Which ratio is greater ?
(11 : 21) or (19 : 28)

Answer

To determine which ratio is greater between (11 : 21) and (19 : 28), we can follow these steps:

### Step 1: Write the Ratios in Fraction Form
Convert the ratios into fractions:
- \( \frac{11}{21} \) and \( \frac{19}{28} \)

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
We need to find the LCM of 21 and 28 to compare the fractions easily.

- The factors of 21 are: \( 3 \times 7 \)
- The factors of 28 are: \( 4 \times 7 \) or \( 2^2 \times 7 \)

To find the LCM, take the highest power of each prime factor:
- For 2: \( 2^2 \)
- For 3: \( 3^1 \)
- For 7: \( 7^1 \)

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84 \]

### Step 3: Convert Both Fractions to Have the Same Denominator
Now, convert both fractions to have the denominator of 84.

For \( \frac{11}{21} \):
- \( \frac{84}{21} = 4 \) (since \( 84 \div 21 = 4 \))
- Multiply the numerator by 4: \( 11 \times 4 = 44 \)
- So, \( \frac{11}{21} = \frac{44}{84} \)

For \( \frac{19}{28} \):
- \( \frac{84}{28} = 3 \) (since \( 84 \div 28 = 3 \))
- Multiply the numerator by 3: \( 19 \times 3 = 57 \)
- So, \( \frac{19}{28} = \frac{57}{84} \)

### Step 4: Compare the Numerators
Now we compare the numerators of the two fractions:
- \( 44 \) (from \( \frac{11}{21} \)) and \( 57 \) (from \( \frac{19}{28} \))

Since \( 57 > 44 \), we conclude that:
\[ \frac{11}{21} < \frac{19}{28} \]

### Final Conclusion
Therefore, the ratio \( (19 : 28) \) is greater than \( (11 : 21) \).

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