If a hundred chickens eat a hundred
bushels of grain in a hundred days, how many bushels will ten chickens eat in
ten days? And, if on the average, one and a half of these chickens lay an egg
and a half in a day and a half, how many days will it take a chicken to lay one
and a half dozen eggs? General Solution: How could you solve this problem for
any number of chickens?
We can treat this as a proportion problem. For the first part, we note that as the number of chickens increases, the number of bushels increases; and, as the number of days increases, the number of bushels also increases. So, we can set up an equation relating all the quantities of the form C*D = k*B. Here, C is the number of chickens, D is the number of days, B is the number of bushels, and k is some constant. We can solve for k using the information provided in the problem. When C = 100 and D = 100 then B = 100. so 100*100 = k*100, so k = 100. Now the equation C*D = 100*B allows us to solve for any of the three variables given the other two. For example if C = 10 and D = 10 then we can see that B = 1.
For the second part, again the number of eggs increases as the number of chickens increases and as the number of days increases, so we have an equation of the form C*D = t*E, where again t is some constant making the whole thing work. The information we are given tells us that when C = 1.5 and D = 1.5, then E = 1.5, so 1.5*1.5 = t*1.5, and we can see that t=1.5. Now to answer their question: if C = 1 (a chicken) and E = 1.5 dozen or 18, what is D? 1*D = 1.5*18. Since, 1.5*18 = 12, we see that D = 12. To answer these problems for any number of chickens we use the general equations we have constructed.
Please let me know if you need any clarification. I'm always happy to answer your questions.
Sep 24th, 2015
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