Three languages 2

Participants of a congress may present their contributions in English, Italian, or Spanish. Each of the 120 participants knows at least two of these languages, and 10 participants speak all three languages. English and Spanish is spoken by exactly as many participants as those who know English and Italian but do not know Spanish. Spanish and Italian but without English is known by the same number of participants as those who speak all three languages.

Determine using a Venn diagram:
English is known by a participants.
English is not known by A participants.
Spanish is known by s participants.
Spanish is not known by S participants.
Italian is known by t participants.
Italian is not known by T participants.
English and Italian is known by q participants.
English and Spanish is known by w participants.
Italian and Spanish is known by e participants.

Final Answer:

a =  110
A =  10
s =  65
S =  55
t =  75
T =  45
q =  65
w =  55
e =  20

Step-by-step explanation:

$$\begin{gathered} z=10 \\ x+10+(x+10)+z=120 \\ x+10+(x+10)+10=120 \\ 2x=90 \\ x=\frac{90}{2}=45 \\ x=45 \\ y=x+10=45+10=55 \\ a=x+10+y=45+10+55=110 \end{gathered}$$

$A=z=10$

$s=x+10+z=45+10+10=65$

$S=y=55$

$t=y+10+z=55+10+10=75$

$T=x=45$

$q=y+10=55+10=65$

$w=x+10=45+10=55$

$e=z+10=10+10=20$

Try another example