How to find LCM and solve fractions| lcm | least common multiple | how to add subtract fractions - Free Printable

Let's solve both fraction problems step by step using the LCM (Least Common Multiple) method, which is a reliable and efficient way to add or subtract fractions with different denominators.

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🔹 Problem 1:

$$
\frac{2}{9} + \frac{5}{12} = ?
$$

#### Step 1: Find the LCM of the denominators (9 and 12)

- Prime factorization:
- $9 = 3^2$
- $12 = 2^2 \times 3$
- LCM = Take the highest powers of all primes:
- $2^2 \times 3^2 = 4 \times 9 = 36$

So, the LCM is 36.

#### Step 2: Convert each fraction to have denominator 36

- $\frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36}$
- $\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}$

#### Step 3: Add the fractions

$$
\frac{8}{36} + \frac{15}{36} = \frac{23}{36}
$$

✔ Final Answer:
$$
\frac{2}{9} + \frac{5}{12} = \frac{23}{36}
$$

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🔹 Problem 2:

$$
\frac{22}{27} - \frac{8}{15} = ?
$$

#### Step 1: Find the LCM of 27 and 15

- Prime factorization:
- $27 = 3^3$
- $15 = 3 \times 5$
- LCM = $3^3 \times 5 = 27 \times 5 = 135$

So, LCM is 135

#### Step 2: Convert each fraction

- $\frac{22}{27} = \frac{22 \times 5}{27 \times 5} = \frac{110}{135}$
- $\frac{8}{15} = \frac{8 \times 9}{15 \times 9} = \frac{72}{135}$

#### Step 3: Subtract

$$
\frac{110}{135} - \frac{72}{135} = \frac{38}{135}
$$

✔ Final Answer:
$$
\frac{22}{27} - \frac{8}{15} = \frac{38}{135}
$$

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✔ Summary:

1. $\frac{2}{9} + \frac{5}{12} = \boxed{\frac{23}{36}}$
2. $\frac{22}{27} - \frac{8}{15} = \boxed{\frac{38}{135}}$

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💡 "Best Trick" Tip (Optional but helpful):

Instead of finding LCM every time, you can use the cross-multiplication method for two fractions:

For $\frac{a}{b} \pm \frac{c}{d}$, the result is:
$$
\frac{a \cdot d \pm c \cdot b}{b \cdot d}
$$

Then simplify.

Example:
$\frac{2}{9} + \frac{5}{12} = \frac{2 \cdot 12 + 5 \cdot 9}{9 \cdot 12} = \frac{24 + 45}{108} = \frac{69}{108} = \frac{23}{36}$

Same answer! But this gives a larger numerator/denominator first — so always simplify.

> So the best trick is: Use cross-multiplication if you're fast at simplifying, but LCM helps avoid large numbers and is clearer for beginners.

Let me know if you'd like a visual or diagram explanation! 😊