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# Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate of them and then gave of what remained to her

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Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate 1/3 of them and then gave 1/4 of what remained to her friend Rachel. Tracy and her mom then each ate 15 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one to five candies, leaving Tracy with three candies. How many candies did Tracy have at the start?

Dec 2, 2018

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$$\text{tracy had }n \text{ candies}\\ \text{she ate }\dfrac 1 3 \text{ of these leaving }\dfrac{2n}{3}\\ \text{she gave }\dfrac 1 4 \text{ of them to Rachel leaving }\dfrac 3 4 \dfrac{2n}{3}=\dfrac n 2\\ \text{Tracy and mom ate 30 candies from the }\dfrac{n}{2} \text{ that tracy had left}\\\text{Tracy's brother took from }1-5 \text{ candies, leaving }3 \text{ left}$$

$$\text{so condensing that down }\\ \dfrac n 2 - 30 -b = 3,~b\in 1,2,\dots 5\\ n=2(33+b)\\ \text{we note that }\dfrac n 3 \in \mathbb{N}\\ \text{so it must be that }b=3,~n=72$$

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Dec 2, 2018
edited by Rom  Dec 2, 2018
edited by Rom  Dec 2, 2018

#1
+3613
+2

$$\text{tracy had }n \text{ candies}\\ \text{she ate }\dfrac 1 3 \text{ of these leaving }\dfrac{2n}{3}\\ \text{she gave }\dfrac 1 4 \text{ of them to Rachel leaving }\dfrac 3 4 \dfrac{2n}{3}=\dfrac n 2\\ \text{Tracy and mom ate 30 candies from the }\dfrac{n}{2} \text{ that tracy had left}\\\text{Tracy's brother took from }1-5 \text{ candies, leaving }3 \text{ left}$$

$$\text{so condensing that down }\\ \dfrac n 2 - 30 -b = 3,~b\in 1,2,\dots 5\\ n=2(33+b)\\ \text{we note that }\dfrac n 3 \in \mathbb{N}\\ \text{so it must be that }b=3,~n=72$$

Rom Dec 2, 2018
edited by Rom  Dec 2, 2018
edited by Rom  Dec 2, 2018