If $p is defined as 0.5(p-30) for any number p. What is p, if $($($p))

Hi all,

Please could you explain this further. Is $ a specific mathematical function (like √)? I've never come across this before. I'm particularly confused by these parts of the above solution. For example, where is the -80 coming from in the first line?

$($p) = $(-50) = 0.5(-80) = -40, and $($($p))) = $(-40) = 0.5(-70) = -35
$($p) = $(-40) = 0.5(-70) = -35, and $($($p))) = $(-35) = 0.5(-65) = -32.5

I don't understand why you can't do the following:

p = -30
$p = 0.5(p-30)
$ x -30 = 0.5(-30-30)
$ x -30 = 0.5(-60)
$ x -30 = -30
$ = 1

Therefore, p = -30 because if you plug $ = 1 and p = -30 into the above, you get the following:

$p = 0.5(p-30)
1 x -30 = 0.5(-30-30)
-30 = 0.5(-60)
-30 = -30

Any advice would be appreciated.

Given that $p = 0.5(p-30) and we need to find the value of p, if $($($p))=-30

Let's solve the problem using two Methods

Method 1: Substitution

Let's take each answer choice and check which one satisfies

A. –70
$($($p))=-30
=> $($($-70))=-30

To find $-70 we need to compare what is after $ in $-70 and $p
=> We need to substitute p with -70 in $p = 0.5(p-30) to get the value of $-70

=> $-70 = 0.5(-70-30) = 0.5*-100 = -50
=> $($($p))= $($-50)
Similarly, $-50 = 0.5(-50-30) = 0.5*-80 = -40
=> $($($p))= $-40 = 0.5(-40-30) = 0.5*-70 = -35 ≠ -30 => FALSE

B. –50
=> $($($-50))=-30
=> $($($-50)) = $($-40) = $-35 = 0.5(-35-30) = 0.5*-65 = -32.5 ≠ -30 => FALSE

C. –30
=> $($($-30))=-30
=> $($($-30)) = $($(0.5(-30-30))) = $($(0.5*-60)) = $($-30) = $-30 = -30 => TRUE
In Test Situation we don't need to solve further but I am solving to complete the solution

D. 10
=> $($($10))=-30
=> $($($10)) = $($(0.5(10-30))) = $($(0.5*-20)) = $($-10) = $((0.5(-10-30))) = $(0.5*-40) = $(-20) = 0.5(-20-30) = 0.5*-50 = -25 ≠ -30 => FALSE

E. 30
=> $($($30)) = $($(0.5(30-30))) = $($(0.5*0)) = $($0) = $((0.5(0-30))) = $(0.5*-30) = $(-15) = 0.5(-15-30) = 0.5*-45 = -22.5 ≠ -30 => FALSE

So, Answer will be C

Method 2: Algebra

$($($p)) = $($(0.5(p-30)))

To find $(0.5(p-30)) we need to compare what is after $ in $(0.5(p-30)) and $p

=> We need to substitute p with 0.5(p-30) in $p = 0.5(p-30) to get the value of $(0.5(p-30))
=> $(0.5(p-30)) = 0.5(0.5(p-30)-30) = 0.5(0.5*p - 0.5*30 - 30) = 0.5(0.5*p - 15 - 30) = 0.5(0.5*p - 45)

=> $($($p)) = $($(0.5(p-30))) = $(0.5(0.5*p - 45))

Similarly, $(0.5(0.5*p - 45)) = 0.5(0.5(0.5*p - 45) - 30) = 0.5(0.5*0.5*p - 0.5*45 - 30) = 0.5(0.25*p - 22.5 - 30) = 0.5(0.25*p - 52.5)

=> $($($p)) = 0.5(0.25*p - 52.5) = -30 (given)
=> Multiplying both sides by 8 we get
8*0.5(0.25*p - 52.5) = 8*-30 = -240
=> 4 * 0.25p - 4*52.5 = -240
=> p - 210 = -240
=> p = -240 + 210 = -30

So, Answer will be C
Hope it helps!

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