Which one is greater out of `(2 : 3)` and `(4 : 7)` ?

Options

A. `2 : 3`
B. `4 : 7`
C. both are equal
D. none of these

Answer

To determine which ratio is greater between \(2 : 3\) and \(4 : 7\), we can follow these steps:

### Step 1: Convert Ratios to Fractions
We start by converting the ratios into fractions:
- \(2 : 3\) can be written as \(\frac{2}{3}\)
- \(4 : 7\) can be written as \(\frac{4}{7}\)

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the least common multiple (LCM) of the denominators (3 and 7) to compare the two fractions easily.
- The LCM of 3 and 7 is \(21\) (since \(3 \times 7 = 21\)).

### Step 3: Convert Fractions to Have the Same Denominator
Now we convert both fractions to have the same denominator of \(21\):
- For \(\frac{2}{3}\):
\[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \]
- For \(\frac{4}{7}\):
\[ \frac{4}{7} = \frac{4 \times 3}{7 \times 3} = \frac{12}{21} \]

### Step 4: Compare the Numerators
Now that both fractions have the same denominator, we can compare their numerators:
- The numerators are \(14\) (from \(\frac{14}{21}\)) and \(12\) (from \(\frac{12}{21}\)).
- Since \(14\) is greater than \(12\), we conclude that:
\[ \frac{14}{21} > \frac{12}{21} \]

### Conclusion
Thus, \(2 : 3\) is greater than \(4 : 7\).