Rational Expressions

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Rational Expressions · Grade 9 · Worksheet 1

1. Matiu has 50 pencils in a box. Matiu gives 13 pencils to a friend. How many pencils does Matiu have left in the box? Answer: 37 Solution: Start with 50 pencils. Remove 13 pencils given away: 50 - 13 = 37. Matiu has 37 pencils remaining.
   Full step-by-step solution

   Step 1: Start with 50 pencils. Step 2: Remove 13 pencils given away: 50 - 13 = 37. Step 3: Matiu has 37 pencils remaining.

2. (x² - 121)/(x + 11) = ? Answer: x - 11 Solution: Recognize that the numerator x² - 121 is a difference of squares Factor the numerator: x² - 121 = (x + 11)(x - 11) Rewrite the expression: (x + 11)(x - 11)/(x + 11) Cancel the common factor (x + 11) from numerator and denominator The simplified expression is x - 11 The answer is x - 11.
   Full step-by-step solution

   Step 1: Recognize that the numerator x² - 121 is a difference of squares Step 2: Factor the numerator: x² - 121 = (x + 11)(x - 11) Step 3: Rewrite the expression: (x + 11)(x - 11)/(x + 11) Step 4: Cancel the common factor (x + 11) from numerator and denominator Step 5: The simplified expression is x - 11 The answer is x - 11.

3. Liam is making a fruit salad for a school event. The recipe calls for (x^2 - 100)/(x + 10) cups of sliced strawberries and (x^2 + 20x + 100)/(x + 10) cups of blueberries. How many total cups of fruit does Liam need for his fruit salad? Answer: 2x Solution: Write the expression for total fruit: (x^2 - 100)/(x + 10) + (x^2 + 20x + 100)/(x + 10) Since both fractions have the same denominator, combine the numerators: (x^2 - 100 + x^2 + 20x + 100)/(x + 10) Simplify the numerator: (2x^2 + 20x)/(x + 10) Factor the numerator: 2x(x + 10)/(x + 10) Cancel…
   Full step-by-step solution

   Step 1: Write the expression for total fruit: (x^2 - 100)/(x + 10) + (x^2 + 20x + 100)/(x + 10) Step 2: Since both fractions have the same denominator, combine the numerators: (x^2 - 100 + x^2 + 20x + 100)/(x + 10) Step 3: Simplify the numerator: (2x^2 + 20x)/(x + 10) Step 4: Factor the numerator: 2x(x + 10)/(x + 10) Step 5: Cancel the common factor (x + 10): 2x Step 6: The total cups of fruit needed is 2x.

4. (9/(x+4)) + (11/(x-5)) = ? Answer: (20x-1)/((x+4)(x-5)) Solution: Identify the denominators: (x+4) and (x-5) The common denominator is (x+4)(x-5) Rewrite each fraction with the common denominator: First fraction: (9/(x+4)) × ((x-5)/(x-5)) = 9(x-5)/((x+4)(x-5)) Second fraction: (11/(x-5)) × ((x+4)/(x+4)) = 11(x+4)/((x+4)(x-5)) Add the numerators: 9(x-5) +…
   Full step-by-step solution

   Step 1: Identify the denominators: (x+4) and (x-5) Step 2: The common denominator is (x+4)(x-5) Step 3: Rewrite each fraction with the common denominator: First fraction: (9/(x+4)) × ((x-5)/(x-5)) = 9(x-5)/((x+4)(x-5)) Second fraction: (11/(x-5)) × ((x+4)/(x+4)) = 11(x+4)/((x+4)(x-5)) Step 4: Add the numerators: 9(x-5) + 11(x+4) = 9x - 45 + 11x + 44 = 20x - 1 Step 5: Write the result: (20x-1)/((x+4)(x-5)) The answer is (20x-1)/((x+4)(x-5)).

5. Mere is making a special fruit punch for a party. The recipe requires (x² - 25)/(x² + 4x - 5) cups of cranberry juice and (x + 5)/(x - 1) cups of orange juice. How many total cups of juice does Mere need to combine for the fruit punch? Express your answer as a single simplified rational expression. Answer: (2x+10)/(x-1) Solution: Write the expression to add: (x² - 25)/(x² + 4x - 5) + (x + 5)/(x - 1) Factor the denominators and numerators where possible: x² - 25 = (x - 5)(x + 5) and x² + 4x - 5 = (x + 5)(x - 1) Rewrite the first fraction: [(x - 5)(x + 5)]/[(x + 5)(x - 1)] = (x - 5)/(x - 1) Now we have: (x - 5)/(x - 1) +…
   Full step-by-step solution

   Step 1: Write the expression to add: (x² - 25)/(x² + 4x - 5) + (x + 5)/(x - 1) Step 2: Factor the denominators and numerators where possible: x² - 25 = (x - 5)(x + 5) and x² + 4x - 5 = (x + 5)(x - 1) Step 3: Rewrite the first fraction: [(x - 5)(x + 5)]/[(x + 5)(x - 1)] = (x - 5)/(x - 1) Step 4: Now we have: (x - 5)/(x - 1) + (x + 5)/(x - 1) Step 5: Since both fractions have the same denominator, add the numerators: [(x - 5) + (x + 5)]/(x - 1) Step 6: Simplify the numerator: (x - 5 + x + 5)/(x - 1) = (2x)/(x - 1) Step 7: The simplified expression is (2x)/(x - 1)

6. (5/(x+5)) + (10/(x-5)) = ? Answer: (15x+25)/((x+5)(x-5)) Solution: Identify the denominators: (x+5) and (x-5) The common denominator is (x+5)(x-5) Rewrite each fraction with the common denominator: 5/(x+5) = 5(x-5)/[(x+5)(x-5)] = (5x-25)/[(x+5)(x-5)] 10/(x-5) = 10(x+5)/[(x+5)(x-5)] = (10x+50)/[(x+5)(x-5)] Add the numerators: (5x-25) + (10x+50) = 15x+25 Write…
   Full step-by-step solution

   Step 1: Identify the denominators: (x+5) and (x-5) Step 2: The common denominator is (x+5)(x-5) Step 3: Rewrite each fraction with the common denominator: 5/(x+5) = 5(x-5)/[(x+5)(x-5)] = (5x-25)/[(x+5)(x-5)] 10/(x-5) = 10(x+5)/[(x+5)(x-5)] = (10x+50)/[(x+5)(x-5)] Step 4: Add the numerators: (5x-25) + (10x+50) = 15x+25 Step 5: Write the result: (15x+25)/[(x+5)(x-5)] The answer is (15x+25)/((x+5)(x-5))

7. (5x/(x² - 25)) + (10/(x + 5)) = ? Answer: (15x - 50)/(x² - 25) Solution: Factor the denominator x² - 25 = (x + 5)(x - 5) Rewrite the expression with factored denominator: 5x/((x + 5)(x - 5)) + 10/(x + 5) The common denominator is (x + 5)(x - 5) Multiply the second term by (x - 5)/(x - 5) to get common denominator: 10(x - 5)/((x + 5)(x - 5)) Now we have: 5x/((x + 5)(x…
   Full step-by-step solution

   Step 1: Factor the denominator x² - 25 = (x + 5)(x - 5) Step 2: Rewrite the expression with factored denominator: 5x/((x + 5)(x - 5)) + 10/(x + 5) Step 3: The common denominator is (x + 5)(x - 5) Step 4: Multiply the second term by (x - 5)/(x - 5) to get common denominator: 10(x - 5)/((x + 5)(x - 5)) Step 5: Now we have: 5x/((x + 5)(x - 5)) + 10(x - 5)/((x + 5)(x - 5)) Step 6: Add the numerators: (5x + 10(x - 5))/((x + 5)(x - 5)) Step 7: Expand the numerator: (5x + 10x - 50)/((x + 5)(x - 5)) Step 8: Combine like terms: (15x - 50)/((x + 5)(x - 5)) Step 9: Write the denominator in standard form: (15x - 50)/(x² - 25) The answer is (15x - 50)/(x² - 25).

8. Kaia is making a special fruit punch for a party. The recipe calls for (x^2 - 81)/(x^2 - 18x + 81) cups of cranberry juice and (x + 9)/(x - 9) cups of orange juice. How many total cups of juice does Kaia need to combine for her fruit punch? Answer: 2 Solution: Write the expression: (x^2 - 81)/(x^2 - 18x + 81) + (x + 9)/(x - 9) Factor each part: x^2 - 81 = (x + 9)(x - 9) and x^2 - 18x + 81 = (x - 9)^2 Rewrite the first fraction: [(x + 9)(x - 9)]/(x - 9)^2 = (x + 9)/(x - 9) Now we have: (x + 9)/(x - 9) + (x + 9)/(x - 9) Since the denominators are the…
   Full step-by-step solution

   Step 1: Write the expression: (x^2 - 81)/(x^2 - 18x + 81) + (x + 9)/(x - 9) Step 2: Factor each part: x^2 - 81 = (x + 9)(x - 9) and x^2 - 18x + 81 = (x - 9)^2 Step 3: Rewrite the first fraction: [(x + 9)(x - 9)]/(x - 9)^2 = (x + 9)/(x - 9) Step 4: Now we have: (x + 9)/(x - 9) + (x + 9)/(x - 9) Step 5: Since the denominators are the same, add the numerators: (x + 9 + x + 9)/(x - 9) = (2x + 18)/(x - 9) Step 6: Factor the numerator: 2(x + 9)/(x - 9) Step 7: Notice this equals 2 × [(x + 9)/(x - 9)] Step 8: Since we started with two equal terms of (x + 9)/(x - 9), the sum is 2 The answer is 2.

9. Matiu is preparing a special drink for his friends. He mixes 4/(x+2) liters of mango juice with 6/(x+2) liters of pineapple juice. How many liters of mixed juice does Matiu have in total? Answer: 10/(x+2) Solution: Write the expression for the total juice: 4/(x+2) + 6/(x+2) Since the denominators are the same, add the numerators: (4 + 6)/(x+2) Simplify the numerator: 10/(x+2) The total amount of mixed juice is 10/(x+2) liters.
   Full step-by-step solution

   Step 1: Write the expression for the total juice: 4/(x+2) + 6/(x+2) Step 2: Since the denominators are the same, add the numerators: (4 + 6)/(x+2) Step 3: Simplify the numerator: 10/(x+2) The total amount of mixed juice is 10/(x+2) liters.