## The puzzle of a market purchase

A farmer bought 100 cows, sheep and chickens at Rs.1000. Cost of cows Rs.50 each, sheep Rs.10 each, and chickens 50 paise each. How many of each purchased?

#### The puzzle of a purchase

From the local market a farmer bought 100 numbers of cows, sheep and chickens at a total cost of Rs.1000. Cost of cows was Rs.50 each, cost of sheep was Rs.10 each and of chickens was 50 paise each.

How many of cows, sheep and chicken did the farmer purchase?

100 paise make 1 rupee and prices are indicative only with no relation to actual prices.

**Recommended time to solve** is 15 minutes.

This is not so tough. Deserves a try from you.

### Solution to the puzzle of a purchase in the local market: Key pattern discoveries

In this type of puzzles, there must be a hidden clue embedded in the numbers. Let us have a close look at the numbers and try to find any useful relationship between the numbers, if it exists.

**First observation:** The total cost is a multiple of Rs.10 and though the prices of both a cow and a sheep are multiples of Rs.10, the price of a chicken is not. It is 1/20th of Rs.10.

It follows,

Conclusion 1:The number of chickens bought must be a multiple of 20 to make the total amount spent a multiple of Rs.10.

As it will turn out soon, this is the critical pattern to lead me to the solution.

**Second observation:** Total number of animals is 100, a multiple of 20. And the number of chickens is also a multiple of 20.

It follows,

Conclusion 2:The number of cows and sheep together must also be a multiple of 20.

This is a very important pattern discovered as well.

#### Solution to the puzzle of a purchase in the local market: A well-judged trial and the solution

**Third observation:** The number of chickens should be the largest, while the number of cows should be large enough to cover most of the expense of Rs.1000.

As a first trial,* I find the largest number of cows to have been bought is 19,* costing Rs.950. One more sheep would take the expense to Rs.960. in this trial,* the number of cows and sheep is 20 that satisfies the multiple of 20 condition as well.*

*With the remaining Rs.40 the farmer could have bought 80 chickens.*

What a good fortune for a trial!

The total number of animals bought becomes 100 and the total amount spent becomes Rs.1000. The condition of the number of chickens to be a multiple of 20 is also satisfied.

**Answer: **19 cows, 1 sheep and 80 chickens were bought.

*If you reduce the number of cows even by 1,* sheep bought must be 2 and leftover money would be Rs.80 with which the farmer could have bought 160 numbers of chickens far exceeding the actual total number of animals.

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