The product of two fractions is `(14)/(15)` and their quotient is `(35)/(24)` The greater fraction is-

Options

A. `(7)/(4)`
B. `(7)/(6)`
C. `(7)/(3)`
D. `(4)/(5)`

Answer

To solve the problem step by step, we need to find the two fractions based on their product and quotient. Let's denote the two fractions as \( x \) and \( y \).

### Step 1: Set up the equations
We know from the problem statement:
1. The product of the two fractions:
\[ x \cdot y = \frac{14}{15} \]
2. The quotient of the two fractions:
\[ \frac{x}{y} = \frac{35}{24} \]

### Step 2: Express \( x \) in terms of \( y \)
From the quotient equation, we can express \( x \) in terms of \( y \):
\[ x = y \cdot \frac{35}{24} \]

### Step 3: Substitute \( x \) in the product equation
Now, substitute \( x \) in the product equation:
\[ \left(y \cdot \frac{35}{24}\right) \cdot y = \frac{14}{15} \]
This simplifies to:
\[ \frac{35}{24}y^2 = \frac{14}{15} \]

### Step 4: Solve for \( y^2 \)
To isolate \( y^2 \), multiply both sides by the reciprocal of \( \frac{35}{24} \):
\[ y^2 = \frac{14}{15} \cdot \frac{24}{35} \]

### Step 5: Simplify the right-hand side
Now, simplify the right-hand side:
\[ y^2 = \frac{14 \cdot 24}{15 \cdot 35} \]
Calculating the numerator and denominator:
- \( 14 \cdot 24 = 336 \)
- \( 15 \cdot 35 = 525 \)

So we have:
\[ y^2 = \frac{336}{525} \]

### Step 6: Simplify the fraction
Now, we can simplify \( \frac{336}{525} \):
- The GCD of 336 and 525 is 21.
- Dividing both by 21 gives:
\[ y^2 = \frac{16}{25} \]

### Step 7: Take the square root
Taking the square root of both sides, we find:
\[ y = \frac{4}{5} \]

### Step 8: Find \( x \)
Now, substitute \( y \) back into the equation for \( x \):
\[ x = \frac{4}{5} \cdot \frac{35}{24} \]
Calculating this gives:
\[ x = \frac{4 \cdot 35}{5 \cdot 24} = \frac{140}{120} = \frac{7}{6} \]

### Step 9: Identify the greater fraction
We have found:
- \( x = \frac{7}{6} \)
- \( y = \frac{4}{5} \)

Since \( \frac{7}{6} > \frac{4}{5} \), the greater fraction is:
\[ \text{Greater fraction} = \frac{7}{6} \]

### Final Answer
The greater fraction is \( \frac{7}{6} \).

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