SOLUTION: There are 25 students in a class. Of these, 14 speak Spanish, 12 speak French, 6 speak French and Spanish, 5 speak German and Spanish, 2 speak all three languages. Each of the 6 w

Algebra ->  Permutations -> SOLUTION: There are 25 students in a class. Of these, 14 speak Spanish, 12 speak French, 6 speak French and Spanish, 5 speak German and Spanish, 2 speak all three languages. Each of the 6 w      Log On


   

Question 1078801: There are 25 students in a class. Of these, 14 speak Spanish, 12 speak French, 6 speak French and
Spanish, 5 speak German and Spanish, 2 speak all three languages. Each of the 6 who speak German
speaks another one of these languages as well.
(a) How many speak both French and German?
(b) How many speak none of these three languages?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let s = number of students who speak spanish
let f = number of students who speak french
let g = number of students who speak german.

let sf = number of students who speak spanish and french
let sg = number of students who speak spanish and german
let fg = number of students who speak french and german

let sfg = number of students who speak spanish and french and german.

all of these categories include subtending categories.

this means that:

s includes sf and sg and sfg
f includes sf and fg and sfg
g includes sg and fg and sfg

you want to purify these categories so that they do not include any subtending categories.

these pure categories will have an o (for only) letter added to the end of them.

you will get:

sfgo = sfg

sfo = sf - sfgo
sgo = sg - sfgo
fgo = fg - sfgo

so = s - sfo - sgo - sgfo
fo = f - sfo - fgo - sgfo
go = g - sgo - fgo - sgfo

your formula for t will be:

t = so + fo + go + sfo + sgo + fgo + sfgo + n

t is the total which is equal to 25
n represent the students who aren't in each category.

you start with:

s = 14
f = 12
g = 6
sf = 6
sg = 5
fg = x
sfg = 2
n = y

fg = x because we don't know what that number is.
n = y because we don't know what that number is either.

first you want to find sfgo.

sfgo = sfg = 2

next you want to find sfo

sfo = sf - sfgo = 6 - 2 = 4

next you want to find sgo

sgo = sg - sfgo = 5 - 2 = 3

next you ant to find fgo

fgo = fg - sfg = x - 2 = x-2

so far you have:

s = 14
f = 12
g = 6
sf = 6
sfo = 4
sg = 5
sgo = 3
fg = x
fgo = x-2
sft = 2
sfgo = 2

now you want to find so.

so = s - sfo - sgo - sgfo = 14 - 4 - 3 - 2 = 5

now you want to find fo.

fo = f - sfo - fgo - sgfo = 12 - 4 - (x-2) - 2 = 8-x

now you want to find go.

go = g - sgo - fgo - sfgo = 6 - 3 - (x-2) - 2 = 3-x

so far you have:

s = 14
so = 5
f = 12
fo = 8-x
g = 6
go = 3-x
sf = 6
sfo = 4
sg = 5
sgo = 3
fg = x
fgo = x-2
sft = 2
sfgo = 2

you know that go must be equal to 0 because you are told that each of the 6 students who speak german speak another language.

sincego = 3-x, this means that x must be equal to 3 because 3-3 = 0

assuming that x = 3, you get:

s = 14
so = 5
f = 12
fo = 8-3 = 5
g = 6
go = 3-3 = 0
sf = 6
sfo = 4
sg = 5
sgo = 3
fg = x
fgo = 3-2 = 1
sft = 2
sfgo = 2

to simplify these results, you end up with:


s = 14
so = 5
f = 12
fo = 5
g = 6
go = 0
sf = 6
sfo = 4
sg = 5
sgo = 3
fg = x
fgo = 1
sft = 2
sfgo = 2

if you let k = the total number of students and if you let n = the number students who don't speak any of the 3 languages, your formula will become:

k = so + fo + go + sfo + sgo + fgo + sgf + n

since the total number of students is 25, then k = 25.

replacing all the variable with their respective numbers, you get:

25 = 5 + 5 + 0 + 4 + 3 + 1 + 2 + n

n is still these because we don't know what that number is yet.

simplifying you get:

25 = 20 + n

solve for n to get n = 5

you wanted to know the number of students who speak german and french plus the number of students who don't speak any of those languages.

that means you want fgo plus sfgo plus n

fgo = 1
sfgo = 2
n = 5

fg is equal to fgo + sfgo, therefore fg = 3

the number of students who speak french and german is 3.

the number of students who do not speak any of those 3 languages is 5.

that's my somewhat educated guess.

i'm reasonably sure it's correct.

how do i know?

i worked the problem back from the solution and came up with the original numbers of the problem.

you were originally given that s = 14, f = 12, g = 6

the table i drew up is shown below: 



                        s               f              g
                      
s only                  5               0              0
f only                  0               5              0
g only                  0               0              0
sf only                 4               4              0
sg only                 3               0              3
fg only                 0               1              1
sgf only                2               2              2


total                   14              12             6


since those are the numbers for s, f, and g originally given, i'm reasonably certain that i got it right.

consequently, i'll go with:

number of students who speak german and french = 3
number of students who don't speak any of the three languages = 5