### Brain Teasers

# Justin's Marbles

Justin has a bag of marbles and nine friends. He gives Mike exactly 1/2 of the marbles. He gives Paul exactly 1/3 of the remaining marbles. He gives Dave exactly 1/4 of the remaining marbles. He gives Bert exactly 1/5 of the remaining marbles. He gives Kyle exactly 1/6 of the remaining marbles. He gives Andy 1/7 of the remaining marbles. He gives Rick 1/8 of the remaining marbles. He gives Fred 1/9 of the remaining marbles. He gives Will 1/10 of the remaining marbles. What is the smallest quantity of marbles that Justin still has in the bag?

### Answer

For each friend that Justin gives marbles, you need to calculate the fraction of the original quantity of marbles that Justin keeps in his possession (keep), the fraction of the original quantity of marbles that Justin gives to that friend (give), and the prime factorization of the denominator of the fraction given to that friend (factor).Mike: keep 1 * 1/2 = 1/2, give 1 * 1/2 = 1/2, factor 2 = 2

Paul: keep 1/2 * 2/3 = 1/3, give 1/2 * 1/3 = 1/6, factor 6 = 2 * 3

Dave: keep 1/3 * 3/4 = 1/4, give 1/3 * 1/4 = 1/12, factor 12 = 2 * 2 * 3

Bert: keep 1/4 * 4/5 = 1/5, give 1/4 * 1/5 = 1/20, factor 20 = 2 * 2 * 5

Kyle: keep 1/5 * 5/6 = 1/6, give 1/5 * 1/6 = 1/30, factor 30 = 2 * 3 * 5

Andy: keep 1/6 * 6/7 = 1/7, give 1/6 * 1/7 = 1/42, factor 42 = 2 * 3 * 7

Rick: keep 1/7 * 7/8 = 1/8, give 1/7 * 1/8 = 1/56, factor 56 = 2 * 2 * 2 * 7

Fred: keep 1/8 * 8/9 = 1/9, give 1/8 * 1/9 = 1/72, factor 72 = 2 * 2 * 2 * 3 * 3

Will: keep 1/9 * 9/10 = 1/10, give 1/9 * 1/10 = 1/90, factor 90 = 2 * 3 * 3 * 5

The lowest common multiple of the denominators of the "give" fractions is:

2 * 2 * 2 * 3 * 3 * 5 * 7 = 2520

The quantity of marbles that Justin still possesses is:

2520 * 1/10 = 252

Justin starts with 2520 marbles.

Mike gets 1/2 of 2520, or 1260. Justin still has 1/2 of 2520, or 1260.

Paul gets 1/3 of 1260, or 420. Justin still has 2/3 of 1260, or 840.

Dave gets 1/4 of 840, or 210. Justin still has 3/4 of 840, or 630.

Bert gets 1/5 of 630, or 126. Justin still has 4/5 of 630, or 504.

Kyle gets 1/6 of 504, or 84. Justin still has 5/6 of 504, or 420.

Andy gets 1/7 of 420, or 60. Justin still has 6/7 of 420, or 360.

Rick gets 1/8 of 360, or 45. Justin still has 7/8 of 360, or 315.

Fred gets 1/9 of 315, or 35. Justin still has 8/9 of 315, or 280.

Will gets 1/10 of 280, or 28. Justin still has 9/10 of 280, or 252.

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