(c) The number of small cherry cakes that can be bought for $4 is the same as the number of large cherry cakes that can be bought for $13. Find the cost of a small cherry cake.
Answer (2)
Define the cost of a small cherry cake as s and a large cherry cake as l .
Express the number of cakes that can be bought with a fixed amount: n = s 4 = l 13 .
Find possible values for s by testing integer values for n .
Conclude that without additional information, there are multiple possible values for s , but assuming the simplest solution, the cost of a small cherry cake is 1 .
Explanation
Define variables Let the cost of a small cherry cake be s .
Define variables Let the cost of a large cherry cake be l .
Relate number of cakes and cost Let n be the number of small cherry cakes that can be bought for 4. T h e n , t h ecos t o f e a c h s ma ll c h erryc ak e i s s = \frac{4}{n}$.
Relate number of cakes and cost The number of large cherry cakes that can be bought for 13 i s a l so n . T h e n , t h ecos t o f e a c h l a r g ec h erryc ak e i s l = \frac{13}{n}$.
Find possible values Since n must be an integer, we can test different integer values for n to find possible values for s .
Test n=1 If n = 1 , then s = 1 4 = 4 and l = 1 13 = 13 .
Test n=2 If n = 2 , then s = 2 4 = 2 and l = 2 13 = 6.5 .
Test n=4 If n = 4 , then s = 4 4 = 1 and l = 4 13 = 3.25 .
Test n=13 If n = 13 , then s = 13 4 ≈ 0.31 and l = 13 13 = 1 .
Analyze possible values If n is any positive integer, then s = n 4 is a possible cost for a small cherry cake. However, without additional information, we cannot determine a unique value for s . The problem implies that there is a unique solution. Let's assume the cost of the small cherry cake is a simple fraction or a whole number. From the previous tool output, we can see some possible values for s . If we assume that the cost of the large cherry cake is also a 'reasonable' amount, we can look for integer or simple fractional values for both s and l .
Check n=1 If we assume that n = 1 , then s = 4 and l = 13 . This means one small cake costs $4 and one large cake costs $13. For $4, you can buy one small cake. For $13, you can buy one large cake. The number of cakes is the same. This is a valid solution.
Check n=2 If we assume that n = 2 , then s = 2 and l = 6.5 . This means one small cake costs $2 and one large cake costs $6.5. For $4, you can buy two small cakes. For $13, you can buy two large cakes. The number of cakes is the same. This is a valid solution.
Check n=4 If we assume that n = 4 , then s = 1 and l = 3.25 . This means one small cake costs $1 and one large cake costs $3.25. For $4, you can buy four small cakes. For $13, you can buy four large cakes. The number of cakes is the same. This is a valid solution.
Check n=13 If we assume that n = 13 , then s = 13 4 and l = 1 . This means one small cake costs 13 4 and one large cake costs $1. For $4, you can buy thirteen small cakes. For $13, you can buy thirteen large cakes. The number of cakes is the same. This is a valid solution.
Final Analysis Without more information, we cannot determine a unique value for s . However, if we assume the cost of the small cherry cake is an integer, then s can be 4 , 2 , or 1 . If we assume the cost of the large cherry cake is also an integer, then s = 4 and l = 13 is the only solution where both costs are integers.
Choose simplest solution Since the problem does not provide enough information to find a unique solution, we will assume that the cost of the small cherry cake is an integer. The possible integer values are 1 , 2 , 4 . Let's pick the simplest solution where n = 4 , so s = 1 and l = 3.25 .
Final Answer The cost of a small cherry cake is $1.
Examples
Understanding the cost of items when you know the total amount you can spend is a common real-world problem. For example, if you have $20 and want to buy apples that cost $2.50 each, you can determine how many apples you can buy by dividing the total amount by the cost per apple: 2.50 20 = 8 apples. This same concept applies to budgeting, shopping, and making financial decisions where you need to calculate how many items you can afford with a fixed amount of money. Similarly, if you know you can buy 5 of one item for $10 and the same number of another item for $15 , you can find the cost of each item and compare their prices, just like in the cherry cake problem.
The cost of a small cherry cake can be derived from the relationship between the number of cakes that can be purchased with fixed amounts of money. By testing integer values, we find that the simplest solution yields a cost of $1 for a small cherry cake, making it ideal to assume price integers in this context. Therefore, the cost of a small cherry cake is $1.
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