العربية

أوروبا

آسيا

← Level 3 – Fraction kid

8 / 30 ⏱ 16 minutes

Expanding and simplifying fractions

The same fraction in a different coat: expanding and simplifying.

One fraction, many outfits

Half a pizza. You cut it into two pieces — you have 2/4. You cut it into four — you have 4/8. It is still the same amount of pizza. Just cut differently.

1/2 = 2/4 = 4/8 = 50/100. One fraction, many write-ups. And switching between them is the skill almost all later fraction counting stands on.

Expanding: you multiply the numerator and the denominator by the same number. From 1/2 you get 2/4 (both times 2). More parts, but smaller — the value does not change.

Simplifying: you divide the numerator and the denominator by the same number. From 6/8 you get 3/4 (both divided by 2). Fewer parts, but bigger — the same value.

Why learn it? Simplifying turns fractions into readable shapes (12/16 tells nobody anything, 3/4 does). And expanding you will need right in lesson ten, when you add fractions with different denominators.

Ema handed in a test and reports to Filip: “I had 12 out of 16 points.” Filip: “And is that a lot, or a little?” Ema writes 12/16 and starts simplifying: “Twelve and sixteen can both be divided by four. 12 : 4 = 3, 16 : 4 = 4.” She shows the result: 3/4. “Three quarters of the points. That is like 75 out of a hundred.” Filip whistles — 3/4 he can picture at once, 12/16 not at all. Same mark, same test, but the simplified fraction suddenly speaks clearly. That is exactly what simplifying is for: translating a fraction into a shape you understand.

💡

Simplify in steps when you do not see the big number at once: 24/36 → divided by 2 → 12/18 → divided by 2 → 6/9 → divided by 3 → 2/3. More small steps is also right.

Both numbers, the same number, always

The golden sentence of this lesson: What you do at the bottom, do at the top too — with the same number.

Expanding (multiplying):

  1. Pick a number, for example 3.
  2. Multiply the numerator and the denominator by it: 2/5 → (2×3)/(5×3) = 6/15.
  3. The value did not change: 2/5 = 6/15.

Why it works: you cut each part into 3 smaller ones. You have 3× more parts, but each is 3× smaller. Times three and divided by three cancel out.

Simplifying (dividing):

  1. Find a number that both numbers can be divided by with no remainder — a common factor.
  2. Divide both by it: 6/8 → (6:2)/(8:2) = 3/4.
  3. Repeat until there is no common factor left (except 1). Then the fraction is in simplest form.

How to look for a common factor: try small numbers — 2 (both even?), 3, 5. With 12/16 you see both can be divided by 4 — or twice in a row by two, the result is the same.

A check that you simplified correctly: the simplified fraction can be expanded back to the original. 3/4 expanded by four is 12/16 ✓.

Expand 3/4 into twelfths

I want denominator 12. I ask: what do I multiply 4 by to get 12? Answer: 3 (because 4 × 3 = 12).

I must multiply the numerator by the same number: 3 × 3 = 9.

Result: 3/4 = 9/12.

A picture check: a chocolate bar into 4 columns, I have 3. I cut each column into 3 squares → 12 squares in total, mine are 9. Still the same piece of chocolate.

Simplify 18/24 as far as it goes

I look for common factors:

Round 1: Both numbers even → I divide by 2: 18/24 → 9/12.

Round 2: 9 and 12 can both be divided by 3: 9/12 → 3/4.

Round 3: 3 and 4 have no common factor (except 1). Done — simplest form.

A faster path: who sees that 18 and 24 can be divided by 6 at once is at 3/4 in one step. Both paths are right, the result is always the same.

Are 2/3 and 10/15 the same fraction?

I try to expand 2/3 so the bottom is 15. What times 3 gives 15? Five.

I expand by five: (2×5)/(3×5) = 10/15.

Yes — 2/3 = 10/15, it is the same fraction in a different outfit.

The other path: simplify 10/15 by five → 2/3. Both paths meet. That is how you spot “twin fractions”: one can be dressed as the other by expanding or simplifying.

⚠️

Never add. Expanding is multiplying, not adding: from 1/2, adding one at the top and the bottom makes 2/3 — and that is a COMPLETELY different fraction (bigger!). Only multiplying and dividing both numbers by the same number keeps the value.

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.

Simplify fractions to simplest form. Write the answer as a fraction with a slash, for example 3/4.

Practise the questions

On paper

By each step write which number you multiply or divide by. Who writes the steps does not make mistakes.

  1. Expand: 1/2 into sixths, 2/5 into twentieths, 3/8 into twenty-fourths. Always write which number you expand by.

  2. Simplify to simplest form: 8/12, 15/20, 24/36. Write the steps one by one.

  3. Ema had 12/16 of the points on a test, Adam 15/20. Simplify both fractions and decide who did the test better — or if it was the same.

  4. Write four different outfits of the fraction 2/3 (expand by different numbers). Then simplify one of them back and check that 2/3 comes out.

Now you 💪

  1. I can expand 3/4 to any denominator that is a multiple of four.
  2. I can simplify 18/24 to simplest form and I know when to stop.
  3. I can explain why the value does not change when you expand — for example by cutting chocolate.
✅

Done when: You expand and simplify a fraction by multiplying/dividing both numbers by the same number and you can tell when a fraction is in simplest form.

What to take from this lesson

© 2026 Ing. Martin Polak / AlgoRhino · شروط استخدام المحتوى