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 break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert each mixed number into an improper fraction.

1. \(9 \frac{1}{3} = 9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}\)
2. \(19 \frac{2}{3} = 19 + \frac{2}{3} = \frac{57}{3} + \frac{2}{3} = \frac{59}{3}\)
3. \(20 \frac{3}{4} = 20 + \frac{3}{4} = \frac{80}{4} + \frac{3}{4} = \frac{83}{4}\)
4. \(19 \frac{1}{4} = 19 + \frac{1}{4} = \frac{76}{4} + \frac{1}{4} = \frac{77}{4}\)

### Step 2: Add the Whole Numbers
Next, we add the whole numbers together:

\[ 9 + 19 + 20 + 19 = 67 \]

### Step 3: Add the Fractions
Now, we will add the fractions. To do this, we need a common denominator. The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

Convert each fraction to have a denominator of 12:

1. \(\frac{28}{3} = \frac{28 \times 4}{3 \times 4} = \frac{112}{12}\)
2. \(\frac{59}{3} = \frac{59 \times 4}{3 \times 4} = \frac{236}{12}\)
3. \(\frac{83}{4} = \frac{83 \times 3}{4 \times 3} = \frac{249}{12}\)
4. \(\frac{77}{4} = \frac{77 \times 3}{4 \times 3} = \frac{231}{12}\)

Now, add these fractions:

\[ \frac{112}{12} + \frac{236}{12} + \frac{249}{12} + \frac{231}{12} = \frac{112 + 236 + 249 + 231}{12} = \frac{828}{12} \]

### Step 4: Simplify the Fraction
Now simplify \(\frac{828}{12}\):

\[ \frac{828 \div 12}{12 \div 12} = \frac{69}{1} = 69 \]

### Step 5: Combine Whole Numbers and Fraction Result
Finally, combine the whole number sum and the fraction sum:

\[ 67 + 69 = 69 \]

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