In a group of people, 12% speak both English and French, and 9% speak neither. If 15% more people speak only English than only French, find the percentage of people who speak only English and only Fre

: Language percentages

Let $E$ be the set of people who speak English and $F$ be the set of people who speak French.
Let $P(X)$ denote the percentage of people in set $X$.

Given information:

1. $P(E \cap F) = 12\%$ (speak both English and French)
2. $P(\text{neither E nor F}) = 9\%$ (cannot speak any of the two)
3. $P(\text{only E}) = P(\text{only F}) + 15\%$ (percentage speaking only English is 15% more than those speaking only French)

From (2), the percentage of people who speak at least one language is:

$$P(E \cup F) = 100\% - P(\text{neither E nor F}) = 100\% - 9\% = 91\%$$

We know that $P(E \cup F) = P(\text{only E}) + P(\text{only F}) + P(E \cap F)$.
Substituting the known values:

$$91\% = P(\text{only E}) + P(\text{only F}) + 12\%$$ $$P(\text{only E}) + P(\text{only F}) = 91\% - 12\% = 79\%$$

Now we have a system of two equations:
I. $P(\text{only E}) = P(\text{only F}) + 15\%$
II. $P(\text{only E}) + P(\text{only F}) = 79\%$

Substitute equation I into equation II:

$$(P(\text{only F}) + 15\%) + P(\text{only F}) = 79\%$$ $$2 \cdot P(\text{only F}) + 15\% = 79\%$$ $$2 \cdot P(\text{only F}) = 79\% - 15\%$$ $$2 \cdot P(\text{only F}) = 64\%$$ $$P(\text{only F}) = \frac{64\%}{2} = 32\%$$

Now find $P(\text{only E})$ using equation I:

$$P(\text{only E}) = 32\% + 15\% = 47\%$$

a) Represent the information on a Venn diagram.
A Venn diagram would show two overlapping circles, one for English (E) and one for French (F), within a rectangle representing the total population (100%).

• The intersection of E and F would contain $12\%$.
• The part of circle E outside the intersection (only E) would contain $47\%$.
• The part of circle F outside the intersection (only F) would contain $32\%$.
• The area outside both circles (neither E nor F) would contain $9\%$.

b) Find the percentage of people who can speak:
i. English

$$P(E) = P(\text{only E}) + P(E \cap F) = 47\% + 12\% = \boxed{59\%}$$

ii. French

$$P(F) = P(\text{only F}) + P(E \cap F) = 32\% + 12\% = \boxed{44\%}$$

iii. Only French

$$P(\text{only F}) = \boxed{32\%}$$

c) Find the number of people who can speak only French if the population of the village is 4000.
Number of people who speak only French = $P(\text{only F}) \times \text{Total Population}$

$$= 32\% \times 4000$$ $$= \frac{32}{100} \times 4000$$ $$= 32 \times 40$$ $$= \boxed{1280 \text{ people}}$$