In short

Fractions trip people up because every operation you perform on the equation also has to keep the denominators straight — and adding \frac{x}{2} to \frac{x}{3} already needs an LCM step before you can even combine them. The clean fix is to do the LCM step once, at the very start, on the whole equation. Find the LCM of every denominator that appears, multiply both sides by it, distribute through every term, and watch every denominator cancel. What remains is an ordinary integer linear equation that you solve the usual way.

You stare at \frac{x}{2} + \frac{x}{3} = 5 and your brain stalls. The same student who can fly through 2x + 3x = 30 in five seconds suddenly slows to a crawl. Why? Because the denominators feel like little obstacles on every line — you have to keep checking whether you broke them, whether you forgot one, whether your common denominator is right.

The trick is to stop fighting the denominators on every step and instead kill them all at once. CBSE Class 8 introduces this as the standard recipe for fraction-laden linear equations, and once you see it work you will never go back to combining fractions term by term. (NCERT Class 8 Chapter 2 builds entire exercise sets around it.)

Why fractions are hard

Look at what an innocent equation like \frac{x}{2} + \frac{x}{3} = 5 actually demands.

To combine the left side, you have to:

  1. Notice the denominators are 2 and 3.
  2. Find their LCM, which is 6.
  3. Rewrite \frac{x}{2} as \frac{3x}{6} and \frac{x}{3} as \frac{2x}{6}.
  4. Add the numerators: \frac{3x + 2x}{6} = \frac{5x}{6}.
  5. Now the equation reads \frac{5x}{6} = 5, and you finally solve.

That is five steps just to add two fractions. Now imagine the equation has three fraction terms, brackets in the numerators, and constants on both sides. Each fraction is a small accounting headache, and the moment you forget to convert one term properly the whole solution wobbles.

There is a deeper reason fractions feel hard: division is the slow operation in arithmetic. Multiplication and addition are quick. Every \frac{1}{n} in your equation is a hidden division, and your brain has to manage all of them at once.

The LCM trick

Here is the move. Do not combine the fractions first. Instead, find the LCM of every denominator in the entire equation — left side and right side together — and multiply both sides by that number.

For \frac{x}{2} + \frac{x}{3} = 5, the denominators are 2 and 3. Their LCM is 6. Multiply both sides by 6:

6 \cdot \left(\frac{x}{2} + \frac{x}{3}\right) = 6 \cdot 5

Distribute on the left:

6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 30 \implies 3x + 2x = 30

Why: the LCM is, by definition, divisible by every denominator. So when you multiply \frac{x}{2} by 6, the 2 in the denominator divides cleanly into 6, leaving 3x. The same happens for \frac{x}{3}, leaving 2x. Every fraction's denominator dies because the LCM was built to kill it.

Now you are looking at 5x = 30, a kindergarten-level linear equation. x = 6. Done.

The whole point: instead of fighting denominators on every line of working, you spend one line annihilating them all simultaneously, and then solve a clean integer equation.

The procedure

Five steps, in order, every time:

Equation with fractions x/2 + x/3 = 5 Identify all denominators { 2, 3 } Compute LCM LCM = 6 Multiply every term by LCM 6·(x/2)+6·(x/3)=6·5 Fractions vanish 3x + 2x = 30 One LCM step replaces dozens of denominator-juggling steps.

Example 1 — the warm-up

Solve \dfrac{x}{2} + \dfrac{x}{3} = 5.

Denominators are 2 and 3. \text{LCM}(2, 3) = 6.

Multiply both sides by 6:

6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 5
3x + 2x = 30
5x = 30 \implies x = 6

Verify by plugging back: \frac{6}{2} + \frac{6}{3} = 3 + 2 = 5. Correct.

Example 2 — fractions with brackets in the numerator

Solve \dfrac{x+1}{4} - \dfrac{2x-3}{6} = 1.

Denominators are 4 and 6. \text{LCM}(4, 6) = 12.

Multiply both sides by 12:

12 \cdot \frac{x+1}{4} - 12 \cdot \frac{2x-3}{6} = 12 \cdot 1
3(x+1) - 2(2x-3) = 12

Why: 12 \div 4 = 3 kills the first denominator, 12 \div 6 = 2 kills the second. The numerators stay intact, now multiplied by those quotients.

Distribute:

3x + 3 - 4x + 6 = 12
-x + 9 = 12 \implies -x = 3 \implies x = -3

Verify: \frac{-3+1}{4} - \frac{2(-3)-3}{6} = \frac{-2}{4} - \frac{-9}{6} = -\frac{1}{2} + \frac{3}{2} = 1. Correct.

Example 3 — the trap (forgetting the constants)

Solve \dfrac{x}{2} + 5 = \dfrac{x}{3} + 7.

Denominators are 2 and 3. \text{LCM} = 6. Multiply every term by 6 — including the 5 and the 7:

6 \cdot \frac{x}{2} + 6 \cdot 5 = 6 \cdot \frac{x}{3} + 6 \cdot 7
3x + 30 = 2x + 42

The wrong move — and a very common one — is to write 3x + 5 = 2x + 7, leaving the constants alone because "they have no fractions to clear". But the rule is to multiply the whole equation by 6, which means every term on both sides. Why: an equation only stays balanced if you do the same thing to both whole sides. Multiplying just the fractions by 6 is doing one operation to part of a side and a different operation (none) to the other part — that breaks the equation.

Continuing correctly:

3x + 30 = 2x + 42 \implies x = 12

Verify: \frac{12}{2} + 5 = 6 + 5 = 11 and \frac{12}{3} + 7 = 4 + 7 = 11. Correct.

That third example is the single biggest source of fraction-equation errors in CBSE Class 8 papers. Treat the LCM multiplication as a global operation on the whole equation, not a local cleanup of the fractions only, and you will never fall into it.

References