my name is sam and im am a 10th grade student. 

A standard deck of 52 cards (26 red, 26 black) is separated into two piles, not necessarily equal in size. The first pile contains seven times as many black cards as red cards. The second pile contains a number of red cards that is a multiple of the number of black cards in that pile. How many red cards are in the first pile?



Hi Sam,

You could solve it algebraically but a guess and check strategy works easily. The first pile has only 3 possibilities since the number of blacks is 7 times the number of reds - possibilities for the first pile of

(b,r): (7,1), (14,2) & (21,3).

This means that the number of blacks and reds in the 2nd piles belong to

(b,r): (19,25), (12,24) & (5,23)

since we know there are 26 of each colour. Only the second option, (12,24) has the number of red cards a multiple of the number of black cards. This identifies the second pile as having 36 cards and the first pile has 16.

Actually I don't like the term guess and check for this technique. The first step is not just a guess, it is an obseervation that there are only three possibilities. Then you check each of them.



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