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Chalk it Out
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Your teacher has a total of 9 chalks. When a chalk reduces to 1/3 of its original size, it gets too small for her to hold for writing and hence she keeps it aside. But your teacher hates wasting things and so, when she realizes that she has enough of these small pieces to join and make another chalk of the same size, she joins them and uses the new chalkstick. If she uses one chalk each day, how many days would the 9 chalks last?
Your teacher uses one chalk each day. Hence the total number of days she uses 9 chalks is 9. Each chalk leaves a fraction of 1/3 its size... so 9 such fractions remain. Since 3 such fractions are joined to give a new chalk, your teacher combines all the fractions to get 3 chalks which can again be used for 3 days. Hence, she has managed to use 9 chalks for (9 + 3) days!
But, what about the leftovers of the chalks used over the last 3 days?? They can be joined to form yet another chalk... which means another day! So, your teacher uses the 9 chalks for a total of 13 days.
Food for thought:
Does the teacher actually use one chalk each day or 2/3 each day? Is there any ambiguity (or lack of clarity) in the problem statement? Interpreting the question posed is an important first step in problem-solving. Re-reading the problem statement often helps!
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