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

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Re: If \$p is defined as 0.5(p-30) for any number p. What is p, if \$(\$(\$p)) [#permalink]
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

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If \$p is defined as 0.5(p-30) for any number p. What is p, if \$(\$(\$p)) [#permalink]
Top Contributor
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

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