Joel made egg and tuna sandwiches in the ratio of 2:3. He then made 15 more tuna sandwiches and the ratio changed to 1:3. How many sandwiches did Joel make in all?
2/3=3/t; solving for t=4.5
when calculating to 15 tuna sandwiches, makes 10 egg sandwiches but this is still 2:3.
1:3 of 15 tuna sandwiches makes 5 egg sandwiches for total of 20 sandwiches.
Not clear on what the 2:3 does for this problem.
t = FINAL tuna
e/(t-15) = 2/3
e/t = 1/3
so finally t = 3 e
e/ (3e -15) = 2/3
3 e = 6 e - 30
e = 10
t = 3 t = 30
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The problem states that Joel made egg and tuna sandwiches in the ratio of 2:3. This means that for every 2 egg sandwiches, he made 3 tuna sandwiches. Let's say he initially made 2x egg sandwiches and 3x tuna sandwiches.
We are given that when he made 15 more tuna sandwiches, the ratio changed to 1:3. This means that for every 1 egg sandwich, he made 3 tuna sandwiches. In this case, he made y egg sandwiches and 3y tuna sandwiches.
We can set up the following equations based on the given information:
2x/3x = 2/3
1y/3y = 1/3
To solve the first equation:
2x/3x = 2/3
Cross-multiplying, we get:
2x * 3 = 2 * 3x
6x = 6x
Canceling out the common terms, we are left with:
2 = 2
This equation is true for any value of x, which indicates that the ratio 2:3 is maintained irrespective of the number of sandwiches made.
To solve the second equation:
1y/3y = 1/3
Multiplying both sides by 3y, we get:
1y = 3y/3
Simplifying, we have:
1y = y
This equation is also true for any value of y, indicating that the ratio 1:3 is maintained when Joel makes 15 more tuna sandwiches.
Based on this information, we can conclude that the ratio does not impact the number of sandwiches Joel made in total. Therefore, we can simply add up the total number of egg and tuna sandwiches made:
Total sandwiches = (2x + y) + (3x + 3y)
= 2x + y + 3x + 3y
= (2x + 3x) + (y + 3y)
= 5x + 4y
Since the value of x and y is not given in the problem, we cannot determine the exact number of sandwiches Joel made in total.
To solve this problem, let's break it down step by step:
1. Initially, Joel made egg and tuna sandwiches in the ratio of 2:3. This means that for every 2 egg sandwiches, he made 3 tuna sandwiches.
2. Let's assume that Joel made a total of x sandwiches initially.
3. Since the ratio of egg to tuna sandwiches is 2:3, we can set up the equation 2/3 = (x-15)/x. This is because after making 15 more tuna sandwiches, the ratio changed to 1:3.
4. Solving the equation, we get 2x = 3(x-15).
5. Expanding the equation, we have 2x = 3x - 45.
6. Subtracting 3x from both sides, we get x = 45.
7. So initially, Joel made 45 sandwiches.
8. Now, let's calculate the number of sandwiches he made after adding the 15 additional tuna sandwiches.
9. Initially, the ratio was 2:3, meaning for every 2 egg sandwiches, there were 3 tuna sandwiches.
10. Since Joel made 45 sandwiches initially, the number of egg sandwiches he made is (2/5) * 45 = 18.
11. After adding 15 more tuna sandwiches, the ratio changes to 1:3. This means for every 1 egg sandwich, there are 3 tuna sandwiches.
12. So, the number of egg sandwiches after adding 15 more tuna sandwiches is (1/4) * 45 = 11.25.
13. Since we can't have a fraction for sandwiches, we will round up to the nearest whole number, which is 12.
14. Therefore, Joel made a total of 45 + 12 = 57 sandwiches in all.