**QUESTION:-**

Neha, a shopaholic, was caught overspending by her friends who decided to teach her a lesson on how to economise. They gave Neha Rs. 100 and asked her to buy no more or no less than 100 items for that amount. Also, the price of those 100 things should be exactly Rs. 100 - not a rupee less, not a rupee more. She was only allowed to buy ball-point pens, pencils and sketch pens. The sketch pens cost Rs. 6 each, the ball-point pens Rs. 3 each and the pencils cost 10 paise each. Their plan was to distribute the 100 things amongst the under-privileged. How many of each thing must she buy in order to satisfy the given conditions? Provide a detailed explanation along with your answer.

*Solution*One sketch pen, 29 ball-point pens and 70 pencils.

Let x be the number of sketch pens, y be the number of ball-point pens and z be the number of pencils she should buy.

From the given conditions, we can conclude that:

x + y + z = 100 ... (1)

6x + 3y + 0.1z = 100 ... (2)

On multiplying the first equation by 6 and then by 3, and, then, subtracting the two equations obtained from the second, we get two more equations:

6x + 6y + 6z = 600

3x + 3y + 3z = 300

3y = 500 – 5.9c ... (3)

3x = 2.9z - 200 ... (4)

Normally, two equations aren’t sufficient to solve equations with three variables.

But, x and y are non-negative integers. So, if 3y is greater than or equal to 0, then 500 - 5.9z is greater than or equal to 0. This means that z is less than or equal to 84.75. Also, if 3x is greater than or equal to 0, then 2.9z - 200 > 0. This means z is greater than or equal to 68.97.

However, since buying pencils is the only way to spend a fraction of a rupee, the number of pencils to be bought must cost an even amount in Rupees.

The only two numbers of pencils, that can be bought between 68.97 and 84.75 that satisfy this condition are 70 and 80.

If we substitute 80 for z in equations (3) & (4), we can solve for x, which equals 10.67, and y, which equals 9.33. But the values of a and b must be integers, so we know this is not the solution.

If we substitute 70 for z in equations (3) & (4), we can solve for x, which equals 1, and y, which equals 29.

So one sketch pen must be bought for Rs. 6, 29 ball-point pens must be bought for Rs. 87, and 70 pencils must be bought for Rs. 7.

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