If `x=(1)/(20.21)+(1)/(21.22)+...+(1)/(29.30)andy=(1)/(30.31)+(1)/(31.32)+...+(1)/(39.40)`, Then what is the value of `(x)/(y)`?

Options

A. `(1)/(2)`
B. 2
C. 4
D. `(1)/(4)`

Answer

To solve the problem step by step, we will first calculate the values of \( x \) and \( y \) separately, and then find the ratio \( \frac{x}{y} \).

### Step 1: Calculate \( x \)

The expression for \( x \) is given as:
\[ x = \frac{1}{20 \cdot 21} + \frac{1}{21 \cdot 22} + \frac{1}{22 \cdot 23} + \ldots + \frac{1}{29 \cdot 30} \]

We can rewrite each term using the formula:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

Thus, we can express \( x \) as:
\[ x = \left( \frac{1}{20} - \frac{1}{21} \right) + \left( \frac{1}{21} - \frac{1}{22} \right) + \left( \frac{1}{22} - \frac{1}{23} \right) + \ldots + \left( \frac{1}{29} - \frac{1}{30} \right) \]

### Step 2: Simplify \( x \)

Notice that this is a telescoping series. Most terms will cancel out:
\[ x = \frac{1}{20} - \frac{1}{30} \]

Now, find a common denominator to simplify:
\[ x = \frac{1}{20} - \frac{1}{30} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

### Step 3: Calculate \( y \)

The expression for \( y \) is given as:
\[ y = \frac{1}{30 \cdot 31} + \frac{1}{31 \cdot 32} + \ldots + \frac{1}{39 \cdot 40} \]

Using the same formula:
\[ y = \left( \frac{1}{30} - \frac{1}{31} \right) + \left( \frac{1}{31} - \frac{1}{32} \right) + \ldots + \left( \frac{1}{39} - \frac{1}{40} \right) \]

### Step 4: Simplify \( y \)

Again, this is a telescoping series:
\[ y = \frac{1}{30} - \frac{1}{40} \]

Finding a common denominator:
\[ y = \frac{4}{120} - \frac{3}{120} = \frac{1}{120} \]

### Step 5: Calculate \( \frac{x}{y} \)

Now we can find the ratio:
\[ \frac{x}{y} = \frac{\frac{1}{60}}{\frac{1}{120}} = \frac{1}{60} \cdot \frac{120}{1} = 2 \]

### Final Answer
Thus, the value of \( \frac{x}{y} \) is:
\[ \boxed{2} \]