If x is the HCF and y is the LCM of \(\frac{3}{5}, \frac{6}{25}, \frac{9}{20}, \frac{27}{50},\) then which one of the following is correct?
To find the correct relationship between \(x\) and \(y\), we need to calculate the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the given fractions:
\(\frac{3}{5}, \frac{6}{25}, \frac{9}{20}, \text{ and } \frac{27}{50}\)
1. Formulas for Fractions
When dealing with fractions, use these standard rules:
\(\text{HCF of fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}\)
\(\text{LCM of fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}\)
Numerators: \(3, 6, 9, 27\)
Denominators: \(5, 25, 20, 50\)
2. Calculate \(x\) (HCF of the fractions)
HCF of Numerators \((3, 6, 9, 27)\): The largest number that divides all of them perfectly is \(3\).
LCM of Denominators \((5, 25, 20, 50)\): * Multiples of \(50\): \(50, 100 \dots\)
\(100\) is perfectly divisible by \(5, 25, 20,\) and \(50\). So, the LCM is \(100\).
\(x = \frac{3}{100}\)
3. Calculate \(y\) (LCM of the fractions)
LCM of Numerators \((3, 6, 9, 27)\): * Multiples of \(27\): \(27, 54 \dots\)
\(54\) is perfectly divisible by \(3, 6, 9,\) and \(27\). So, the LCM is \(54\).
HCF of Denominators \((5, 25, 20, 50)\): The largest number that divides all of them perfectly is \(5\).
\(y = \frac{54}{5}\)
4. Find the Relationship Between \(y\) and \(x\)
We want to express \(y\) in terms of \(x\) (i.e., \(y = k \cdot x\)). Let's find \(k\) by dividing \(y\) by \(x\):
\(k = \frac{y}{x} = \frac{\frac{54}{5}}{\frac{3}{100}}\)
\(k = \frac{54}{5} \times \frac{100}{3}\)
Simplify the expression:
\(\frac{54}{3} = 18\)
\(\frac{100}{5} = 20\)
\(k = 18 \times 20 = 360\)
Therefore, \(y = 360x\).
Correct Answer
Option 4 (\(y = 360x\))
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