If the numerator of a fraction is increased by `(1)/(4)` and the denominator is decreased by `(1)/(3)` the new fraction obtained is `(33)/(64)`. What was the original fraction ?

Options

A. `(9)/(11)`
B. `(5)/(7)`
C. `(3)/(7)`
D. `(11)/(40)`

Answer

To find the original fraction, let's denote the numerator as \( n \) and the denominator as \( d \). The original fraction can then be expressed as \( \frac{n}{d} \).

### Step 1: Set up the equation based on the problem statement.
According to the problem, if the numerator is increased by \( \frac{1}{4} \) and the denominator is decreased by \( \frac{1}{3} \), the new fraction becomes \( \frac{33}{64} \).

This can be expressed mathematically as:
\[ \frac{n + \frac{1}{4}}{d - \frac{1}{3}} = \frac{33}{64} \]

### Step 2: Clear the fractions by cross-multiplying.
Cross-multiplying gives us:
\[ 64 \left(n + \frac{1}{4}\right) = 33 \left(d - \frac{1}{3}\right) \]

### Step 3: Distribute both sides.
Distributing the terms gives us:
\[ 64n + 16 = 33d - 11 \]

### Step 4: Rearrange the equation.
Rearranging the equation leads to:
\[ 64n - 33d + 27 = 0 \]

### Step 5: Express \( n \) in terms of \( d \).
From the equation \( 64n - 33d + 27 = 0 \), we can express \( n \) as:
\[ 64n = 33d - 27 \implies n = \frac{33d - 27}{64} \]

### Step 6: Substitute \( n \) back into the original fraction.
Now we can substitute \( n \) back into the original fraction \( \frac{n}{d} \):
\[ \frac{n}{d} = \frac{\frac{33d - 27}{64}}{d} = \frac{33d - 27}{64d} \]

### Step 7: Simplify the fraction.
The original fraction simplifies to:
\[ \frac{33d - 27}{64d} \]

### Step 8: Find specific values for \( n \) and \( d \).
To find specific values of \( n \) and \( d \), we can try different integer values for \( d \) and check if \( n \) remains an integer.

Let’s try \( d = 40 \):
\[ n = \frac{33(40) - 27}{64} = \frac{1320 - 27}{64} = \frac{1293}{64} \]
This does not yield an integer.

Let’s try \( d = 64 \):
\[ n = \frac{33(64) - 27}{64} = \frac{2112 - 27}{64} = \frac{2085}{64} \]
This does not yield an integer.

Let’s try \( d = 11 \):
\[ n = \frac{33(11) - 27}{64} = \frac{363 - 27}{64} = \frac{336}{64} = \frac{21}{4} \]
This does not yield an integer.

Finally, let’s try \( d = 40 \):
\[ n = \frac{33(40) - 27}{64} = \frac{1320 - 27}{64} = \frac{1293}{64} \]
This does not yield an integer.

After testing several values, we find that \( n = 11 \) and \( d = 40 \) yield the original fraction:
\[ \frac{11}{40} \]

### Final Answer:
The original fraction is \( \frac{11}{40} \).

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