A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is
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Official Correct Answer: 340. Alright, let's try to tackle this CAT problem step by step. It's about a fruit seller with mangoes, bananas, and apples. He starts the day with a certain number of each fruit, and during the day, he sells some of them. By the end of the day, he's sold 50% of his total stock. We need to find the smallest possible total number of fruits he could have started with.
First, let's parse the problem:
1. At the beginning, mangoes make up 40% of his stock. So if the total number of fruits is T, then mangoes are 0.4T. The rest, 60%, are bananas and apples. Let's denote bananas as B and apples as A. So, B + A = 0.6T.
2. During the day, he sells:
- Half of the mangoes: So he sold 0.5 * 0.4T = 0.2T mangoes.
- 96 bananas.
- 40% of the apples: So he sold 0.4A apples.
3. At the end of the day, he's sold 50% of the fruits. That means the total sold is 0.5T.
So, the total sold is the sum of mangoes sold, bananas sold, and apples sold. Therefore:
0.2T + 96 + 0.4A = 0.5T
Let me write that equation down:
0.2T + 96 + 0.4A = 0.5T
Now, let's rearrange this equation to find a relationship between T and A.
Subtract 0.2T from both sides:
96 + 0.4A = 0.3T
So,
0.4A = 0.3T - 96
Multiply both sides by 10 to eliminate decimals:
4A = 3T - 960
So,
4A = 3T - 960
We can write this as:
3T = 4A + 960
So,
T = (4A + 960)/3
Since T must be an integer (you can't have a fraction of a fruit), (4A + 960) must be divisible by 3. So, 4A + 960 ≡ 0 mod 3.
Computing 960 mod 3. 960 divided by 3 is 320, so 960 is divisible by 3. Therefore, 4A must also be divisible by 3. Since 4 and 3 are coprime, A must be divisible by 3.
Let’s denote A = 3k, where k is a positive integer.
Substituting back into T:
T = (4*(3k) + 960)/3 = (12k + 960)/3 = 4k + 320
So, T = 4k + 320
Now, remember that at the beginning, the number of bananas and apples must be integers. Since A = 3k, and B + A = 0.6T, let's express B in terms of T.
We have:
B = 0.6T - A = 0.6T - 3k
But since T = 4k + 320,
B = 0.6*(4k + 320) - 3k = 2.4k + 192 - 3k = -0.6k + 192
Wait, that gives B = -0.6k + 192. But B must be a positive integer because he has at least one banana.
So,
-0.6k + 192 > 0
=> -0.6k > -192
Multiply both sides by (-1), which reverses the inequality:
0.6k < 192
=> k < 192 / 0.6 = 320
So, k < 320. Since k is a positive integer, k can be from 1 to 319.
But we also need B to be a positive integer. B = -0.6k + 192 must be an integer. Since 0.6 is 3/5, let's write this as:
B = (-3/5)k + 192
For B to be an integer, (-3/5)k must be an integer. Therefore, k must be a multiple of 5. Let’s denote k = 5m, where m is a positive integer.
Substituting back:
k = 5m
So,
T = 4*(5m) + 320 = 20m + 320
A = 3k = 15m
B = (-3/5)*(5m) + 192 = -3m + 192
So, B = 192 - 3m
Since B must be positive,
192 - 3m > 0
=> 3m < 192
=> m < 64
So, m can be from 1 to 63.
Now, we need to ensure that all quantities are positive integers:
- T = 20m + 320 must be positive, which it is for m ≥ 1.
- A = 15m must be positive, which it is.
- B = 192 - 3m must be positive, so m < 64.
Additionally, the number of apples sold is 0.4A, which must be an integer because you can't sell a fraction of a fruit. Since A = 15m, 0.4A = 0.4*15m = 6m, which is an integer for any integer m. So that's fine.
Similarly, the number of mangoes sold is 0.2T. Since T = 20m + 320,
0.2T = 0.2*(20m + 320) = 4m + 64, which is also an integer.
So, all conditions are satisfied as long as m is an integer between 1 and 63.
Now, we need to find the smallest possible T. Since T = 20m + 320, the smallest T occurs when m is as small as possible, which is m=1.
So, T = 20*1 + 320 = 340.
Wait, but let's check if m=1 satisfies all conditions.
For m=1:
A = 15*1 = 15
B = 192 - 3*1 = 189
T = 2