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Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Let us consider a fraction whose denominator is smaller than the squar...
Given:
Let the fraction be represented as x/y, where y = x^2 - 1.
To find:
The fraction x/y.
Solution:
Step 1: Setting up the equation
If we add 2 to the numerator and denominator, the fraction will exceed 31. This can be represented as:
(x + 2)/(y + 2) > 31
If we subtract 3 from the numerator and denominator, the fraction will be smaller than 101. This can be represented as:
(x - 3)/(y - 3) < />
Step 2: Finding the value of y
From the given information, we have y = x^2 - 1.
Step 3: Simplifying the equations
Let's simplify the first equation:
(x + 2)/(y + 2) > 31
(x + 2)/(x^2 + 2) > 31
(x + 2) > 31(x^2 + 2)
x + 2 > 31x^2 + 62
31x^2 - x + 60 < />
Now, let's simplify the second equation:
(x - 3)/(y - 3) < />
(x - 3)/(x^2 - 4) < />
(x - 3) < 101(x^2="" -="" />
(x - 3) < 101x^2="" -="" />
101x^2 - x - 401 > 0
Step 4: Solving the quadratic inequalities
To solve the quadratic inequalities, we need to find the roots of the quadratic equations:
31x^2 - x + 60 = 0
101x^2 - x - 401 = 0
Using the quadratic formula, we can find the roots:
For 31x^2 - x + 60 = 0:
x = (-(-1) ± √((-1)^2 - 4(31)(60))) / (2(31))
x = (1 ± √(1 + 7440)) / 62
x = (1 ± √7441) / 62
For 101x^2 - x - 401 = 0:
x = (-(-1) ± √((-1)^2 - 4(101)(-401))) / (2(101))
x = (1 ± √(1 + 161604)) / 202
x = (1 ± √161605) / 202
Step 5: Determining the possible values of x
From the above equations, we can see that x can take two possible values: (1 + √7441) / 62 and (1 + √161605) / 202.
Step 6: Finding the denominator y
Since y = x^2 - 1, we can substitute the values of x to find the corresponding values of y.
For x = (1 + √7441) / 62:
y = ((1 + √7441) / 62)^2 - 1
For x = (1 + √161605) / 202:
y = ((
Let the fraction be represented as x/y, where y = x^2 - 1.
To find:
The fraction x/y.
Solution:
Step 1: Setting up the equation
If we add 2 to the numerator and denominator, the fraction will exceed 31. This can be represented as:
(x + 2)/(y + 2) > 31
If we subtract 3 from the numerator and denominator, the fraction will be smaller than 101. This can be represented as:
(x - 3)/(y - 3) < />
Step 2: Finding the value of y
From the given information, we have y = x^2 - 1.
Step 3: Simplifying the equations
Let's simplify the first equation:
(x + 2)/(y + 2) > 31
(x + 2)/(x^2 + 2) > 31
(x + 2) > 31(x^2 + 2)
x + 2 > 31x^2 + 62
31x^2 - x + 60 < />
Now, let's simplify the second equation:
(x - 3)/(y - 3) < />
(x - 3)/(x^2 - 4) < />
(x - 3) < 101(x^2="" -="" />
(x - 3) < 101x^2="" -="" />
101x^2 - x - 401 > 0
Step 4: Solving the quadratic inequalities
To solve the quadratic inequalities, we need to find the roots of the quadratic equations:
31x^2 - x + 60 = 0
101x^2 - x - 401 = 0
Using the quadratic formula, we can find the roots:
For 31x^2 - x + 60 = 0:
x = (-(-1) ± √((-1)^2 - 4(31)(60))) / (2(31))
x = (1 ± √(1 + 7440)) / 62
x = (1 ± √7441) / 62
For 101x^2 - x - 401 = 0:
x = (-(-1) ± √((-1)^2 - 4(101)(-401))) / (2(101))
x = (1 ± √(1 + 161604)) / 202
x = (1 ± √161605) / 202
Step 5: Determining the possible values of x
From the above equations, we can see that x can take two possible values: (1 + √7441) / 62 and (1 + √161605) / 202.
Step 6: Finding the denominator y
Since y = x^2 - 1, we can substitute the values of x to find the corresponding values of y.
For x = (1 + √7441) / 62:
y = ((1 + √7441) / 62)^2 - 1
For x = (1 + √161605) / 202:
y = ((
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Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer?
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Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer?.
Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 31; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 101. Find the fraction. a)3/8 b)4/15 c)5/24 d)None Correct answer is option 'D'. Can you explain this answer?.
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