A woman bought a number of items for $48. She realized that if she had bought 10 more items for the same money, she would have paid $20 less per item. How many items did she buy?

Answer 1

i: items

p: price per item

We start with the equations:

(1) i*p = 48

(2) (10 + i) * (p - 20)=48

Then to solve:

p = 48/i Solving the first equation

(10 + i)*(48/i - 20) = 48 Substitution

480/i - 200 + 48 - 20i = 48 FOIL

At this point, we have to make the equation into a quadratic equation and then solve appropriately:

480/i - 200 - 20i = 0 -48 on both sides

480 - 200i - 20i^2 = 0 Multiply by i on both sides

-20 (-24 + 10i + i^2) = 0 Factor out -20

-24 + 10i + i^2 = 0 Divide both sides by -20

i^2 + 10i - 24 = 0 Commutative property

(i+12)*(i-2) = 0 Factoring

i = -12 and i = 2 Zero product property

Because i=-12 isn't a reasonable number of items, it's an extraneous solution. Therefore, i=2.

She originally both 2 items for $24 each, but if she would've bought 12 items, they would've been $4 each.