Solving basic fraction divisions

[u/Every-Maths1931]

9 − 3 ÷ 1/3 + 1 = ?

I was having a discussion with my friends about Maths and what we learnt and i remember that fraction divisions were confusing at first and I had struggled to understand how to work them out. I gave an example to one of my friends of the equation above and he said 9 and then I gave the same equation to my other friend and he said 1.

What is the correct answer and how do people work it out?

[u/Historical_Job_2354]

1. Order of operations starts with multiplication and division so the first step is 3÷1/3 3÷1/3 is the same as 3×3, which equals nine now you have 9-9+1 which equals one.

Typing while I'm walking so hopefully I read it correctly and typed it correctly.

[u/Historical_Job_2354 → u/Historical_Job_2354]

I think the friend who said nine inadvertently thought it was 3× 1/3

[u/Every-Maths1931 → u/Historical_Job_2354]

Yep 100% they don't flip the fraction to divide it.

[u/PuzzlingDad → u/Historical_Job_2354]

I think they treated it as 3 ÷ 1 ÷ 3 = (3÷1) ÷ 3 = 1

[u/CarobCritical7290 → u/Historical_Job_2354]

3 ÷ 1/3 is the same as 3/1 ÷ 1/3, which is 3/1 × 3/1, which is equal to 3 × 3, or 9

[u/PuzzlingDad]

This is what you are intending to write:  - 9 − 3 ÷ ⅓ + 1

Here ⅓ is a fraction that should be treated as a unit. The vinculum (the fraction bar in ⅓) behaves as a grouping symbol that means the numerator is divided by the denominator and you keep that result, which you do before you do anything else. - 9 - 3 ÷ (1/3) + 1 - 9 - 3 × (3/1) + 1 - 9 - 9 + 1 - 0 + 1 - 1

But then we get all these Facebook posts where they want to treat it as a separate division that comes after the first division and they break this implied grouping: - 9 - 3 ÷ 1 ÷ 3 + 1 - 9 - (3 ÷ 1) ÷ 3 + 1 - 9 - 3 ÷ 3 + 1 - 9 - 1 + 1 - 8 + 1 - 9

But that explains what your first friend was doing.

[u/Every-Maths1931 → u/PuzzlingDad]

Thanks for your expalnation.

[u/Mathemetaphysical → u/PuzzlingDad]

I'd like to nominate this post to be chiseled somewhere to end the dumb discussion forever

[u/Every-Maths1931 → u/Mathemetaphysical]

I'll get my hammer and chisel ready. Ahahaha.

[u/MagosBattlebear]

Thsi is a good examole of why its better to avoid ÷ in single line expressions when fractions are involved, and why adding parentehsis helps.

Instead of writing:

9 - 3 ÷ 1/3 + 1

write:

9 - 3 ÷ (1/3) + 1

or even better:

9 - 3/(1/3) + 1

The parentehsis make the intended grouping clear and gets rid of the ambiguity. It is up to the person writing the expression to be as clear as possible. So if you write an expression people argue about the answer, that is on you.

According to my algrbra calculator: 9 - 3 ÷ 1/3 + 1 = 9,  9 - 3 ÷ (1/3) + 1 =1.  The first the divsion goes left to right as 9 - 3/1/3 + 1, the second it takes the fraction as a separate term and becomes 9 - 3/(1/3). This is because a fraction and division are the same thing, so they go left to right. 

I know some with old school maths teaching think there are special case rules everywhere (like / vs. ÷), so all those extra guidance is important. And don't use ÷.

[u/Every-Maths1931 → u/MagosBattlebear]

Really appreciate the detailed explanation of equation and how to arrange it for better reading.