To solve the problem \( 3 + \frac{1 \frac{4}{15}}{1} + \frac{3}{20} \), we will follow these steps:
### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \( 1 \frac{4}{15} \) can be converted to an improper fraction. The formula to convert a mixed number to an improper fraction is:
\[
\text{Improper Fraction} = \left(\text{Whole Number} \times \text{Denominator} + \text{Numerator}\right) / \text{Denominator}
\]
For \( 1 \frac{4}{15} \):
\[
\text{Improper Fraction} = \left(1 \times 15 + 4\right) / 15 = \frac{15 + 4}{15} = \frac{19}{15}
\]
So, we can rewrite the expression as:
\[
3 + \frac{19}{15} + \frac{3}{20}
\]
### Step 2: Convert whole number to a fraction
Next, we need to express the whole number \( 3 \) as a fraction with a common denominator. We can express \( 3 \) as:
\[
3 = \frac{3 \times 60}{60} = \frac{180}{60}
\]
### Step 3: Find the least common multiple (LCM)
Now, we need to find the LCM of the denominators \( 15 \) and \( 20 \).
- The multiples of \( 15 \) are \( 15, 30, 45, 60, 75, 90, \ldots \)
- The multiples of \( 20 \) are \( 20, 40, 60, 80, 100, \ldots \)
The LCM of \( 15 \) and \( 20 \) is \( 60 \).
### Step 4: Convert each fraction to have the common denominator
Now we convert each fraction to have the common denominator of \( 60 \):
1. For \( \frac{19}{15} \):
\[
\frac{19}{15} = \frac{19 \times 4}{15 \times 4} = \frac{76}{60}
\]
2. For \( \frac{3}{20} \):
\[
\frac{3}{20} = \frac{3 \times 3}{20 \times 3} = \frac{9}{60}
\]
### Step 5: Add the fractions
Now we can add all the fractions together:
\[
\frac{180}{60} + \frac{76}{60} + \frac{9}{60} = \frac{180 + 76 + 9}{60} = \frac{265}{60}
\]
### Step 6: Simplify the fraction
Next, we simplify \( \frac{265}{60} \). We can divide both the numerator and the denominator by \( 5 \):
\[
\frac{265 \div 5}{60 \div 5} = \frac{53}{12}
\]
### Step 7: Convert to a mixed number
Finally, we convert \( \frac{53}{12} \) to a mixed number:
1. Divide \( 53 \) by \( 12 \):
- \( 12 \) goes into \( 53 \) \( 4 \) times (since \( 12 \times 4 = 48 \)).
- The remainder is \( 53 - 48 = 5 \).
So, we can write:
\[
\frac{53}{12} = 4 \frac{5}{12}
\]
### Final Answer
Thus, the final answer is:
\[
4 \frac{5}{12}
\]
