² - 9 as (x + 3)(x - 3) Step 3: Rewrite the expression: [(x + 3)(x - 3)/(x + 3)] × [(x + 3)/(x - 3)] Step 4: Cancel the common (x + 3) factors: (x - 3)/(1) × (1)/(x - 3) Step 5: Cancel the common (x - 3) factors: 1/1 Step 6: The simplified expression is 1 Therefore, the total number of cups of sugar is 1.

7. (12/(x+7)) + (2/(x-2)) = ? Answer: (14x+10)/((x+7)(x-2)) Solution: Identify the denominators: (x+7) and (x-2). The common denominator is (x+7)(x-2). Rewrite the first fraction: 12/(x+7) = [12(x-2)]/[(x+7)(x-2)] = (12x-24)/[(x+7)(x-2)].
   Full step-by-step solution

   Step 1: Identify the denominators: (x+7) and (x-2). Step 2: The common denominator is (x+7)(x-2). Step 3: Rewrite the first fraction: 12/(x+7) = [12(x-2)]/[(x+7)(x-2)] = (12x-24)/[(x+7)(x-2)]. Step 4: Rewrite the second fraction: 2/(x-2) = [2(x+7)]/[(x+7)(x-2)] = (2x+14)/[(x+7)(x-2)]. Step 5: Add the numerators: (12x-24) + (2x+14) = 14x - 10. Step 6: Combine over the common denominator: (14x-10)/[(x+7)(x-2)]. Step 7: Factor the numerator: 2(7x-5). The denominator cannot be factored further with the numerator, so the simplified form is (14x-10)/[(x+7)(x-2)].