Which of the following rational numbers lie between $\frac{-7}{10} and \frac{-1}{4}$?

Question

Accepted Answer

To determine which rational numbers lie between $\frac{-7}{10}$ and $\frac{-1}{4}$, we need to compare each option's numbers with these two boundary values.

First, let's convert $\frac{-7}{10}$ and $\frac{-1}{4}$ to decimal form for easier comparison:
- $\frac{-7}{10} = -0.7$
- $\frac{-1}{4} = -0.25$

Now, let's evaluate each option:

**Option 1: $\frac{5}{-8}$, $\frac{-2}{3}$, $\frac{-3}{5}$**

- $\frac{5}{-8} = -0.625$, which is between $-0.7$ and $-0.25$.
- $\frac{-2}{3} = -0.6666\ldots$, which is also between $-0.7$ and $-0.25$.
- $\frac{-3}{5} = -0.6$, which is between $-0.7$ and $-0.25$.

All numbers in Option 1 lie between $\frac{-7}{10}$ and $\frac{-1}{4}$, making it the correct choice.

**Option 2: $\frac{-3}{5}$, 0, $\frac{-1}{5}$**

- $\frac{-3}{5} = -0.6$, which is between $-0.7$ and $-0.25$.
- 0 is not between $-0.7$ and $-0.25$ as it is greater than $-0.25$.
- $\frac{-1}{5} = -0.2$, which is not between $-0.7$ and $-0.25$ as it is greater than $-0.25$.

Option 2 is incorrect because 0 and $\frac{-1}{5}$ do not lie between $\frac{-7}{10}$ and $\frac{-1}{4}$.

**Option 3: $\frac{-2}{3}$, $\frac{-3}{5}$, $\frac{-3}{4}$**

- $\frac{-2}{3} = -0.6666\ldots$, which is between $-0.7$ and $-0.25$.
- $\frac{-3}{5} = -0.6$, which is between $-0.7$ and $-0.25$.
- $\frac{-3}{4} = -0.75$, which is not between $-0.7$ and $-0.25$ as it is less than $-0.7$.

Option 3 is incorrect because $\frac{-3}{4}$ does not lie between $\frac{-7}{10}$ and $\frac{-1}{4}$.

**Option 4: $\frac{5}{-8}$, $\frac{2}{3}$, $\frac{-3}{5}$**

- $\frac{5}{-8} = -0.625$, which is between $-0.7$ and $-0.25$.
- $\frac{2}{3} = 0.6666\ldots$, which is not between $-0.7$ and $-0.25$ as it is greater than $-0.25$.
- $\frac{-3}{5} = -0.6$, which is between $-0.7$ and $-0.25$.

Option 4 is incorrect because $\frac{2}{3}$ does not lie between $\frac{-7}{10}$ and $\frac{-1}{4}$.

In conclusion, Option 1 is the correct answer because all its numbers lie between $\frac{-7}{10}$ and $\frac{-1}{4}$, while the other options contain numbers that do not satisfy this condition.