Compare the ratios (5:6) and (3:4).

Options

A. ` (5:6) = (3:4)`.
B. ` (5:6) lt (3:4)`.
C. ` (5:6) gt (3:4)`.
D. none of these

Answer

To compare the ratios \(5:6\) and \(3:4\), we can follow these steps:

### Step 1: Convert Ratios to Fractions
First, we convert the ratios into fractions:
- The ratio \(5:6\) can be written as the fraction \(\frac{5}{6}\).
- The ratio \(3:4\) can be written as the fraction \(\frac{3}{4}\).

### Step 2: Find a Common Denominator
To compare the two fractions, we need to have a common denominator. The denominators here are \(6\) and \(4\). The least common multiple (LCM) of \(6\) and \(4\) is \(12\).

### Step 3: Convert Each Fraction to Have the Common Denominator
Now, we will convert both fractions to have the denominator of \(12\):
- For \(\frac{5}{6}\):
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]
- For \(\frac{3}{4}\):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

### Step 4: Compare the Two Fractions
Now that both fractions have the same denominator, we can compare the numerators:
- \(\frac{10}{12}\) and \(\frac{9}{12}\)

Since \(10\) is greater than \(9\), we can conclude that:
\[ \frac{10}{12} > \frac{9}{12} \]

### Step 5: State the Comparison of Ratios
Thus, we can say:
\[ 5:6 > 3:4 \]

### Final Conclusion
The ratio \(5:6\) is greater than the ratio \(3:4\).

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