Compound fractions · Martinとコード

A fraction inside a fraction is not a monster. The main fraction bar means divide — and you already know that.

A fraction over a fraction? That is just hidden division

Sometimes you meet a fraction that has another fraction in its numerator or denominator. Like two thirds over four fifths, written one above the other like a layered cake. It is called a compound fraction.

It looks scary. But it holds one secret: the fraction bar means divide. The main, longest bar in the middle says “divide what is on top by what is on the bottom”.

So a compound fraction with 2/3 on top and 4/5 on the bottom is ordinary division 2/3 : 4/5. And dividing fractions you handled last time: flip and multiply.

The only real job is spotting which bar is the main one. It is usually the longest, and the equals sign sits next to it. Once you find it, rewrite the stacked writing into one line with a colon — and the monster is gone.

Ema found in her older brother’s textbook a question: a stacked fraction across three lines. “That is for secondary school, right?” she asks Adam. Adam shrugs and then notices: “Wait, that long bar in the middle… that is just divide.” They rewrite the beast into a line: (3/4) : (1/2). Flip, multiply: 3/4 × 2/1 = 6/4 = 3/2. Ema shakes her head: “It looked like an hour of counting and it is two lines.” Adam: “Half of maths is rewriting a thing so it looks like something you already know.”

💡

First find the main fraction bar (the longest one) and rewrite the whole thing into one line with a colon. Only then calculate.

How to tame a compound fraction

Step 1: find the main bar. It is the longest and it splits the upper floor from the lower one. Everything above it is the dividend, everything below it is the divisor.

Step 2: rewrite into a line. Upper floor : lower floor. If a floor has a mixed number or a sum, first merge it into one fraction. For example the upper floor 1 + 1/2 rewrite as 3/2.

Step 3: divide by flipping. Flip the divisor and multiply. Cancelling crossways pays off here too.

Step 4: simplify the result to simplest form, or convert to a mixed number when that is a nicer answer (7/2 = 3 and 1/2).

A full example: a compound fraction with 5/6 on top and 10/9 on the bottom. Rewrite: 5/6 : 10/9. Flip: 5/6 × 9/10. Cancel crossways: five against ten (1 and 2), nine against six (3 and 2). Left is 1/2 × 3/2 = 3/4.

You will meet compound fractions in the rule of three, in physics (speed = distance over time, where both can be fractions) and in recipes. It is not new material — it is a test of whether dividing fractions has landed.

Upper floor 3/4, lower floor 9/8

Step 1: the main bar splits 3/4 on top and 9/8 on the bottom.

Step 2: I rewrite into a line: 3/4 : 9/8.

Step 3: I flip the divisor and multiply: 3/4 × 8/9. I cancel crossways: three against nine (left 1 and 3), eight against four (left 2 and 1).

Step 4: 1/1 × 2/3 = 2/3.

Result: 2/3. Check: I divide by a number bigger than 1 (9/8), so the result must be smaller than 3/4. And 2/3 is smaller (8/12 < 9/12). That checks out.

A whole number on top: 6 over 3/5

The upper floor is ordinary six, the lower floor the fraction 3/5.

Step 1: I rewrite into a line: 6 : 3/5.

Step 2: I convert six to a fraction: 6/1 : 3/5.

Step 3: I flip and multiply: 6/1 × 5/3. I cancel: six against three, left 2 and 1.

Step 4: 2/1 × 5/1 = 10.

Result: 10. A thought for a check: how many times do three fifths fit into six wholes? Into one whole they fit about 1.7 times, into six about ten times. That checks out.

A sum on a floor: on top 1/2 + 1/3, on the bottom 5/6

Step 1: first I tidy the upper floor. The common denominator of 2 and 3 is 6: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Step 2: I rewrite into a line: 5/6 : 5/6.

Step 3: a number divided by itself is 1. (If you do not believe it: 5/6 × 6/5 = 30/30 = 1.)

Result: 1. The lesson: before you start flipping and multiplying, merge each floor into a single fraction. It saves you a mess.

⚠️

Do not count the floors out of order. First merge the upper floor into one fraction, then the lower one, only then divide. If you start cancelling numbers across the main bar while there are still sums on a floor, you make nonsense.

Practice: now you do it 🎲

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

A compound fraction is division — train it here. Write the answer as a fraction a/b.

Practise the questions

On paper: unpack the layered cake

Write all four steps for each question: floors, line, flip, result.

Solve the compound fraction: on top 2/5, on the bottom 4/15. Do not forget cancelling crossways.
Solve: on top 8, on the bottom 2/3. First rewrite the whole number as a fraction.
Solve: on top 1/4 + 1/2, on the bottom 3/8. First add the upper floor.
Invent your own compound fraction whose result is 2. Check it by calculating.

Now you 💪

I can spot the main fraction bar and I know it means divide.
Before I divide, I merge each floor into a single fraction.
I can write the result in simplest form or as a mixed number.

✅

Done when: Three compound fractions are solved with all the steps, and your own invented fraction really gives 2 after checking.

What to take from this lesson

The main fraction bar = divide. A compound fraction is division of two fractions.
First merge the floors, then rewrite into a line, then flip and multiply.
Rewrite a whole number on a floor as a fraction with denominator 1.
A number divided by itself is always 1.
A compound fraction is not new material — it is dividing fractions in costume.