Adding fractions with different denominators · البرمجة مع مارتن

A half plus a quarter

How much is 1/2 + 1/4? Definitely not 2/6. Half a pizza and a quarter of a pizza together is almost a whole pizza — and 2/6 is a third. Something does not add up.

The problem: halves and quarters are parts of different sizes. And you can only add the same things — you know that from decimal numbers (korunas with korunas) and from the last lesson (eighths with eighths).

The fix: dress the fractions in the same denominators. For that you have exactly the right tool from lesson eight — expanding. From 1/2 you make 2/4 and suddenly you add quarters with quarters: 2/4 + 1/4 = 3/4. Done.

The whole magic of today’s lesson is finding a common denominator — a number you can expand both fractions onto. The simplest candidate: a number that is in the times table of both denominators. For 2 and 4 it is 4. For 3 and 4 it is 12.

This is the most important lesson about fractions. Give it time.

Tereza is baking a cake with mum. The recipe wants 1/2 a cup of milk in the batter and 1/4 of a cup in the icing. “How much milk should I measure in total?” Tereza asks. She tries adding the ones and the two with the four: “Two sixths? That is less than the half on its own, that cannot be true.” Mum takes a measuring cup: “Half a cup is two quarters.” Tereza fills in: “Two quarters plus one quarter… three quarters of a cup!” She measures 3/4 and it sits exactly. From then on she knows: first the same parts, only then adding.

💡

The fastest common denominator: look whether the bigger denominator is a multiple of the smaller one. With 1/2 + 1/4, 4 = 2 × 2 — you only expand one fraction.

Method: find, expand, add

Step 1: Find a common denominator. You look for a number that is in the times table of both denominators. For 3 and 4: multiples of three are 3, 6, 9, 12… and multiples of four 4, 8, 12… → common denominator 12. When you are stuck, the product of the denominators always works (3 × 4 = 12) — sometimes it is bigger than you need, but it is never wrong.

Step 2: Expand both fractions onto that denominator. Exactly like in lesson eight — both numbers by the same multiple:

1/3 into twelfths: times 4 → 4/12
1/4 into twelfths: times 3 → 3/12

Step 3: Add like last time. Same denominators you already can: 4/12 + 3/12 = 7/12.

Step 4: Simplify when you can. 7/12 is already in simplest form.

Subtracting: exactly the same method, you just subtract in step 3.

The most common easy case: one denominator is a multiple of the other (2 and 4, 3 and 6, 5 and 10). Then you expand only one fraction and leave the other — half the work. Always look for that before you start multiplying the denominators together.

Work out 1/2 + 1/4

Find: 4 is a multiple of 2 → common denominator 4. I only expand the half.

Expand: 1/2 = 2/4 (times 2 at the top and the bottom).

Add: 2/4 + 1/4 = 3/4.

Simplify: 3/4 is already simplest form.

A common-sense check: a half and a quarter together is less than a whole, but more than a half. 3/4 sits exactly. (And 2/6 from the start? A third — smaller than the half on its own. Clear nonsense.)

Work out 2/3 + 1/4

Find: 3 and 4 — neither is a multiple of the other. I take the product: 3 × 4 = 12.

Expand both:

2/3 = 8/12 (times 4)
1/4 = 3/12 (times 3)

Add: 8/12 + 3/12 = 11/12.

Check: 2/3 is more than a half, 1/4 is a piece on top — the result should be just under a whole. 11/12 = a whole minus one twelfth. It sits beautifully.

Work out 5/6 − 1/2

Subtracting, the same method.

Find: 6 is a multiple of 2 → common denominator 6.

Expand: 1/2 = 3/6. I leave the fraction 5/6.

Subtract: 5/6 − 3/6 = 2/6.

Simplify: 2/6 = 1/3 (divided by two).

A story for it: in the kettle there is 5/6 of a litre of tea, you pour half a litre into a flask. A third of a litre is left.

⚠️

Do not add “top with top, bottom with bottom”. 1/2 + 1/4 is not 2/6. Until fractions have different denominators, you must not add anything — first dress both in the same parts, only then add the numerators.

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.

Add fractions — sometimes with the same, sometimes with different denominators. First a common denominator, then add. Write the answer as a fraction with a slash, for example 7/12.

Practise the questions

On paper

Write all four steps: find — expand — add — simplify. Shortcuts in a month, now build the habit.

Work out: 1/2 + 1/6, then 1/3 + 1/4, then 3/5 + 3/10. Under each one underline the common denominator.
Work out: 3/4 − 1/2, then 7/10 − 2/5, then 5/6 − 1/3. Simplify the results to simplest form.
Tereza’s cake again: 1/2 a cup of milk in the batter, 1/4 in the icing and 1/8 for brushing. How much of a cup of milk in total?
Adam read 1/3 of a book on Monday and 1/4 of the book on Tuesday. What part of the book is left? (The whole book = 1 = 12/12.)

Now you 💪

I can find a common denominator for 3 and 4 and for 2 and 6 — and I know when it is enough to expand only one fraction.
I can go through the whole find–expand–add–simplify method without skipping a step.
I can spot nonsense like 1/2 + 1/4 = 2/6 and say why it is wrong.

✅

Done when: You add and subtract fractions with different denominators: you find a common denominator, expand, add the numerators and simplify the result.

What to take from this lesson

Different denominators = parts of different sizes. Those cannot be added.
Method: find a common denominator → expand → add the numerators → simplify.
A common denominator is in the times table of both denominators. The product of the denominators always works.
When one denominator is a multiple of the other, you only expand one fraction.
Check with an estimate: a half + a quarter must sit between a half and a whole.