Simplyfy :
`5 5/9-:3 1/3`

Options

A. `3/5`
B. `5/3`
C. `1 2/3`
D. both (b) and (c)

Answer

To simplify the expression \(5 \frac{5}{9} \div 3 \frac{1}{3}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

For \(5 \frac{5}{9}\):
- Multiply the whole number (5) by the denominator (9): \(5 \times 9 = 45\)
- Add the numerator (5): \(45 + 5 = 50\)
- So, \(5 \frac{5}{9} = \frac{50}{9}\)

For \(3 \frac{1}{3}\):
- Multiply the whole number (3) by the denominator (3): \(3 \times 3 = 9\)
- Add the numerator (1): \(9 + 1 = 10\)
- So, \(3 \frac{1}{3} = \frac{10}{3}\)

### Step 2: Rewrite the Division as Multiplication
Now we rewrite the division of fractions as multiplication by the reciprocal of the second fraction:

\[ \frac{50}{9} \div \frac{10}{3} = \frac{50}{9} \times \frac{3}{10} \]

### Step 3: Multiply the Fractions
Now we multiply the fractions:

\[ \frac{50 \times 3}{9 \times 10} = \frac{150}{90} \]

### Step 4: Simplify the Resulting Fraction
Next, we simplify \(\frac{150}{90}\):
- Find the greatest common divisor (GCD) of 150 and 90, which is 30.
- Divide both the numerator and the denominator by 30:

\[ \frac{150 \div 30}{90 \div 30} = \frac{5}{3} \]

### Step 5: Convert to Mixed Number
Finally, we can convert \(\frac{5}{3}\) back to a mixed number:
- Divide 5 by 3, which gives us 1 with a remainder of 2.
- Thus, \(\frac{5}{3} = 1 \frac{2}{3}\).

So, the simplified form of \(5 \frac{5}{9} \div 3 \frac{1}{3}\) is:

\[ 1 \frac{2}{3} \]