A thrifty shop-owner made a collection of 10 rupee, 5 rupee and 2 rupee coins. One night the shop-owner had an idea of dividing his coins into...read on...
Coin Collection Division Puzzle: What is the minimum amount of money the shop-owner had?
A thrifty shop-owner made a collection of 10 rupee, 5 rupee and 2 rupee coins. One night while enjoying his collection, the shop-owner had a sudden idea of dividing his collection of coins into equal portions.
That night he could divide the coins in equal numbers of each type of coin in each of 8 bags. Repeating the exercise, next night he divided the same collection in equal number of each denomination into each of seven bags.
The third night also he could divide the coins in equal number of each denomination now into each of six bags.
What is the minimum amount of money the shop-owner had in his coin collection?
Solution thoughts
The facts
- Each denomination had same number of coins. ignore three denominations. Work on only one denomination. This will work because number of each type of coin was same in each bag every time. We have to find this minimum number.
- The shop-owner divided this equal number of say, 10 rupee coins, equally in each of 8 bags the first night.
- It means: the number could be divided by 8. In other words, the number has 8 as a factor.
- Same way, the number must have had factors of 7 and 6 as well.
Question is: what can a minimum such number be?
The clue
The minimum and equal number of coins of each denomination has 6, 7, and 8 as factors.
Elementary mathematics
At first it may seem, the minimum number will be the product of 6, 7, and 8 with no other new factor. But the product 336 will have a factor of 2 common between 6, 7 and 8.
Why should we consider a number with a superfluous extra factor of 2? Half of 336, that is 168, can very well be divided equally into 8 portions, 7 portions or 6 portions.
This is use of the concept of Least Common Multiple of 6, 7 and 8.
In the formal method to find LCM of a few numbers, the product of the numbers are divided by the product of the factors that are common between the numbers. In this case, one factor of 2 is common.
Factors of 6, 7 and 8 and number of coins of each denomination in each bag: Solution
6 = 2 x 3
7 = 7
8 = 2 x 2 x 2.
Common factor is one number of 2.
Minimum number that can be divided by 6, by 7 or by 8 equally is half of the product of 6, 7, and 8, that is 168.
The first night, number of coins of each denomination in each of 8 bags was: 3 x 7 = 21.
The second night, number of coins of each denomination in each of 7 bags was: 3 x 2 x 2 x 2 = 24.
The third night, number of coins of each denomination in each of 6 bags was: 7 x 2 x 2 = 28.
Minimum amount of money in the coin collection was: Rs. 168 (10 + 5 + 2) = Rs. 2856.
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