Question

All sacks of sugar have the same weight. All sacks of flour also have the same weight, but not necessarily the same as the weight of the sacks of sugar. Suppose that two sacks of sugar together with three sacks of flour weigh no more than $40$ pounds, and that the weight of a sack of flour is no more than $5$ pounds more than the weight of two sacks of sugar. What is the largest possible weight (in pounds) of a sack of flour?

Answer 1

The largest possible weight of a sack of flour is 10 pounds.

Step-by-step explanation:

We can solve this problem using a system of inequalities. Let's define the variables:

S = weight of a sack of sugar (in pounds)

F = weight of a sack of flour (in pounds)

Based on the given information, we can write the following inequalities:

1. 2S + 3F ≤ 40 (the total weight of 2 sacks of sugar and 3 sacks of flour must be no more than 40 pounds)
2. 
   
3. 
   
4. F ≤ S + 5 (the weight of a sack of flour is no more than 5 pounds more than the weight of two sacks of sugar)

To find the largest possible weight of a sack of flour (F), we need to maximize the value of F while satisfying these inequalities.

Let's solve the system of inequalities:

1. From inequality (2), we can substitute F = S + 5 into inequality (1):
2. 
   
3. 
   
4. 2S + 3(S + 5) ≤ 40
5. 
   
6. 
   
7. 2S + 3S + 15 ≤ 40
8. 
   
9. 
   
10. 5S + 15 ≤ 40
11. 
    
12. 
    
13. 5S ≤ 25
14. 
    
15. 
    
16. S ≤ 5
17. 
    
18. 
    
19. From S ≤ 5, we can substitute S = 5 into inequality (2) to find the maximum value of F:
20. 
    
21. 
    
22. F ≤ 5 + 5
23. 
    
24. 
    
25. F ≤ 10

Therefore, the largest possible weight of a sack of flour is 10 pounds.