What will come in place of question mark (?) in the following questions ?
`3(1)/(5) + 5(1)/(2)-3(2)/(3) = ? + 1(1)/(4)`.

Options

A. `3(47)/(60)`
B. `4(5)/(12)`
C. `3(3)/(4)`
D. `3(5)/(6)`

Answer

To solve the equation \( 3\frac{1}{5} + 5\frac{1}{2} - 3\frac{2}{3} = ? + 1\frac{1}{4} \), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
1. Convert \( 3\frac{1}{5} \) to an improper fraction:
\[ 3\frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5} \]

2. Convert \( 5\frac{1}{2} \) to an improper fraction:
\[ 5\frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2} \]

3. Convert \( 3\frac{2}{3} \) to an improper fraction:
\[ 3\frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]

4. Convert \( 1\frac{1}{4} \) to an improper fraction:
\[ 1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation using the improper fractions:
\[ \frac{16}{5} + \frac{11}{2} - \frac{11}{3} = ? + \frac{5}{4} \]

### Step 3: Find a common denominator
To solve this, we need a common denominator for \( 5, 2, 3, \) and \( 4 \). The least common multiple (LCM) of these denominators is \( 60 \).

### Step 4: Convert each fraction to have the common denominator
1. Convert \( \frac{16}{5} \):
\[ \frac{16}{5} = \frac{16 \times 12}{5 \times 12} = \frac{192}{60} \]

2. Convert \( \frac{11}{2} \):
\[ \frac{11}{2} = \frac{11 \times 30}{2 \times 30} = \frac{330}{60} \]

3. Convert \( \frac{11}{3} \):
\[ \frac{11}{3} = \frac{11 \times 20}{3 \times 20} = \frac{220}{60} \]

4. Convert \( \frac{5}{4} \):
\[ \frac{5}{4} = \frac{5 \times 15}{4 \times 15} = \frac{75}{60} \]

### Step 5: Substitute back into the equation
Now we can substitute these values back into the equation:
\[ \frac{192}{60} + \frac{330}{60} - \frac{220}{60} = ? + \frac{75}{60} \]

### Step 6: Simplify the left side
Combine the fractions on the left side:
\[ \frac{192 + 330 - 220}{60} = ? + \frac{75}{60} \]
Calculating the numerator:
\[ 192 + 330 = 522 \]
\[ 522 - 220 = 302 \]
So we have:
\[ \frac{302}{60} = ? + \frac{75}{60} \]

### Step 7: Isolate the question mark
To isolate \( ? \), we can subtract \( \frac{75}{60} \) from both sides:
\[ ? = \frac{302}{60} - \frac{75}{60} = \frac{302 - 75}{60} = \frac{227}{60} \]

### Step 8: Convert back to a mixed number
To convert \( \frac{227}{60} \) back to a mixed number:
1. Divide \( 227 \) by \( 60 \):
- \( 60 \) goes into \( 227 \) three times (since \( 3 \times 60 = 180 \)).
- The remainder is \( 227 - 180 = 47 \).

Thus, we have:
\[ ? = 3\frac{47}{60} \]

### Final Answer
The value that comes in place of the question mark is \( 3\frac{47}{60} \).

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