Question

How do you solve `3/4+x=5/4`?

Answer 1

`x=1/2`

Explanation:

We start with `3/4+x=5/4`. Our goal is to solve for `x`, which means that we need to subtract `3/4` on both sides, leaving us with `x=5/4-3/4`. To subtract equations, we first must make sure the demoninators are the same. In our case they are, so now we just deal with the numerators. Treat it like `5-3`, which is `2`. Now we slap on the denominator and get `2/4`. That can be simplified to `1/2`, which means that `x=1/2`.

Answer 2

`x=1/2`

`color(purple)("Solution split into 2 parts.")`
`color(purple)("Part 1: Detailed explanation about adding and subtracting fractions.")`

`color(purple)("Part 2: Answering your question.")`

Explanation:

`color(blue)("Insight into addition and subtraction of fractions")`

Hold the thought for a moment that when you apply the process of, say, `6-2` you are manipulating counts.

Now consider a fraction (rational number). We have the structure of:

`" "("count")/("size indicator of what you are counting")`

Using the proper names for these we have:

`" " ("numerator")/("denominator")`

Size indicator tells you how many of what you are counting are needed to make a whole of something (a complete 1)

So
For `1/2` it takes 2 of them to make a whole but we have got a count of 1 of them.

For `2/16` it takes 16 of them to make a whole but we have got a count of 2 of them

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Ok! Lets go back to `6-2`

Write this as `6/1-2/1" "larr" not normally done this way"`

The denominators (size indicators) are the same so we can `ul("directly")` apply the subtraction of the counts

,...............................................................................................................
So if you wish to add subtract the counts you need to make the size indicators the same. Otherwise you are trying to do the equivalent operation as the following example.

`2/("box of apples")" "-" "3/("single apples")`

You need to convert the size indicator of "box of apples" to the size indicator of "single apples" before you can determine how many apples you are left with.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

`3/4+x=5/4`

Subtract `color(blue)(3/4)` from both sides

`color(brown)(3/4color(blue)(-3/4)+x=5/4color(blue)(-3/4)`

But `3/4-3/4=0`

`0+x=5/4-3/4`

`x=5/4-3/4`

The size indicators (denominators) are the same so we can directly subtract the counts.

`x=(5-3)/4 = 2/4`

But `2/4` is equivalent to `1/2`

`x=1/2`

Answer 3

`x =1/2`

Explanation:

Fractions are numbers which many students find difficult to work with. Luckily with equations which have fractions, we can get rid of any fractions immediately by a multiplying by the LCM of the denominators. (same as the LCD)

Note that: if you multiply a fraction by a multiple of the denominator, the denominator can cancel.
For example:

`8/3 xx6 = 8/cancel3 xxcancel6^2 =16`

`2/5 xx15 = 2/cancel5xxcancel15^3 = 6`

In this question both the denominators are 4
.
Multiply the WHOLE equation by 4

`color(red)(4xx) 3/4 +color(red)(4xx)x = color(red)(4xx)5/4`

`color(red)(cancel4xx) 3/cancel4 +color(red)(4xx)x = color(red)(cancel4xx)5/cancel4`

`3+4x = 5`

`4x = 5-3 = 2`

`x = 1/2`