In a group of people, 12% speak both English and French, and 5% speak neither. If 15% more speak only English than only French, what percentage speak only English and what percentage speak only French

Let $E$ represent people who speak English and $F$ represent people who speak French.
Given information:

• Percentage of people who speak both English and French, $P(E \cap F) = 12\%$.
• Percentage of people who speak neither language, $P(E \cup F)' = 5\%$.
• Percentage of people who speak only English, $P(E \text{ only})$, is $15\%$ more than those who speak only French, $P(F \text{ only})$. So, $P(E \text{ only}) = P(F \text{ only}) + 15\%$.

The sum of all distinct regions must equal $100\%$:

$$P(E \text{ only}) + P(F \text{ only}) + P(E \cap F) + P(E \cup F)' = 100\%$$

Substitute the known values:

$$P(E \text{ only}) + P(F \text{ only}) + 12\% + 5\% = 100\%$$ $$P(E \text{ only}) + P(F \text{ only}) + 17\% = 100\%$$ $$P(E \text{ only}) + P(F \text{ only}) = 100\% - 17\%$$ $$P(E \text{ only}) + P(F \text{ only}) = 83\%$$

Now we have a system of two equations:

1. $P(E \text{ only}) + P(F \text{ only}) = 83\%$
2. $P(E \text{ only}) = P(F \text{ only}) + 15\%$

Substitute equation (2) into equation (1):

$$(P(F \text{ only}) + 15\%) + P(F \text{ only}) = 83\%$$ $$2 \cdot P(F \text{ only}) + 15\% = 83\%$$ $$2 \cdot P(F \text{ only}) = 83\% - 15\%$$ $$2 \cdot P(F \text{ only}) = 68\%$$ $$P(F \text{ only}) = \frac{68\%}{2}$$ $$P(F \text{ only}) = 34\%$$

Now find $P(E \text{ only})$ using equation (2):

$$P(E \text{ only}) = 34\% + 15\%$$ $$P(E \text{ only}) = 49\%$$

The percentages for each region are:

• Only English: $49\%$
• Only French: $34\%$
• Both English and French: $12\%$
• Neither English nor French: $5\%$

A Venn diagram would show two overlapping circles, one for English (E) and one for French (F), within a rectangle representing the total population. The percentages for each region are:

• Region E only: \boxed{\text{49%}}
• Region F only: \boxed{\text{34%}}
• Region E and F (intersection): \boxed{\text{12%}}
• Region outside E and F (neither): \boxed{\text{5%}}

b) i) English:
The percentage of people who can speak English is the sum of those who speak only English and those who speak both.

$$P(E) = P(E \text{ only}) + P(E \cap F)$$ $$P(E) = 49\% + 12\%$$ P(E) = \boxed{\text{61%}}

b) ii) French:
The percentage of people who can speak French is the sum of those who speak only French and those who speak both.

$$P(F) = P(F \text{ only}) + P(E \cap F)$$ $$P(F) = 34\% + 12\%$$ P(F) = \boxed{\text{46%}}

The total population of the village is 840.
The percentage of people who can speak only French is $34\%$.
Number of people who can speak only French = $34\%$ of 840.

$$\text{Number} = 0.34 \times 840$$ $$\text{Number} = \boxed{\text{285.6}}$$