What per cent is the least rational number of the greatest rational number, if `(1)/(2),(2)/(5),(1)/(3)` and `(5)/(9)` are arranged in ascending order?

Answer

To solve the problem step by step, we need to find out what percent the least rational number is of the greatest rational number from the given set of numbers: \( \frac{1}{2}, \frac{2}{5}, \frac{1}{3}, \frac{5}{9} \).

### Step 1: Arrange the numbers in ascending order
First, we need to compare the fractions. To do this, we can convert them to have a common denominator or convert them to decimal form.

1. Convert \( \frac{1}{2} = 0.5 \)
2. Convert \( \frac{2}{5} = 0.4 \)
3. Convert \( \frac{1}{3} \approx 0.333 \)
4. Convert \( \frac{5}{9} \approx 0.555 \)

Now we can arrange them in ascending order:
- \( \frac{1}{3} \) (0.333)
- \( \frac{2}{5} \) (0.4)
- \( \frac{1}{2} \) (0.5)
- \( \frac{5}{9} \) (0.555)

So, the ascending order is:
\[ \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{5}{9} \]

### Step 2: Identify the least and greatest rational numbers
From the ordered list, we can identify:
- Least rational number: \( \frac{1}{3} \)
- Greatest rational number: \( \frac{5}{9} \)

### Step 3: Calculate what percent the least number is of the greatest number
We need to find what percent \( \frac{1}{3} \) is of \( \frac{5}{9} \).

Using the formula for percentage:
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]
Here, the part is \( \frac{1}{3} \) and the whole is \( \frac{5}{9} \).

Substituting the values:
\[ \text{Percentage} = \left( \frac{\frac{1}{3}}{\frac{5}{9}} \right) \times 100 \]

### Step 4: Simplify the fraction
To simplify \( \frac{\frac{1}{3}}{\frac{5}{9}} \):
\[ \frac{\frac{1}{3}}{\frac{5}{9}} = \frac{1}{3} \times \frac{9}{5} = \frac{9}{15} = \frac{3}{5} \]

### Step 5: Calculate the percentage
Now we calculate:
\[ \text{Percentage} = \left( \frac{3}{5} \right) \times 100 = 60\% \]

### Final Answer
Thus, the least rational number \( \frac{1}{3} \) is \( 60\% \) of the greatest rational number \( \frac{5}{9} \).