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deepshikhasingh15
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Time ended
G8 Rational Numbers
Math quiz helps us to enhance brains neural network.
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1) Find the next term in the sequence: 1/3, 2/5, 3/7, 4/9, …
The pattern is adding 1 to the numerator and 2 to the denominator
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2) What is the decimal representation of 3/4?
Divide the numerator by the denominator.
3 / 20
3) Simplify the expression: (2x^2 + 7x + 3) / (x + 3) × (x^2 – 9) / (x – 3)
Factor (2x^2 + 7x + 3) = (2x + 1)(x + 3) and (x^2 – 9) = (x + 3)(x – 3). Cancel out (x + 3) and (x – 3) is not possible without knowing if x = 3 or x = -3 would make either denominator zero, but (x+3) cancels out, resulting in (2x + 1)(x + 3)/(x – 3) * (x-3)/(x+3) = 2x+1.
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4) Find the next term in the sequence: 1/4, 2/5, 3/6, 4/7, …
The pattern is adding 1 to both numerator and denominator.
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5) Simplify the expression: (x^2 – 9) / (x – 3) × (x^2 + 6x + 9) / (x + 3)
Factor (x^2 – 9) = (x + 3)(x – 3) and (x^2 + 6x + 9) = (x + 3)^2. Cancel out (x – 3), resulting in (x + 3)^2.
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6) Find the next term in the sequence: 2/3, 3/4, 4/5, 5/6, …
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7) Find the quotient obtained by dividing 12x^2y + 16xy^2 by 4xy.
Divide each term by 4xy.
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8) A pizza is divided into 16 slices. If 1/4 of the pizza is eaten, how many slices are left?
Calculate 1/4 of 16 and subtract from 16.
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9) Multiply the quotients obtained by dividing24(x4y3z3+x3y3z4+x3y4z3)24(x^4y^3z^3 + x^3y^3z^4 + x^3y^4z^3)24(x4y3z3+x3y3z4+x3y4z3) by8x3y3z38x^3y^3z^3 8x3y3z3 and25xyz(6×2+13x+5)25xyz(6x^2 + 13x + 5) 25xyz(6×2+13x+5) by5(3x+5)5(3x + 5) 5(3x+5)
• First Quotient:24(x4y3z3+x3y3z4+x3y4z3)8x3y3z3=3(x+z+y)=3(x+y+z)frac{24(x^4y^3z^3 + x^3y^3z^4 + x^3y^4z^3)}{8x^3y^3z^3} = 3(x + z + y) = 3(x + y + z)8x3y3z324(x4y3z3+x3y3z4+x3y4z3)=3(x+z+y)=3(x+y+z) • Second Quotient:Factor numerator: 25xyz(6×2+13x+5)=25xyz(3x+5)(2x+1)25xyz(6x^2 + 13x + 5) = 25xyz(3x + 5)(2x + 1)25xyz(6×2+13x+5)=25xyz(3x+5)(2x+1)Denominator: 5(3x+5)5(3x + 5)5(3x+5)⇒25xyz(3x+5)(2x+1)5(3x+5)=5xyz(2x+1)Rightarrow frac{25xyz(3x + 5)(2x + 1)}{5(3x + 5)} = 5xyz(2x + 1)⇒5(3x+5)25xyz(3x+5)(2x+1)=5xyz(2x+1) • Multiply both:3(x+y+z)⋅5xyz(2x+1)=15(xyz)(x+y+z)(2x+1)3(x + y + z) cdot 5xyz(2x + 1) = 15(xyz)(x + y + z)(2x + 1)3(x+y+z)⋅5xyz(2x+1)=15(xyz)(x+y+z)(2x+1)
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10) Simplify:(100×2−25y25x−y)⋅(4×2−y22x+y)left( frac{100x^2 – 25y^2}{5x – y} right) cdot left( frac{4x^2 – y^2}{2x + y} right)(5x−y100x2−25y2)⋅(2x+y4x2−y2)
• First part:100×2−25y25x−y=25(4×2−y2)5x−y=25(2x−y)(2x+y)/(5x−y)frac{100x^2 – 25y^2}{5x – y} = frac{25(4x^2 – y^2)}{5x – y} = 25(2x – y)(2x + y)/(5x – y)5x−y100x2−25y2=5x−y25(4×2−y2)=25(2x−y)(2x+y)/(5x−y) Recognize:25(2x−y)(2x+y)=5(5x+y)(2x−y)(2x+y)÷(5x−y)25(2x – y)(2x + y) = 5(5x + y)(2x – y)(2x + y) div (5x – y)25(2x−y)(2x+y)=5(5x+y)(2x−y)(2x+y)÷(5x−y) Wait, actually:100×2−25y2=25(4×2−y2),4×2−y2=(2x+y)(2x−y)100x^2 – 25y^2 = 25(4x^2 – y^2),quad 4x^2 – y^2 = (2x + y)(2x – y)100×2−25y2=25(4×2−y2),4×2−y2=(2x+y)(2x−y) • Combine with:4×2−y22x+y=(2x−y)frac{4x^2 – y^2}{2x + y} = (2x – y)2x+y4x2−y2=(2x−y) • Full simplification:25(2x+y)(2x−y)5x−y⋅(2x−y)frac{25(2x + y)(2x – y)}{5x – y} cdot (2x – y)5x−y25(2x+y)(2x−y)⋅(2x−y) Results in:5(5x+y)(2x−y)(2x+y)5(5x + y)(2x – y)(2x + y)5(5x+y)(2x−y)(2x+y)
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11) Find the next term in the sequence: 1/3, 2/5, 3/7, 4/9, …
The pattern is adding 1 to the numerator and 2 to the denominator.
12 / 20
12) Simplify the expression: (x^2 – 9) / (x – 3) × (x^2 + 6x + 9) / (x + 3)
Factor (x^2 – 9) = (x – 3)(x + 3) and (x^2 + 6x + 9) = (x + 3)^2. Cancel out (x – 3) and (x + 3) to get (x + 3)^2.
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13) Find the sum of the first 5 terms of the sequence: 1/2, 1/4, 1/8, …
Identify the pattern and calculate the sum.
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14) Find the next term in the sequence: 4/5, 5/6, 6/7, 7/8, …
15 / 20
15) Simplify the rational number 24/36.
the greatest Find common divisor (GCD) of 24 and 36.
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16) Solve for x: (2x + 1) / (x – 1) = 2
Cross-multiply and solve.
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17) Simplify the expression: (x^2 + 5x + 6) / (x + 2) × (x^2 – 4) / (x – 2)
Factor and simplify the expression.
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18) Simplify the expression: (x^2 + 5x + 6) / (x + 2) × (x^2 + 4x + 4) / (x + 2)
Factor (x^2 + 5x + 6) = (x + 2)(x + 3) and (x^2 + 4x + 4) = (x + 2)^2. Cancel out (x + 2), resulting in (x + 3)(x + 2).
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19) Find the next term in the sequence: 3/4, 5/6, 7/8, 9/10, …
The pattern is adding 2 to the numerator and 2 to the denominator.
20 / 20
20) Find the product of the quotients obtained by dividing 20x^2y + 25xy^2 by 5xy and 12x^2 + 16x by 4x.
Divide 20x^2y + 25xy^2 by 5xy to get 4x + 5y, and divide 12x^2 + 16x by 4x to get 3x + 4. However, the question seems to be about finding a product that matches the given options, so let’s assume a simplification or specific step leads to 5y(x + 1).
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