Question

How do you subtract `\frac { 31} { 126} - \frac { 31} { 144}`?

Answer 1

`31/1008`

Explanation:

First we need a common denominator. For that find the least common multiple (LCM) of `126` and `144`:

`126 = 2xx3xx3xx7`

`144 = 2xx2xx2xx2xx3xx3`

So their LCM is:

`2xx2xx2xx2xx3xx3xx7 = 1008`

...being the smallest number that includes all of their prime factors in at least the multiplicity that they occur in each number.

So:

`31/126-31/144 = (31xx8)/(126xx8)-(31xx7)/(144xx7)`

`color(white)(31/126-31/144) = (31xx8)/1008-(31xx7)/1008`

`color(white)(31/126-31/144) = (31xx(8-7))/1008`

`color(white)(31/126-31/144) = 31/1008`

Answer 2

`31/1008`

Explanation:

`color(blue)("Preamble")`

`2xx4xx126=1008` as the common denominator.

`color(brown)("Two very important facts")`

`color(brown)("Fact 1 :")`

A fraction's structure is `("count")/("size indicator")->("numerator")/("denominator")`

You can not DIRECTLY add or subtract the counts unless the size indicators are the same.

`color(brown)("Fact 2 :")`
Multiply a number by 1 and you do not change its intrinsic value. However, 1 comes in many forms so you can change the way a number looks without changing its intrinsic value.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`
You need a common multiple of 126 and 144. Ok lets try something.
Look at the last digits. We have 6 and 4.

`color(magenta)(2)xx6=12 and 12-:4=3" A whole number so it may work."`

So lets test: `(126xx2)-:144=1.75` Not a whole number so fails.

However `color(magenta)(4)xx1.75=7` which is a whole number so lets look at

`(color(magenta)(2xx4)xx126)-:144=7` this works so I choose to use the common denominator of: `2xx4xx126 = 1008`

`" "color(green)([31/126color(red)(xx1)] - [31/144color(red)(xx1)]`

`" "color(green)([31/126color(red)(xx8/8)] - [31/144color(red)(xx7/7)]`

`" "color(green)([248/1008]-[217/1008])`

`" "31/1008`

Answer 3

`31/1008`

Explanation:

`31/126-31/144`

`:.=(248-217)/1008`

`:.=31/1008`

check:

`31/126=0.246031746` ,`31/144=0.215277777`

`0.246031746-0.215277777=color(red)0.030753968`

`31/1008=color(red)0.030753968`