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A fraction is such that if it is squared and then the numerator is divided by 2, while the denominator is increased by 25%, the new fraction thus obtained is 4 times the original fraction. What is the new fraction?
- a)8
- b)None of these
- c)50
- d)10
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A fraction is such that if it is squared and then the numerator is div...
According to the given information, if we square the fraction and perform the specified operations, we get:
((a/b)^2) = (a^2)/(b^2)
Numerator after division by 2: (a^2)/2
Denominator after increasing by 25%: (5/4)b
Numerator after division by 2: (a^2)/2
Denominator after increasing by 25%: (5/4)b
The new fraction obtained is 4 times the original fraction, so we can set up the following equation:
4(a/b) = (a^2)/2 / ((5/4)b)
To simplify, let's multiply both sides of the equation by 2b and 4 to eliminate the fractions:
8ab = a^2 / (5/4)
8ab = (4/5) * a^2
8ab = (4/5) * a^2
Now, let's solve for a/b:
8ab = (4/5) * a^2
40ab = 4a^2
10b = a
40ab = 4a^2
10b = a
Substituting the value of a into the original fraction, we get:
New fraction = (10b)/b = 10
Therefore, the new fraction is 10.
Most Upvoted Answer
A fraction is such that if it is squared and then the numerator is div...
Understanding the Problem
To solve the problem, we need to find a fraction \( \frac{a}{b} \) such that when squared, and manipulated as described, results in a new fraction that is four times the original fraction.
Setting Up the Equation
1. The original fraction is \( \frac{a}{b} \).
2. Squaring the fraction gives \( \left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2} \).
3. Dividing the numerator by 2 results in \( \frac{a^2/2}{b^2} \).
4. Increasing the denominator by 25% means multiplying by 1.25, leading to:
\[ \text{New fraction} = \frac{a^2/2}{b^2 \times 1.25} = \frac{a^2/2}{(5/4)b^2} = \frac{2a^2}{5b^2} \]
Condition Given
According to the problem, this new fraction equals 4 times the original fraction:
\[
\frac{2a^2}{5b^2} = 4 \times \frac{a}{b}
\]
Cross-Multiplying
Cross-multiplying gives:
\[
2a^2 = 20 \frac{ab^2}{1} \implies 2a^2 = 20ab \implies a^2 = 10ab
\]
Rearranging the Equation
Rearranging gives:
\[
a^2 - 10ab = 0 \implies a(a - 10b) = 0
\]
This implies \( a = 0 \) or \( a = 10b \). Since we're looking for a non-zero fraction, we take \( a = 10b \).
Finding the New Fraction
Substituting \( a = 10b \) back into the original fraction:
\[
\frac{10b}{b} = 10
\]
Thus, the new fraction is \( 10 \).
Conclusion
The correct answer is option D: 10.
To solve the problem, we need to find a fraction \( \frac{a}{b} \) such that when squared, and manipulated as described, results in a new fraction that is four times the original fraction.
Setting Up the Equation
1. The original fraction is \( \frac{a}{b} \).
2. Squaring the fraction gives \( \left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2} \).
3. Dividing the numerator by 2 results in \( \frac{a^2/2}{b^2} \).
4. Increasing the denominator by 25% means multiplying by 1.25, leading to:
\[ \text{New fraction} = \frac{a^2/2}{b^2 \times 1.25} = \frac{a^2/2}{(5/4)b^2} = \frac{2a^2}{5b^2} \]
Condition Given
According to the problem, this new fraction equals 4 times the original fraction:
\[
\frac{2a^2}{5b^2} = 4 \times \frac{a}{b}
\]
Cross-Multiplying
Cross-multiplying gives:
\[
2a^2 = 20 \frac{ab^2}{1} \implies 2a^2 = 20ab \implies a^2 = 10ab
\]
Rearranging the Equation
Rearranging gives:
\[
a^2 - 10ab = 0 \implies a(a - 10b) = 0
\]
This implies \( a = 0 \) or \( a = 10b \). Since we're looking for a non-zero fraction, we take \( a = 10b \).
Finding the New Fraction
Substituting \( a = 10b \) back into the original fraction:
\[
\frac{10b}{b} = 10
\]
Thus, the new fraction is \( 10 \).
Conclusion
The correct answer is option D: 10.
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A fraction is such that if it is squared and then the numerator is divided by 2, while the denominator is increased by 25%, the new fraction thus obtained is 4 times the original fraction. What is the new fraction?a)8b)None of thesec)50d)10Correct answer is option 'D'. Can you explain this answer?
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