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23 Jul 2022, 08:34
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1) Count Dracula's physician recently told him the count of Red Blood Cells (RBCs) in his blood was too high, primarily due to his daily binge sucking. Being the blood-thirsty monster that he is, the Count decided not to reduce his intake, but to dilute human blood with Pig blood instead. The Count conducted tests to find that 3 liters of human blood contains 108 million RBCs, while 2 liters of pig blood contains only 40 million RBCs. If the Count needs to have 12 liters of blood daily to survive, how many liters of Pig blood should he drink such that his intake of RBs is the prescribed 320 million RBCs/day?
2) A researcher runs an experiment 13 times, getting 13 different resultant values. The results are only integer values, no result was repeated, and are such that the sum of these 13 different values is zero. What is the least number of these results that must be positive?
2) A researcher runs an experiment 13 times, getting 13 different resultant values. The results are only integer values, no result was repeated, and are such that the sum of these 13 different values is zero. What is the least number of these results that must be positive?
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23 Jul 2022, 09:29
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1. So he needs 12 litres of blood, and 320 mil RBCs per day.
Human blood has 108 mil RBCs in 3 litres, which is 36 mil RBCs per litre, which is 432 mil RBCs in 12 litres, more than what his doctor has prescribed.
Pig blood has 40 mil RBCs in 2 litres, which is 20 mil RBCs per litre, which is 240 mil RBCs in 12 litres, less than prescribed.
So he's diluting human blood with pig's blood, as in mixing the two, so it becomes a problem of proportion. You'll see that if he mixes 5 litres of human blood (5x36 = 180 mil RBCs) with 7 litres of pig blood (7x20 = 140 mil RBCs) he gets to the required dosage of 320 mil RBCs in 12 litres of blood. So the answer is 7 litres of pig blood.
2. If the results are only integer values (negative, positive or zero), and no fractions. For the sum to be zero, the total positive value must equal the total negative value so that they cancel each other out, or the results must be zero. But since the values are different and no result was repeated, maximum one result value can be zero, the others must give equal positive and negative values. However, there is no constraint on the magnitude of the result, so even one large positive could cancel out many small negatives (e.g. the results could be 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, 66 where the positive 66 cancels the sum of the others and the result is zero). So the answer is that at least 1 result should be positive.
Human blood has 108 mil RBCs in 3 litres, which is 36 mil RBCs per litre, which is 432 mil RBCs in 12 litres, more than what his doctor has prescribed.
Pig blood has 40 mil RBCs in 2 litres, which is 20 mil RBCs per litre, which is 240 mil RBCs in 12 litres, less than prescribed.
So he's diluting human blood with pig's blood, as in mixing the two, so it becomes a problem of proportion. You'll see that if he mixes 5 litres of human blood (5x36 = 180 mil RBCs) with 7 litres of pig blood (7x20 = 140 mil RBCs) he gets to the required dosage of 320 mil RBCs in 12 litres of blood. So the answer is 7 litres of pig blood.
2. If the results are only integer values (negative, positive or zero), and no fractions. For the sum to be zero, the total positive value must equal the total negative value so that they cancel each other out, or the results must be zero. But since the values are different and no result was repeated, maximum one result value can be zero, the others must give equal positive and negative values. However, there is no constraint on the magnitude of the result, so even one large positive could cancel out many small negatives (e.g. the results could be 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, 66 where the positive 66 cancels the sum of the others and the result is zero). So the answer is that at least 1 result should be positive.
23 Jul 2022, 14:11
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1) Count Dracula's physician recently told him the count of Red Blood Cells (RBCs) in his blood was too high, primarily due to his daily binge sucking. Being the blood-thirsty monster that he is, the Count decided not to reduce his intake, but to dilute human blood with Pig blood instead. The Count conducted tests to find that 3 liters of human blood contains 108 million RBCs, while 2 liters of pig blood contains only 40 million RBCs. If the Count needs to have 12 liters of blood daily to survive, how many liters of Pig blood should he drink such that his intake of RBs is the prescribed 320 million RBCs/day?
2) A researcher runs an experiment 13 times, getting 13 different resultant values. The results are only integer values, no result was repeated, and are such that the sum of these 13 different values is zero. What is the least number of these results that must be positive?
2) A researcher runs an experiment 13 times, getting 13 different resultant values. The results are only integer values, no result was repeated, and are such that the sum of these 13 different values is zero. What is the least number of these results that must be positive?
1.
3L of human contains 108 RBCs. That's 36 RBCs per L.
2L of pig contains 40 RBC. That's 20 RBCs per L.
Let H be the number of liters of human blood and P be the number of liters of pig blood.
36H+20P = 320
The total number of liters is 12, so H+P = 12. That means H = 12-P. Let's plug in 12-P for H.
36(12-P)+20P = 320
432-36P+20P = 320
-16P = -112
P = 7
2.
All of the numbers except for one can be negative. The one positive number just needs to be big enough (and not bigger) to offset all the negatives. That is, any negative number is just a measure of the distance that number is to the left of 0 on the number line. We can add up a whole bunch of those distances and still be able to find one positive number that is as far to the right of zero as the sum of all those other distances combined. For example
-235747252
-2358745
-9897
-1
-293857
-1357
-902938093
-3
-3982587
-9
-45435
-135327
1145512563
Or:
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
78
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