Two fractions are such that their product is -9/10 and sum is `-13//40.` What are the two fractions. (a) 1/5 , − 9/4 (b) 1/5 , − 9/2 (c) 4/5 , -9/8 (d) -2/5 , -9/4

Options

A. `1//5,-9//4`
B. `1//5,-9//2`
C. `4//5,-9//8`
D. `-2//5, -9//4`

Answer

To solve the problem, we need to find two fractions \( a \) and \( b \) such that:

1. Their product is \( ab = -\frac{9}{10} \)
2. Their sum is \( a + b = -\frac{13}{40} \)

### Step 1: Set up the equations
From the information given, we can write the two equations:
- \( ab = -\frac{9}{10} \) (1)
- \( a + b = -\frac{13}{40} \) (2)

### Step 2: Express one variable in terms of the other
From equation (1), we can express \( a \) in terms of \( b \):
\[ a = -\frac{9}{10b} \]

### Step 3: Substitute into the sum equation
Now, substitute \( a \) from the above expression into equation (2):
\[ -\frac{9}{10b} + b = -\frac{13}{40} \]

### Step 4: Clear the fractions
To eliminate the fractions, multiply the entire equation by \( 40b \) (the least common multiple of the denominators):
\[ 40b \left(-\frac{9}{10b}\right) + 40b(b) = 40b \left(-\frac{13}{40}\right) \]
This simplifies to:
\[ -360 + 40b^2 = -13b \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 40b^2 + 13b - 360 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to \( 40 \times -360 = -14400 \) and add to \( 13 \). The numbers \( 180 \) and \( -72 \) work:
\[ 40b^2 + 180b - 72b - 360 = 0 \]
Grouping gives us:
\[ (40b^2 + 180b) + (-72b - 360) = 0 \]
Factoring by grouping:
\[ 20b(2b + 9) - 72(2b + 9) = 0 \]
Factoring out \( (2b + 9) \):
\[ (2b + 9)(20b - 72) = 0 \]

### Step 7: Solve for \( b \)
Setting each factor to zero gives:
1. \( 2b + 9 = 0 \) → \( b = -\frac{9}{2} \)
2. \( 20b - 72 = 0 \) → \( b = \frac{72}{20} = \frac{18}{5} = 4.5 \)

### Step 8: Find corresponding \( a \)
Using \( ab = -\frac{9}{10} \):
- If \( b = -\frac{9}{2} \):
\[ a = -\frac{9}{10 \times -\frac{9}{2}} = -\frac{9 \times 2}{10 \times 9} = -\frac{2}{10} = -\frac{1}{5} \]
Thus, the pair is \( \left(-\frac{1}{5}, -\frac{9}{2}\right) \).

- If \( b = \frac{18}{5} \):
\[ a = -\frac{9}{10 \times \frac{18}{5}} = -\frac{9 \times 5}{10 \times 18} = -\frac{45}{180} = -\frac{1}{4} \]
This does not yield a valid fraction based on the options provided.

### Final Answer
The two fractions are:
\[ \left(-\frac{1}{5}, -\frac{9}{2}\right) \]
Thus, the correct option is (b) \( \frac{1}{5}, -\frac{9}{2} \).