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The sale of tickets

The sale of tickets

A charity sold 140 lottery tickets for a charity. In the end a total of € 2001 was raised. In the week prior to the draw, given that there were enough tickets left to sell, it was decided to sell them at half the price.

How much money was raised for the sale of tickets in the last week?

Solution

We know they are sold x tickets to Y euros and 140-x tickets to and / 2 euros
Where we get the following equation that we will simplify:

xy + (140-x) y / 2 = 2001
xy + (140y-xy) / 2 = 2001
2xy + 140y-xy = 4002
xy + 140y = 4002
y (x + 140) = 4002
y = 4002 / (x + 140)

We decompose 4002 = 2 * 3 * 23 * 29
x + 140 must be a divisor number of 4002, so we will have to find out what it is.
The dividers of 4002 are: 2, 3, 23, 29, 6, 46, 58, 69, 87, 138, 174
Since all are less than 140 except the last one,
x + 140 = 174 x = 34; y = 4002/174 = 23

Then 34 * 23 = 782 euros were collected at the initial price and 2001 - 782 = 1219 euros in the last week.