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MathWizard likes to play a fun number trick on her friends. She tells them to think of a number. She then tells them to subtract their number from 2 and multiply the result by 5. To this product she tells them to add half the difference when 24 is subtracted from 12 times their original number.

(a) If I use the number 8 and follow MathWizard's steps, what number will I get?

(b) If you correctly apply MathWizard's steps to your favorite number, and then tell MathWizard your result, how can MathWizard quickly figure out what number you started with?

Be sure to write a complete solution to both parts of this problem. You want your final argument to be an explanation, in words and equations, showing what MathWizard's final strategy is and why it works.

Guest Jan 10, 2018

#1
+7153
+1

(a)

 Subtract  8  from  2  : 2 - 8 Multiply by  5  : 5(2 - 8) Add  $$\frac12$$(12 * 8 - 24)  : 5(2 - 8) + $$\frac12$$(12 * 8 - 24)

So

our result   =   5(2 - 8) + $$\frac12$$(12 * 8 - 24)   =   -30 + 36   =   6

(b)

Notice that no matter what number we chose to begin with,

we can calculate our result with this equation:

result   =   5(2 - x) + $$\frac12$$(12x - 24)    , where  x  is the number we chose.

Let's simplify this and solve for  x .

result   =   10 - 5x + 6x - 12

result   =   x - 2

Add  2  to both sides of the equation.

result + 2   =  x

In other words...

the number we chose   =  our result + 2  .

To find the number we chose, MathWizard adds  2  to our result.

hectictar  Jan 10, 2018
#1
+7153
+1

(a)

 Subtract  8  from  2  : 2 - 8 Multiply by  5  : 5(2 - 8) Add  $$\frac12$$(12 * 8 - 24)  : 5(2 - 8) + $$\frac12$$(12 * 8 - 24)

So

our result   =   5(2 - 8) + $$\frac12$$(12 * 8 - 24)   =   -30 + 36   =   6

(b)

Notice that no matter what number we chose to begin with,

we can calculate our result with this equation:

result   =   5(2 - x) + $$\frac12$$(12x - 24)    , where  x  is the number we chose.

Let's simplify this and solve for  x .

result   =   10 - 5x + 6x - 12

result   =   x - 2

Add  2  to both sides of the equation.

result + 2   =  x

In other words...

the number we chose   =  our result + 2  .

To find the number we chose, MathWizard adds  2  to our result.

hectictar  Jan 10, 2018
#2
+1

Nice Job, Well Done

Guest Jan 11, 2018
#3
+7153
+1

Thanks!!

hectictar  Jan 11, 2018