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| Alexei And Boris (Posted on 2006-03-28) |
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Alexei and Boris both have a whole number of chocolates, lollipops and toffees, and the product of the number of each boy's of chocolates, lollipops and toffees is 336. It is known that:
(A) Each boy has fewer chocolates than lollipops.
(B) For each boy, the product of the number of chocolates and lollipops equals the total number of candies he has.
(C) Alexei has more lollipops than toffees.
Determine the number of chocolates, lollipops and toffees possessed by each of Alexei and Boris.
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Submitted by K Sengupta
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Rating: 3.0000 (5 votes)
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Solution:
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The number of chocolates, lollipops and toffees possessed by Alexei are:
( # chocolate, # lollipop, # toffee ) = (2,14,12).
The number of chocolates, lollipops and toffees possessed by Boris are :
( # chocolate, # lollipop, # toffee ) = ( 4,6,14).
EXPLANATION:
Let the number of chocolates, number of lollipops and the number of toffees be respectively denoted by P,Q and R.
Accordingly, by conditions of the problem:
(i) PQR = 336 ;
(ii) R = PQ - P - Q ;
(iii) P < Q ;
Hence, PQ(PQ - P - Q) = 336. But, from (i) we obtain :
PQ = 336/R,so that P + Q = (336/R) - R, with P < Q in terms of (iii).
Since, each of P and Q is a positive integer, it follows that 336/R > R giving R^2 < 336 < 361 , so that R < 19.
Now the positive divisors of 336 less than 19 are 1,2,3,4,6,7,8,12,14 and 16.
Subsituting the aforementioned values for R, we observe that the system of equations:
P+Q = (336/R) - R ; PQ =336/R possesses positive integral solutions only when R=12,14.
Substituting R =12 and 14 in turn we observe that:
( P+Q = 16; PQ = 28) and,
( P+Q =10; PQ = 24)
The solution of the aforementioned system of equations with the restriction P < Q are respectively given by:
(P,Q) = (2,14) and (P,Q)= (4,6) so that:
(P,Q,R) = (2,14,12)and (4,6,14)
Since the number of lollipops (Q) possessed by Alexei is greater than the number of toffees(R) in his possession:
We obtain the number of chocolates, lollipops and toffees possessed by Alexei as:
( # chocolate, # lollipop, # toffee ) = (2,14,12).
This gives the number of chocolates, lollipops and toffees possessed by Boris as:
( # chocolate, # lollipop, # toffee ) = ( 4,6,14).
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