`(5.42xx6+5.42xx24)/(32.71xx32.71-27.29xx27.29)+(6.54xx6.54-3.46xx3.46)/(3.08xx5+3.08xx45)` is equal to

Options

A. `0.3`
B. `0.4`
C. `0.7`
D. `2.5`

Answer

To solve the expression

\[ \frac{(5.42 \times 6 + 5.42 \times 24)}{(32.71 \times 32.71 - 27.29 \times 27.29)} + \frac{(6.54 \times 6.54 - 3.46 \times 3.46)}{(3.08 \times 5 + 3.08 \times 45)} \]

we will break it down step by step.

### Step 1: Simplify the Numerator of the First Fraction

The first part of the numerator is

\[ 5.42 \times 6 + 5.42 \times 24 \]

We can factor out \(5.42\):

\[ 5.42 \times (6 + 24) = 5.42 \times 30 \]

### Step 2: Simplify the Denominator of the First Fraction

The denominator is

\[ 32.71 \times 32.71 - 27.29 \times 27.29 \]

This is a difference of squares, which can be factored as:

\[ (a^2 - b^2) = (a - b)(a + b) \]

Let \(a = 32.71\) and \(b = 27.29\):

\[ (32.71 - 27.29)(32.71 + 27.29) = 5.42 \times 60 \]

### Step 3: Combine the First Fraction

Now we can substitute back into the first fraction:

\[ \frac{5.42 \times 30}{5.42 \times 60} \]

The \(5.42\) cancels out:

\[ \frac{30}{60} = \frac{1}{2} \]

### Step 4: Simplify the Numerator of the Second Fraction

The numerator of the second fraction is:

\[ 6.54 \times 6.54 - 3.46 \times 3.46 \]

This is also a difference of squares:

\[ (6.54 - 3.46)(6.54 + 3.46) = 3.08 \times 10 \]

### Step 5: Simplify the Denominator of the Second Fraction

The denominator is:

\[ 3.08 \times 5 + 3.08 \times 45 \]

We can factor out \(3.08\):

\[ 3.08 \times (5 + 45) = 3.08 \times 50 \]

### Step 6: Combine the Second Fraction

Now we can substitute back into the second fraction:

\[ \frac{3.08 \times 10}{3.08 \times 50} \]

The \(3.08\) cancels out:

\[ \frac{10}{50} = \frac{1}{5} \]

### Step 7: Combine Both Fractions

Now we have:

\[ \frac{1}{2} + \frac{1}{5} \]

### Step 8: Find a Common Denominator

The least common multiple of \(2\) and \(5\) is \(10\). We convert both fractions:

\[ \frac{1}{2} = \frac{5}{10} \quad \text{and} \quad \frac{1}{5} = \frac{2}{10} \]

### Step 9: Add the Fractions

Now we can add:

\[ \frac{5}{10} + \frac{2}{10} = \frac{7}{10} \]

### Step 10: Convert to Decimal

Finally, we convert \(\frac{7}{10}\) to decimal:

\[ 0.7 \]

Thus, the final answer is:

\[ \boxed{0.7} \]