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

3 Answers
May 27, 2018

#~~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"#

May 27, 2018

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.

May 27, 2018

#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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