Math CentralQuandaries & Queries


Question from Carmen, a student:

Ann has a bag of marbles. she gave one half to her friend Tina and then one third to her friend Tom. Now left with 15. How many did she had originally?

We have two responses for you

Hi Carmen,

She gave away $\frac12$ of her marbles and then $\frac13$ of her marbles so she gave away $\frac12 + \frac13$ of her marbles. What is $\frac12 + \frac13$? She has left $1 - \left( \frac12 + \frac13 \right)$ of he marbles which is 15 marbles.

Can you complete the problem now?




This question is ambiguous - did Tom get 1/3 of the original number or, as the word "then" suggests, 1/3 of what was left?

In the first case, you would model the problem by the following equation, where the numbers in parenthesis refer to proportions of the original bag:

(1 - 1/2 - 1/3) N = 15

("1" is the original bagful, "1/2" the proportion of the bagful given to Tina, and "1/3" the proportion of the bagful given to Tom.)

In the second case, with each fraction referring to her current supply, the equation would be

(1 - 1/2)(1 - 1/3) N = 15

(The first parenthesized expression represents what proportion of her marbles she did not give to Tina, and the second represents what proportion of those she did not give to Tom.)

Now, here's the funny thing. Both of these seem to have answers - but only one is possible in the real world, where marbles cannot be divided. So we know which answer must be right! Which one?

(Hint: in each case, how many marbles would Ann have after the first gift?)

Good Hunting!


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