1. **Problem statement:** (a) Given the Venn diagram with sets for vegetables (V) and cash crops (C), and expressions for the number of people in each region, find the value of $y$ given the ratio of people who grow neither crop to those who grow vegetables is 1:2. (b) Find the probability of selecting a person who grows only one type of crop. (c) Annie bought 17 articles (books and pens) costing 25000 total, with books costing 2000 each and pens 1000 each. Find the fraction of pens to total articles. --- 2. **Step (a): Find $y$** - Number of people who grow vegetables $n(V) = (2y + 3) + y = 3y + 3$ - Number of people who grow neither crop $= y + 3$ - Given ratio: neither : vegetables $= 1 : 2$ Set up the ratio: $$\frac{y + 3}{3y + 3} = \frac{1}{2}$$ Cross multiply: $$2(y + 3) = 3y + 3$$ Simplify: $$2y + 6 = 3y + 3$$ Rearrange: $$6 - 3 = 3y - 2y$$ $$3 = y$$ So, $y = 3$. --- 3. **Step (b): Probability of selecting a person who grows only one type of crop** - Only vegetables: $2y + 3 = 2(3) + 3 = 6 + 3 = 9$ - Only cash crops: $3y - 3 = 3(3) - 3 = 9 - 3 = 6$ - Total number of people in the universal set $\Sigma$: $$ (2y + 3) + y + (3y - 3) + (y + 3) = 2y + 3 + y + 3y - 3 + y + 3 = 7y + 3$$ Substitute $y=3$: $$7(3) + 3 = 21 + 3 = 24$$ - Number who grow only one type = $9 + 6 = 15$ - Probability = $\frac{15}{24} = \frac{5}{8}$ --- 4. **Step (c): Fraction of pens to total articles** - Let number of books = $b$, pens = $p$ - Total articles: $b + p = 17$ - Total cost: $2000b + 1000p = 25000$ - From first equation: $b = 17 - p$ - Substitute into cost equation: $$2000(17 - p) + 1000p = 25000$$ Simplify: $$34000 - 2000p + 1000p = 25000$$ $$34000 - 1000p = 25000$$ $$-1000p = 25000 - 34000 = -9000$$ $$p = \frac{9000}{1000} = 9$$ - Fraction of pens to total articles: $$\frac{p}{17} = \frac{9}{17}$$ --- **Final answers:** (a) $y = 3$ (b) Probability of selecting a person who grows only one type of crop = $\frac{5}{8}$ (c) Fraction of pens to total articles = $\frac{9}{17}$