Equations with fractions · البرمجة مع مارتن

How to get rid of fractions in one move: multiply the whole equation by the common denominator.

Fractions gone in one move

The equation x/3 + 2 = 6 looks unpleasant. A fraction next to x makes people nervous. Good news: there is a move that sweeps all the fractions away in one step.

Multiply the whole equation by the denominator. Both sides, every term. With x/3 + 2 = 6 multiply by three: x + 6 = 18. The fraction disappeared and an equation is left that you have been solving for three lessons.

Why is that allowed? Still the scales rule: when you triple both pans, the balance stays.

When there are more denominators — say x/2 + x/3 = 10 — multiply by the common denominator. For 2 and 3 that is 6: the smallest number that both divide. You remember fractions from year 7; here that hard work comes back as an advantage.

Underline today’s one rule twice: you multiply every term of the equation. Even the one that has no fraction. A skipped term = a rocked scale.

Tereza got her report and grandma sent her money. She saved a third, spent 200 Kč on a book and was left with exactly 400 Kč. “How much did grandma send?” Filip writes: x − x/3 − 200 = 400. “Ugh, a fraction.” Tereza: “So chase it out. Times three, everything.” 3x − x − 600 = 1200 → 2x = 1800 → x = 900. Check: a third of 900 is 300, minus 200 for the book… 400 left. It fits. “Grandma sent nine hundred. And we only saw the fraction at the start.”

💡

Before multiplying, circle EVERY term of the equation. Then multiply circle by circle. The most common mistake is not in the fractions — it is in the term without a fraction that got forgotten.

Method: multiply by the common denominator

1. Find the common denominator. Denominators are the numbers at the bottom of the fractions. For 2 and 3 the common denominator is 6, for 4 and 6 it is 12. It is enough to take the smallest number that all the denominators divide.

2. Multiply EVERY term of both sides by it. x/2 + x/3 = 10, I multiply by six:

6 · x/2 + 6 · x/3 = 6 · 10.

3. Cancel the fractions. 6 · x/2 = 3x (six divided by two), 6 · x/3 = 2x. You get 3x + 2x = 60.

4. Solve as before. 5x = 60 → x = 12.

5. Check in the original equation: 12/2 + 12/3 = 6 + 4 = 10. It fits.

Read a fraction like x/3 as “x divided by three” — that is why multiplying by three cancels it: 3 · x/3 = x.

When there is a decimal and a fraction in the equation at the same time, feel free to convert the decimal to a fraction or the other way — pick what you like better. An equation can stand both, just be consistent.

Example 1: x/4 = 7

One fraction, denominator 4. I multiply the whole equation by four:

4 · x/4 = 4 · 7

x = 28.

Check: 28/4 = 7. It fits.

The shortest possible show of the whole idea: multiplying by the denominator cancels the fraction, because “divided by four times four” cancel.

Example 2: x/5 + 3 = 9

Denominator 5. I multiply by five — WATCH, the three and the nine too:

5 · x/5 + 5 · 3 = 5 · 9

x + 15 = 45.

I subtract 15: x = 30.

Check: 30/5 + 3 = 6 + 3 = 9. It fits.

Anyone who multiplies only the fraction and leaves “+ 3 = 9” gets x + 3 = 9 and x = 6. Check: 6/5 + 3 = 4.2. It does not fit — and that is exactly how the mistake is spotted.

Example 3: x/2 − x/6 = 4

Denominators 2 and 6. Common denominator: 6.

I multiply by six: 6 · x/2 − 6 · x/6 = 6 · 4.

I cancel: 3x − x = 24.

I collect: 2x = 24 → x = 12.

Check: 12/2 − 12/6 = 6 − 2 = 4. It fits.

Two fractions, one move, no common denominators inside the fractions — multiplying dealt with them all at once.

⚠️

If you multiply the equation by six, you multiply EVERY term — even the one without a fraction. x/2 + 1 = 4 times 6 is 3x + 6 = 24, not 3x + 1 = 24. A forgotten term is mistake number one with fraction equations.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. البداية with ten. When it goes well, add more.

Practise adding fractions — finding a common denominator is the same muscle you need with equations.

Practise the questions

On paper

Circle the terms, multiply, cancel, solve. And a check into the original equation with the fractions.

Solve: x/3 = 9, x/7 + 2 = 5. For the second one circle all three terms before multiplying.
Solve x/2 + x/4 = 6. Write which common denominator you chose.
Solve 2x/3 − 1 = 7 and do a check.
Tereza spent half her pocket money on a cinema ticket and 50 Kč on popcorn; she had 70 Kč left. Write the equation x/2 + 50 + 70 = x (or your own version), solve and check.

Now you 💪

I find the common denominator of more fractions.
I multiply every term of the equation, even the one without a fraction.
After multiplying I can cancel the fractions into whole terms.

✅

Done when: You have four fraction equations solved with checks and a run of five correct in both practices.

What to take from this lesson

You remove fractions from an equation by multiplying by the common denominator.
Every term of both sides is multiplied — no exception.
x/3 is “x divided by three”; times three returns it to x.
After cancelling the fractions you solve an ordinary equation.
The check is تم in the original equation with the fractions.