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Posted: Aug 5, 2011 8:39am
This was in a math problem contest for 10th graders.
It's not hard, but it's kind of fun in that it seems at first glance to give too little information.
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
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Posted: Aug 5, 2011 12:36pm
For fun, there's still too much info in the problem. Which info is redundant?
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
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stanalger
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St. Louis, MO
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Posted: Aug 6, 2011 12:26pm
We didn't need to know Angela's name.
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stanalger
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St. Louis, MO
966 Posts
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Posted: Aug 6, 2011 12:27pm
Neat problem. Here's a more serious answer:
We only need to know that each drank the same amount of liquid. The serving size (8 oz? 12 oz? 16 oz? 20 oz? trenta?) doesn't matter.
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Posted: Aug 6, 2011 1:21pm
Exactly. Care to explain your reasoning?
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
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stanalger
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St. Louis, MO
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Posted: Aug 8, 2011 11:35am
"Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee."
This means she drank a sixth of the total amount of coffee and MORE THAN a sixth of the total amount of milk. It follows that she drank MORE THAN a sixth of the total amount of liquid.
And since she drank a quarter of the total amount of milk and LESS THAN a quarter of the total amount of coffee, she drank LESS THAN a quarter of the total amount of liuid.
As long as each family member drank the same total amount of liquid, each serving must have contained 1/x of the total amount of liquid, where x is the number of family members. Since x must be a whole number, (1/6) < (1/x) < (1/4) implies x=5.
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Let T=the total amount of liquid. Each serving must have contained (1/5)*T. Since we were given that (1/5)*T = 8 oz, we know that T = 40 oz.
If each serving size had been 3*pi oz (or ANY other value), the value of T would change, but x, the number of family members, would still have to be equal to 5.
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TomasB
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Posted: Aug 16, 2011 5:16am
Am I totally wrong, or is "The amounts of coffee and milk varied from cup to cup, but were never zero" redundant too? Is that info used in Stan's clever solution?
As I see it, once she has her liquid, the others can fill their cups with the correct amount any way they want. Even a full cup of pure coffee or a full cup of milk.
/Tomas
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Posted: Aug 16, 2011 6:25am
Agreed, that was my thought as well, unless Angela has no milk or no coffee.
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
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stanalger
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St. Louis, MO
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Posted: Aug 16, 2011 12:50pm
Agreed.
We need only be told that ANGELA'S drink contained both coffee and milk. Other family members may drink straight coffee or straight milk.
(If Angela drank her coffee black, then the family consists of 6 people, each of them drinking black coffee. If Angela drank straight milk, then the family consists of 4 people, each of them drinking straight milk.)
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