Why does this fraction flip?

Question

Can someone please explain the logic as to why this fraction flips?

1/1-1/4 = 4/3

Why does it equal 4/3 when my calculator says 3/4?

Thank you

Answer 1

HINT: write $$\frac{1}{1}-\frac{1}{4}=\frac{4}{4}-\frac{1}{4}=\cdots$$

Answer 2

The answer isn't 4/3, because subtracting a positive number from 1 won't give you a number larger than 1. Why do you think your calculator might be wrong?

Answer 3

It's an issue of order of operation :

The correct way to calculate this is :

$$1/1-1/4 = {1\over 1}-{1\over 4} = {3\over 4}$$

But someone ( or a calculator ) who lacks the order of operation rules might do it like this :

$$1/1-1/4 = {1\over (1-{1\over 4})} = {1\over {3\over 4}} = {4\over 3}$$

It is my opinion that one should always use the $a\over b$ syntax or use parenthesis to avoid any misinterpretation when using inline symbols like "$/$"

Answer 4

This is a case where good formatting and the order of operations can be very helpful. If we disregard the order of operations, what you have written can be interpreted in several different ways:

1. 1/1-1/4 could be a list of keystrokes on a calculator. If you were to enter this string in to a typical calculator, you would get something like

   1 / 1   = 1
   - 1     = 0
   / 4     = 0
   

   thus the result would be zero. Parentheses or formatting would make this more clear. This computation could be written as \begin{equation*} \frac{\frac{1}{1} - 1}{4} \quad\text{or}\quad ((1/1)-1) / 4. \end{equation*}

2. If we respect the order of operations, then multiplication and division happen before addition and subtraction (maybe you have heard one of the mnemonics PEMDAS or GEMDAS?). In this case, the fractions are evaluated first, then the subtraction happens. With parentheses or good formatting, this might be rendered as $(1/1) - (1/4)$, or \begin{equation} \frac{1}{1} - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4}. \end{equation}

3. Finally, it might have meant $1/(1-(1/4))$, which would be written as \begin{equation} \frac{1}{1-\frac{1}{4}}= \frac{1}{\frac{4}{4} - \frac{1}{4}} = \frac{1}{\frac{4-1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}. \end{equation}

This problem highlights how important it is to keep track of when you are performing different operations. Without further guidance, rely on the order of operations. If you can, use more parentheses. There is no such thing as too many parentheses (probably (okay, maybe there is ;))).

Answer 5

Do not put your faith in correctly functioning calculators. Even when they function correctly you need to keep your brain in gear.

The notation $\dfrac 1 1 - \dfrac 1 4$ means $\left(\dfrac 1 1\right) - \left( \dfrac 1 4 \right).$

But if you enter $1\div1-1\div 4,$ most calculators will treat it as $\left( \left( 1\div 1\right) - 1\right) \div 4.$

But you probably did it in some way that caused the calculator to treat it as $1\div\left( 1 - \left( 1\div 4 \right) \right).$