Question

A class that has 35 students, 24 of them finished the year. How much is that in percent?

Answer 1

`~~68.57%" to 2 dec. places"`

Explanation:

`"express as a fraction and multiply by "100%`

`"that is "24/35xx100%`

`=(24xx100)/35%~~68.57%" to 2 dec. places"`

Answer 2

68.571428...%

Explanation:

defination of percent:
how many in a one hundred?

take this for example:
there are 5 balls, 2 of them is red, how much is that in percent?

we know there are 2 red balls in 5 balls
so there is 2 in a 5 : "`2/5`"
what if the scale is the same and you have 100 balls in total?
this is what you do:
`(2xx20)/(5xx20)=40/100`
so there is 40 red balls in 100 of them, so the percentage of 2 red balls in 5 balls is 40%
40% means "`40/100`"

caution:
half means `50/100=1/2=50%`
`1/2%`is not half, by the way
`1/2% < 1% <50%`

back to your question:
24 students finished the year, they are 35 of them, what if the scale doesn't change and you have 100 students in total?

`24/35=(24xxk)/(35xxk)`
in this case, we let `35xxk = 100`, so `k = 100/35`
`24/35=(24xxk)/(35xxk)=(24xx(100/35))/(35xx(100/35))=(68.571428...)/100`

so the percentage is `68.571428...%`

to calculate percentage very quick on a calculator:

`("what you want?")/("how many in total?")xx100%`

i hope this would help you.

Answer 3

`68.57%` to 2 decimal places

If you round off a value ALWAYS stipulate the decimal places. It is all to do with being aware of the level of precision.

Explanation:

`color(blue)("The teaching bit")`

Percentage is just another fraction. However, it is a special fraction in that the bottom number (denominator) is always 100.

There are two ways of writing a fraction and they both mean `ul("exactly")` the same thing.

Example: suppose we had 20 percent. This can be written as:
`20%" or "20/100`

The thing is; `20/100` is the same as `20xx1/100`. Now we directly compare the two:

`20 color(white)("ddd") %`

`20color(white)("d")obrace(xx1/100)`

As these are EXACTLY the same thing then % is the same as `xx1/100` including the multiply.
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`color(blue)("Answering the question using first principles")`

Of the whole we have `color(red)(24)/color(green)(35)`

We need to change the bottom number of 32 into 100 to get our percentage.

So to change the bottom number we do this: `color(green)(cancel(35))xx100/cancel(35)=100`

To maintain proportionality of the fraction what we do to the bottom we also do to the top for multiply or divide. So we have:

`(color(red)(24)xx100/35)/(color(green)(35)xx100/35) = ubrace([color(red)(24)xx100/35])xxubrace([1/(color(green)(35)xx100/35)] )`
`color(white)("ddddddddddd") 68.5714 ...color(white)("d")xxcolor(white)("ddd")1/100`

But `xx1/100` is the same as % so we have:

`68.57%` to 2 decimal places
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`color(blue)("Answering the question using shortcut method")`

`24/35 ->(24-:35)`

Multiply by 100

`(24-:35)xx100`

Stick a % on the end

`(24-:35)xx100 % =68.57%` to 2 decimal places

Until recently I thought this was not really correct. Since then I have changed my mind. Mathematically this is `ul("very correct")` but I do not wish to confuse matters by explaining why. The use of the symbol % is very significant.
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