Question

Sylvie sells cakes. small cakes cost $3 each, medium cakes cost $5 each and large cakes cost $7 dollars each. Adam has $41 to buy 11 cakes for his work colleagues. He will buy the cakes from Sylvie and will spend all the money. List all the possible combinations of cakes that adam could buy.

Answer 1

To solve for all the possible combinations, we need to find sets of three integers for small, medium, and large cakes such that their total price is $41, and they add up to 11 cakes. This involves using diophantine equations to explore combinations that meet the price and quantity constraints.

Step-by-step explanation:

The student asks for all possible combinations of small, medium, and large cakes that can be bought with $41, considering the prices ($3, $5, $7 respectively) and the total number of cakes needed, which is 11. This is a classic example of a problem that can be solved using the diophantine equations in mathematics, catering to combinations and permutations based on given constraints.

We need to find combinations of the variables s (small cakes), m (medium cakes), and l (large cakes) that satisfy the following system of equations:

• 3s + 5m + 7l = 41 (Total cost equation)
• s + m + l = 11 (Total number of cakes equation)

It's a matter of iterating over possible values of s, m, and l while keeping the equations consistent. We are looking to find positive integer solutions because you can't purchase a fraction of a cake. Through a systematic approach, we can list out all potential combinations that meet the criteria.

Bold example:

One possible combination could be buying 2 small cakes, 3 medium cakes, and 6 large cakes since (2*3)+(3*5)+(6*7) equals $41 and 2+3+6 equals 11 cakes in total.

Answer 2

Adam can choose from five different combinations of small, medium, and large cakes that total 11 cakes and use all of his $41.

Step-by-step explanation:

Adam has $41 to buy 11 cakes. Small cakes cost $3 each, medium cakes cost $5 each, and large cakes cost $7 each. To find all the possible combinations of cakes that Adam could buy, we need to solve the equation:

3s + 5m + 7l = 41 And s + m + l = 11

where s, m, and l are the number of small, medium, and large cakes, respectively. Through various combinations that satisfy both the price and quantity constraints, we can determine all possible combinations which are:

• 1 small, 5 medium, 5 large
• 3 small, 4 medium, 4 large
• 5 small, 3 medium, 3 large
• 7 small, 2 medium, 2 large
• 9 small, 1 medium, 1 large

These combinations all add up to 11 cakes and total $41.