Try: P=15, A = 4002/15 - 140 = 126.8 ...NO

Try: P=16, A = 4002/16 - 140 = 110.125 ...NO

...

Try: P=23, A = 4002/23 - 140 = 34 ...YES

And that was the only one that gave me a whole number for "A"

TEST:

$23 × 34 + $11.50 × 106 = $782 + $1219 = $2001 ...OK

And the full-priced tickets raised $782

]]>A = Tickets sold at full price

B = Tickets sold at half price

AP + B(P/2) = 2001

A+B=140, so B = 140-A

Substitute B = 140-A: AP + (140-A)(P/2) = 2001

Rearrange: AP + 140P/2 -AP/2 = 2001

Simplify: AP/2 + 140P/2 = 2001

Then: A + 140 = 4002/P

Then: A = 4002/P - 140

Now, we know that "P" must be a whole dollar amount, and also "A" must obviously be a whole number.

So, we can try different values of "P" and reject all values that don't give a whole number for "A"

Also we can narrow our search for "P" knowing that 140 tickets were sold:

- if 140 tickets were sold at full price, then the full price would be: 2001/140 = $14.3

- if 140 tickets were sold at half price, then the full price would be: $28.6

So, just test out the values of "P" between $15 and $28, and find any value(s) where A is a whole number.

I will have a go at this later

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