Which of the following is correct?

Options

A. a) `(2)/(3) lt (3)/(5) lt (11)/(15)`
B. b) `(3)/(5) lt (2)/(3) lt (11)/(5)`
C. c) `(11)/(15) lt (3)/(5) lt (2)/(3)`
D. d) `(3)/(5) lt (11)/(15) lt (2)/(3)`

Answer

To determine which of the given options is correct, we will use the cross multiplication method to compare the fractions. Let's break down the solution step by step.

### Step 1: Identify the Fractions
We have three options to compare:
1. Option A: \( \frac{2}{3} \), \( \frac{3}{5} \), and \( \frac{11}{15} \)
2. Option B: \( \frac{3}{5} \), \( \frac{2}{3} \), and \( \frac{11}{5} \)

### Step 2: Analyze Option A
We will compare \( \frac{2}{3} \) and \( \frac{3}{5} \) first.

#### Cross Multiply:
- For \( \frac{2}{3} \) and \( \frac{3}{5} \):
- \( 2 \times 5 = 10 \)
- \( 3 \times 3 = 9 \)

Since \( 10 > 9 \), we conclude that:
\[ \frac{2}{3} > \frac{3}{5} \]

Next, we compare \( \frac{3}{5} \) and \( \frac{11}{15} \).

#### Cross Multiply:
- For \( \frac{3}{5} \) and \( \frac{11}{15} \):
- \( 3 \times 15 = 45 \)
- \( 5 \times 11 = 55 \)

Since \( 45 < 55 \), we conclude that:
\[ \frac{3}{5} < \frac{11}{15} \]

### Conclusion for Option A:
From the comparisons, we see that:
- \( \frac{2}{3} > \frac{3}{5} \) (True)
- \( \frac{3}{5} < \frac{11}{15} \) (True)

However, since \( \frac{2}{3} \) is greater than \( \frac{3}{5} \), this means that the first part of the option is incorrect.

### Step 3: Analyze Option B
Now, we will analyze Option B: \( \frac{3}{5} \), \( \frac{2}{3} \), and \( \frac{11}{5} \).

#### Compare \( \frac{3}{5} \) and \( \frac{2}{3} \):
- Cross Multiply:
- \( 3 \times 3 = 9 \)
- \( 5 \times 2 = 10 \)

Since \( 9 < 10 \), we conclude that:
\[ \frac{3}{5} < \frac{2}{3} \]

Next, we compare \( \frac{2}{3} \) and \( \frac{11}{5} \).

#### Cross Multiply:
- For \( \frac{2}{3} \) and \( \frac{11}{5} \):
- \( 2 \times 5 = 10 \)
- \( 3 \times 11 = 33 \)

Since \( 10 < 33 \), we conclude that:
\[ \frac{2}{3} < \frac{11}{5} \]

### Conclusion for Option B:
From the comparisons, we see that:
- \( \frac{3}{5} < \frac{2}{3} \) (True)
- \( \frac{2}{3} < \frac{11}{5} \) (True)

Thus, both parts of Option B are correct.

### Final Answer:
The correct option is **B**.

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