Question - Finding Largest Possible Number of Items in Set F
Solution:
The image shows a Venn diagram with two sets, F and G, intersecting. Within set F, but not in the intersection, the expression "9u + 12" is written. Within set G, but not in the intersection, the expression "8u + 3" is written. Within the intersection, the expression "7u + 3" is given. The question asks: Given that there are fewer than 94 items in total, what is the largest possible number of items in set F?In order to find the largest possible number of items in set F, considering there are fewer than 94 items in total, we will first express the total number of items combining both sets, F and G, including their intersection.The total number of items in set F is the sum of the distinct items in F and the items F shares with G, which would be (9u + 12) + (7u + 3).The total number of items in set G is the sum of the distinct items in G and the items G shares with F, which would be (8u + 3) + (7u + 3).Since the sets F and G overlap in the intersection, we should not double count the intersection when calculating the grand total. Hence, the intersection (7u + 3) is added just once to the grand total.So the total number of items is:(9u + 12) + (8u + 3) + (7u + 3) - (7u + 3)The `7u + 3` from the intersection is added twice but subtracted once to account correctly for those items that are both in F and G.Simplifying, the total is:(9u + 12) + (8u + 3) + (7u + 3) - (7u + 3)= 9u + 12 + 8u + 3= 17u + 15Now, we use the information given in the question that there are fewer than 94 items in total:17u + 15 < 94Subtracting 15 from both sides, we get:17u < 79Dividing by 17, we find:u < 79 / 17u < 4.647...Since u must be a whole number (as it represents the count of items), the largest possible value for u is 4.Now, we can find the largest possible number of items in set F by plugging in u = 4 into our expression for set F:9u + 12 = 9(4) + 12= 36 + 12= 48Therefore, the largest possible number of items in set F, given that there are fewer than 94 items in total, is 48.
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