The product of two rational numbers is -$\frac{40}{3}$. If one of the two numbers is -$\frac{5}{2}$, then the reciprocal of the other number lies between :

Question

Accepted Answer

To solve the problem, we need to find the reciprocal of the other rational number when the product of two rational numbers is given as -$\frac{40}{3}$ and one of the numbers is -$\frac{5}{2}$.

Let's denote the unknown rational number as $x$. According to the problem, the product of the two numbers is:

$$
-\frac{5}{2} \times x = -\frac{40}{3}
$$

To find $x$, we can solve for it by dividing both sides of the equation by -$\frac{5}{2}$:

$$
x = \frac{-\frac{40}{3}}{-\frac{5}{2}}
$$

Simplifying the division of fractions:

$$
x = \frac{-40}{3} \times \frac{-2}{5} = \frac{40 \times 2}{3 \times 5} = \frac{80}{15}
$$

Simplifying $\frac{80}{15}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

$$
x = \frac{16}{3}
$$

Now, we need to find the reciprocal of $x$, which is:

$$
\text{Reciprocal of } x = \frac{3}{16}
$$

Next, we determine which interval the reciprocal $\frac{3}{16}$ falls into. We convert the fractions in the options to decimals for easier comparison:

1. $\frac{3}{20} = 0.15$ and $\frac{9}{50} = 0.18$
2. $\frac{9}{50} = 0.18$ and $\frac{1}{5} = 0.2$
3. $\frac{1}{5} = 0.2$ and $\frac{1}{4} = 0.25$
4. $\frac{1}{4} = 0.25$ and $\frac{1}{3} = 0.333...$

The decimal value of $\frac{3}{16}$ is:

$$
\frac{3}{16} = 0.1875
$$

Comparing $0.1875$ with the intervals:

- It is greater than $0.18$ and less than $0.2$, which means it lies between $\frac{9}{50}$ and $\frac{1}{5}$.

Therefore, the reciprocal of the other number, $\frac{3}{16}$, lies between $\frac{9}{50}$ and $\frac{1}{5}$, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

1. **Option 1:** $\frac{3}{20}$ and $\frac{9}{50}$ corresponds to the interval $0.15$ to $0.18$. Since $0.1875$ is greater than $0.18$, it does not lie in this interval.

2. **Option 3:** $\frac{1}{5}$ and $\frac{1}{4}$ corresponds to the interval $0.2$ to $0.25$. Since $0.1875$ is less than $0.2$, it does not lie in this interval.

3. **Option 4:** $\frac{1}{4}$ and $\frac{1}{3}$ corresponds to the interval $0.25$ to $0.333...$. Since $0.1875$ is less than $0.25$, it does not lie in this interval.

Thus, Option 2 is the only correct choice, as it accurately contains the reciprocal value $\frac{3}{16}$.