We are asked to find the
LCM (Least Common Multiple) of the following fractions:
$$
\frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{11}{12}
$$
---
🔍 Step 1: Understand LCM of Fractions
The
LCM of fractions is given by the formula:
$$
\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}
$$
>
✔ Important: This formula applies when we want the
least common multiple of fractions, which represents the smallest number that is a multiple of all the given fractions.
---
Step 2: Identify Numerators and Denominators
Given fractions:
- $\frac{2}{3}$ → numerator: 2, denominator: 3
- $\frac{4}{9}$ → numerator: 4, denominator: 9
- $\frac{5}{6}$ → numerator: 5, denominator: 6
- $\frac{11}{12}$ → numerator: 11, denominator: 12
So,
- Numerators: 2, 4, 5, 11
- Denominators: 3, 9, 6, 12
---
Step 3: Find LCM of Numerators
Find
LCM of 2, 4, 5, 11
Break them into prime factors:
- $2 = 2$
- $4 = 2^2$
- $5 = 5$
- $11 = 11$
Take the highest powers of all primes:
- $2^2$, $5^1$, $11^1$
$$
\text{LCM of numerators} = 2^2 \times 5 \times 11 = 4 \times 5 \times 11 = 220
$$
---
Step 4: Find HCF (Highest Common Factor) of Denominators
Denominators: 3, 9, 6, 12
Prime factorizations:
- $3 = 3$
- $9 = 3^2$
- $6 = 2 \times 3$
- $12 = 2^2 \times 3$
Common prime factor: only
3, and the lowest power is $3^1$
$$
\text{HCF of denominators} = 3
$$
---
Step 5: Apply the Formula
$$
\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{220}{3}
$$
---
✔ Final Answer:
$$
\boxed{\frac{220}{3}}
$$
This is the
least common multiple of the given fractions. It is the smallest number that is a multiple of each of the fractions.
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📝 Optional: Verify Conceptually
To verify, suppose we want a number $x$ such that:
- $x$ is divisible by $\frac{2}{3}$, $\frac{4}{9}$, $\frac{5}{6}$, $\frac{11}{12}$
That means:
- $x = k_1 \cdot \frac{2}{3} = k_2 \cdot \frac{4}{9} = \dots$
But in terms of multiples, it's easier to think in terms of
common multiples of fractions — and the standard method we used is correct.
---
✔ Conclusion:
$$
\boxed{\frac{220}{3}}
$$
Parent Tip: Review the logic above to help your child master the concept of fractions lowest common multiple.
