What is the value of `9 (1)/(3) + 19 (2)/(3) + 20 (3)/(4) + 19 (1)/(4)` ?

Options

A. 67
B. 65
C. 59
D. 69

Answer

To solve the expression \( 9 \frac{1}{3} + 19 \frac{2}{3} + 20 \frac{3}{4} + 19 \frac{1}{4} \), we will convert the mixed numbers into improper fractions and then perform the addition step by step.

### Step 1: Convert mixed numbers to improper fractions
1. **Convert \( 9 \frac{1}{3} \)**:
\[ 9 \frac{1}{3} = 9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3} \]

2. **Convert \( 19 \frac{2}{3} \)**:
\[ 19 \frac{2}{3} = 19 + \frac{2}{3} = \frac{57}{3} + \frac{2}{3} = \frac{59}{3} \]

3. **Convert \( 20 \frac{3}{4} \)**:
\[ 20 \frac{3}{4} = 20 + \frac{3}{4} = \frac{80}{4} + \frac{3}{4} = \frac{83}{4} \]

4. **Convert \( 19 \frac{1}{4} \)**:
\[ 19 \frac{1}{4} = 19 + \frac{1}{4} = \frac{76}{4} + \frac{1}{4} = \frac{77}{4} \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the improper fractions:
\[ \frac{28}{3} + \frac{59}{3} + \frac{83}{4} + \frac{77}{4} \]

### Step 3: Combine like terms
1. **Combine the fractions with the same denominator**:
\[ \frac{28}{3} + \frac{59}{3} = \frac{28 + 59}{3} = \frac{87}{3} \]
\[ \frac{83}{4} + \frac{77}{4} = \frac{83 + 77}{4} = \frac{160}{4} = 40 \]

### Step 4: Add the results
Now we have:
\[ \frac{87}{3} + 40 \]
To add these, we need a common denominator. The common denominator between 3 and 1 (for 40) is 3:
\[ 40 = \frac{40 \times 3}{1 \times 3} = \frac{120}{3} \]
Now we can add:
\[ \frac{87}{3} + \frac{120}{3} = \frac{87 + 120}{3} = \frac{207}{3} \]

### Step 5: Simplify the final fraction
Now we simplify \( \frac{207}{3} \):
\[ \frac{207}{3} = 69 \]

### Final Answer
The value of \( 9 \frac{1}{3} + 19 \frac{2}{3} + 20 \frac{3}{4} + 19 \frac{1}{4} \) is \( 69 \).