We can write a formula for the shares Ed bought as:
n * p = $2000n⋅p=$2000 Where:
- nn is the number of shares Ed bought
- pp is the the price Ed paid per share
- $2000 is the total Ed paid for all the shares
We can write an equation for what would of happened if Ed had waited one week for the price per share to drop $10 per share when he could of bought 10 more shares as:
(n + 10)(p - $10) = $2000(n+10)(p−$10)=$2000
Or
np - $10n + 10p - $100 = $2000np−$10n+10p−$100=$2000
Step 1" We can solve the first equation for pp:
(n * p)/color(red)(n) = ($2000)/color(red)(n)n⋅pn=$2000n
(color(red)(cancel(color(black)(n))) * p)/cancel(color(red)(n)) = ($2000)/n
p = ($2000)/n
Step 2: Substitute ($2000)/n for p in the second equation and solve for n:
np - $10n + 10p - $100 = $2000 becomes:
(n * ($2000)/n) - $10n + (10 * ($2000)/n) - $100 = $2000
(color(red)(cancel(color(black)(n))) * ($2000)/color(red)(cancel(color(black)(n)))) - $10n + ($20000)/n - $100 = $2000
$2000 - $10n + ($20000)/n - $100 = $2000
$2000 - $100 - $10n + ($20000)/n = $2000
$2000 - $100 - $10n + ($20000)/n = $2000
$1900 - $10n + ($20000)/n = $2000
$1900 - color(red)($1900) - $10n + ($20000)/n = $2000 - color(red)($1900)
0 - $10n + ($20000)/n = $100
-$10n + ($20000)/n = $100
$10(-n + (2000)/n) = $100
($10(-n + (2000)/n))/color(red)($10) = ($100)/color(red)($10)
(color(red)(cancel(color(black)($10)))(-n + (2000)/n))/cancel(color(red)($10)) = 10
-n + (2000)/n = 10
color(red)(n)(-n + (2000)/n) = color(red)(n) xx 10
(color(red)(n) xx -n) + (color(red)(n) xx (2000)/n) = 10n
-n^2 + 2000 = 10n
-n^2 - color(red)(10n) + 2000 = 10n - color(red)(10n)
-n^2 - 10n + 2000 = 0
color(red)(-1)(-n^2 - 10n + 2000) = color(red)(-1) xx 0
n^2 + 10n - 2000 = 0
(n + 50)(n - 40) = 0
Solution 1:
n + 50 = 0
n + 50 - color(red)(50) = 0 - color(red)(50)
n + 0 = -50
n = -50
Solution 2:
n - 40 = 0
n - 40 + color(red)(40) = 0 + color(red)(40)
n - 0 = 40
n = 40
The solution is: Ed bought 40 shares.
Solution 1 is an extraneous solution because Ed could not have bought a negative 50 (or -50) shares.
