Which is greater: `3(1)/(3)` or `(33)/(10)?`

Options

A. `3(1)/(3)`
B. `(33)/(10)`
C. bothare equal
D. none of these

Answer

To determine which is greater between \( 3 \frac{1}{3} \) and \( \frac{33}{10} \), we can follow these steps:

### Step 1: Convert Mixed Number to Improper Fraction
First, we need to convert the mixed number \( 3 \frac{1}{3} \) into an improper fraction.

\[ 3 \frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]

### Step 2: Compare the Two Fractions
Now we have two fractions to compare: \( \frac{10}{3} \) and \( \frac{33}{10} \). To compare them, we can find a common denominator.

### Step 3: Find the Least Common Multiple (LCM)
The denominators are 3 and 10. The least common multiple (LCM) of 3 and 10 is 30.

### Step 4: Convert Both Fractions to Have the Same Denominator
Now we will convert both fractions to have a denominator of 30.

For \( \frac{10}{3} \):
\[ \frac{10}{3} \times \frac{10}{10} = \frac{100}{30} \]

For \( \frac{33}{10} \):
\[ \frac{33}{10} \times \frac{3}{3} = \frac{99}{30} \]

### Step 5: Compare the Numerators
Now we compare the numerators of both fractions:
- \( 100 \) (from \( \frac{100}{30} \))
- \( 99 \) (from \( \frac{99}{30} \))

Since \( 100 > 99 \), we find that:
\[ \frac{100}{30} > \frac{99}{30} \]

### Conclusion
Thus, \( 3 \frac{1}{3} \) is greater than \( \frac{33}{10} \).

### Final Answer
\( 3 \frac{1}{3} > \frac{33}{10} \)

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