`(9)/(10)+(3)/(11)+(7)/(15)=?`

Options

A. `1(217)/(330)`
B. `1(221)/(330)`
C. `1(211)/(330)`
D. `1(197)/(330)`

Explanation

To solve the equation \( \frac{9}{10} + \frac{3}{11} + \frac{7}{15} \), we will follow these steps:

### Step 1: Identify the denominators
The denominators of the fractions are 10, 11, and 15.

### Step 2: Find the Least Common Multiple (LCM)
To add these fractions, we need a common denominator. We will find the LCM of 10, 11, and 15.

- The prime factorization of 10 is \( 2 \times 5 \).
- The prime factorization of 11 is \( 11 \) (it is a prime number).
- The prime factorization of 15 is \( 3 \times 5 \).

The LCM is found by taking the highest power of each prime:
- From 10: \( 2^1 \) and \( 5^1 \)
- From 11: \( 11^1 \)
- From 15: \( 3^1 \)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^1 \times 5^1 \times 11^1 = 2 \times 3 \times 5 \times 11 = 330 \]

### Step 3: Rewrite each fraction with the common denominator
Now we will convert each fraction to have the denominator of 330.

1. For \( \frac{9}{10} \):
\[ \frac{9}{10} = \frac{9 \times 33}{10 \times 33} = \frac{297}{330} \]

2. For \( \frac{3}{11} \):
\[ \frac{3}{11} = \frac{3 \times 30}{11 \times 30} = \frac{90}{330} \]

3. For \( \frac{7}{15} \):
\[ \frac{7}{15} = \frac{7 \times 22}{15 \times 22} = \frac{154}{330} \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ \frac{297}{330} + \frac{90}{330} + \frac{154}{330} = \frac{297 + 90 + 154}{330} \]

Calculating the numerator:
\[ 297 + 90 = 387 \]
\[ 387 + 154 = 541 \]

So, we have:
\[ \frac{541}{330} \]

### Step 5: Final answer
Thus, the final answer is:
\[ \frac{541}{330} \]

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