Question

The L.C.M. of \(\frac{7}{3},\ \frac{5}{6},\ \frac{14}{9}\) is

• A. \(\frac{70}{18}\)
• B. \(\frac{70}{9}\)
• C. \(\frac{70}{3}\)
• D. 70

Hint:

For fractions, remember: \[ \text{LCM}=\frac{\text{LCM of numerators}}{\text{HCF of denominators}} \] while \[ \text{HCF}=\frac{\text{HCF of numerators}}{\text{LCM of denominators}}. \]

Answer 1

The Correct Option is C

Concept: The L.C.M. of fractions is calculated using \[ \text{LCM of fractions} = \frac{\text{LCM of numerators}} {\text{HCF of denominators}} \]

Step 1: Find the LCM of the numerators.
Numerators are \[ 7,\ 5,\ 14 \] Prime factorization: \[ 7=7,\qquad 5=5,\qquad 14=2\times7 \] Therefore, \[ \text{LCM}(7,5,14) = 2\times5\times7 = 70 \]

Step 2: Find the HCF of the denominators.
Denominators are \[ 3,\ 6,\ 9 \] Factors: \[ 3=3,\qquad 6=2\times3,\qquad 9=3^2 \] Common factor is only \(3\). Hence, \[ \text{HCF}(3,6,9)=3 \]

Step 3: Calculate the required LCM.
\[ \text{LCM} = \frac{70}{3} \] Therefore, \[ \boxed{\frac{70}{3}} \]