Ella spent her whole allowance of $2.00, plus the 16¢ she had left over from last week, on
bubble gum. If the pieces of gum had been a penny cheaper, she would have received three
more pieces than she did. How many pieces did she actually buy?
@mathstudent55

Question

Ella spent her whole allowance of $2.00, plus the 16¢ she had left over from last week, on 
bubble gum. If the pieces of gum had been a penny cheaper, she would have received three 
more pieces than she did. How many pieces did she actually buy? 
@mathstudent55

mathstudent55 · Answer

Let's call the number of pieces of bubble gum she actually got x.

Since she spent $2.16 on them, the price per piece of bubble gum is

$\dfrac{$2.16}{x} $

If the gum were a penny cheaper, then the price per piece would have been

$\dfrac{2.16}{x} - 0.01$

calculusxy · Answer

Then?

mathstudent55 · Answer

Ok, now we see how much she spent at her price, and how much she would have spent at the cheaper price.

mathstudent55 · Answer

Each piece really cost $ \dfrac{2.16}{x} $ , and she bought x of them, so she spent

$ \dfrac{2.16}{x} \times x$

At the lower price, she would have been able to buy 3 more than x, so she would have spent this:

$(\dfrac{2.16}{x} - 0.01)(x + 3) $

mathstudent55 · Answer

These two amounts represent the same amount of 2.16.

The first one is the way she actually spent it.
The second one is the way it would be if each piece of gum were 1 cent cheaper, and she had bought 3 more.
Now you set these two amounts equal.

calculusxy · Answer

The problem is that I don't know how to evaluate this for the amount of gums that she bought. It's like very confusing.

mathstudent55 · Answer

Now we have an equation that we need to solve.

$\dfrac{2.16}{x} \times x = (\dfrac{2.16}{x} - 0.01)(x + 3) $

$2.16 = (\dfrac{2.16}{x} - 0.01)(x + 3) $

Multiply both sides by x:

$2.16x = (2.16 - 0.01x)(x + 3) $

$2.16x = 2.16x + 6.48 - 0.01x^2 - 0.03x$

$0.01x^2 + 0.03x - 6.48 = 0$

$x^2 + 3x - 648 = 0$

Now we can use the quadratic formula to solve the equation.

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

$x = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(-648)}}{2(1)} $

$x = \dfrac{-3 \pm \sqrt{9 +2592}}{2} $

$ x = \dfrac{-3 \pm \sqrt{2601}}{2} $

$ x = \dfrac{-3 \pm 51}{2} $

$x = 24$   or   $x = -27$

Since the number of bubble gum pieces can't be -27, it has to be 24.

mathstudent55 · Answer

Now let's see if the number 24 pieces of gum makes sense.

2.16/24 = 0.09
The price of the gum was 9 cents each

If the price were one cent less, it would be 0.08.
2.16/0.08 = 27

Sure enough, if the price were 1 cent less, she would have been able to buy 27 which is 3 more than the 24 she got at 9 cents each.

The answer is "She bought 24 pieces of gum."

loser66 · Answer

This is another way to approach 
first off, $2 + 16 cents = 216 cents
let x is the cost of 1 piece of gum, so for 216 cents, Ella can buy n pieces and n = $\dfrac{216}{x}$ (a)
if the cost is 1 cent cheaper than it is, it means the cost is (x-1), Ella can buy n +3 pieces of gum with 216 cents. So, we have       $n+3 = \dfrac{216}{x-1}$ (b)
 
from (b)$\rightarrow$ $n = \dfrac{216}{x-1} -3$ and this n = n from (a)
We have equation $\dfrac{216}{x}=\dfrac{216}{x-1}-3=\dfrac{216 - 3(x-1)}{x-1}$
$$216(x-1)=x(216-3x+3)\216x-216=216x-3x^2+3x\3x^2-3x-216=0\x^2-x-72=0\(x-9)(x+8)=0\x=9~~or~~x=-8~~\text{reject, because the cost can't be negative}$$
So, the cost of a piece of gum is 9cents. 
Therefore , with 216cents, Ella can buy $\dfrac{216}{9}=24$ pieces of gum.
if the cost is 1cent cheaper than it is , it means  8 cents, she can buy $\dfrac{216}{8}=27$pieces