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A fraction is equivalent to 4/5. If 8 is added to the numerator and 6 is subtracted from the denominator, the resultant fraction is equivalent to 4/3. Find the difference between the numerator and the denominator of the original fraction.
- a)5
- b)6
- c)8
- d)4
Correct answer is option 'B'. Can you explain this answer?
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A fraction is equivalent to 4/5. If 8 is added to the numerator and 6...
Let the fraction be 4a/5a
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New fraction = (4a+8)/(5a-6) = 4/3
⇒ 12a + 24 = 20a - 24
⇒ 8a = 48
⇒ a = 6
Hence the fraction is 24/30
Difference between numerator & denominator = 30 - 24 = 6
Most Upvoted Answer
A fraction is equivalent to 4/5. If 8 is added to the numerator and 6...
Understanding the Problem
We need to find a fraction equivalent to \( \frac{4}{5} \) and, when modified, equals \( \frac{4}{3} \). Let's represent the original fraction as \( \frac{a}{b} \).
Step 1: Setting Up Equations
1. **First Equation**:
Since \( \frac{a}{b} = \frac{4}{5} \), we can express it as:
\[
5a = 4b \quad (1)
\]
2. **Second Equation**:
When 8 is added to the numerator and 6 is subtracted from the denominator, we get:
\[
\frac{a + 8}{b - 6} = \frac{4}{3}
\]
Cross-multiplying gives us:
\[
3(a + 8) = 4(b - 6) \quad (2)
\]
Step 2: Simplifying the Equations
From equation (2):
\[
3a + 24 = 4b - 24
\]
Rearranging this, we get:
\[
3a - 4b = -48 \quad (3)
\]
Step 3: Solving the System of Equations
Now we have two equations:
- \( 5a - 4b = 0 \) from (1)
- \( 3a - 4b = -48 \) from (3)
Subtracting (1) from (3):
\[
(3a - 4b) - (5a - 4b) = -48 \\
-2a = -48 \\
a = 24
\]
Substituting \( a = 24 \) in (1):
\[
5(24) = 4b \\
120 = 4b \\
b = 30
\]
Step 4: Finding the Difference
The original fraction is \( \frac{24}{30} \). The difference between the numerator and the denominator is:
\[
30 - 24 = 6
\]
Conclusion
Thus, the difference between the numerator and the denominator of the original fraction is \( 6 \), confirming that option 'B' is the correct answer.
We need to find a fraction equivalent to \( \frac{4}{5} \) and, when modified, equals \( \frac{4}{3} \). Let's represent the original fraction as \( \frac{a}{b} \).
Step 1: Setting Up Equations
1. **First Equation**:
Since \( \frac{a}{b} = \frac{4}{5} \), we can express it as:
\[
5a = 4b \quad (1)
\]
2. **Second Equation**:
When 8 is added to the numerator and 6 is subtracted from the denominator, we get:
\[
\frac{a + 8}{b - 6} = \frac{4}{3}
\]
Cross-multiplying gives us:
\[
3(a + 8) = 4(b - 6) \quad (2)
\]
Step 2: Simplifying the Equations
From equation (2):
\[
3a + 24 = 4b - 24
\]
Rearranging this, we get:
\[
3a - 4b = -48 \quad (3)
\]
Step 3: Solving the System of Equations
Now we have two equations:
- \( 5a - 4b = 0 \) from (1)
- \( 3a - 4b = -48 \) from (3)
Subtracting (1) from (3):
\[
(3a - 4b) - (5a - 4b) = -48 \\
-2a = -48 \\
a = 24
\]
Substituting \( a = 24 \) in (1):
\[
5(24) = 4b \\
120 = 4b \\
b = 30
\]
Step 4: Finding the Difference
The original fraction is \( \frac{24}{30} \). The difference between the numerator and the denominator is:
\[
30 - 24 = 6
\]
Conclusion
Thus, the difference between the numerator and the denominator of the original fraction is \( 6 \), confirming that option 'B' is the correct answer.
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A fraction is equivalent to 4/5. If 8 is added to the numerator and 6 is subtracted from the denominator, the resultant fraction is equivalent to 4/3. Find the difference between the numerator and the denominator of the original fraction.a)5b)6c)8d)4Correct answer is option 'B'. Can you explain this answer?
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