Problem Solving

This problem threw me off the first time around. Now I understand how to set up the equation but I'm having trouble with the algebra. First the problem:


[Image of 4 integers A, B, C, and D on a number line]
The integers A, B, C, and D shown on the number line above are all equally spaced. If C and D are equal to 5^12 and 5^13, respectively, what is the value of A?

• 1. 5^11
  2. 5^10
  3. -5^12
  4. (-7)5^12
  5. (-12)5^13

I can set up the equation: [spoiler]D minus three times the difference between C and D (or) 5^13 - 3(5^13-5^12) = A.[/spoiler] But after that, I'm lost with the algebra.

Below is the algebra for the right answer:
[spoiler]5^13 - 3(5^13 - 5^12)
5^13 - 3[5^12(5 - 1)]
5^13 - 3(5^12)(4)
5(5^12)- 12(5^12)
(-7)5^12

The correct answer is D.[/spoiler]

Can someone explain what operations are taking place in the first 2 lines?

Thanks in advance.

The integers are equally spaced.

A+x=B

B+x=C

C+x=D

x is an difference between 2 integers.

Now C=5^12 and D=5^13

5^12+X=5^13

x=5^13-5^12=5^12(5-1)=4.5^12

Since the integers are equally spaced.

A+3x=D

A+3(4.5^12)=5^13

A+12.5^12=5^13

A=5^13-12.5^12=5^12(5-12)

=-7.5^12

Hope this clarify.

saritalr wrote:5^13 - 3(5^13 - 5^12)
5^13 - 3[5^12(5 - 1)]
5^13 - 3(5^12)(4)
5(5^12)- 12(5^12)
(-7)5^12

The correct answer is D.

Can someone explain what operations are taking place in the first 2 lines?

Thanks in advance.

In the second line, you factor out 5^12 from (5^13 - 5^12), which gives you 5^12*(5-1).

Does that help?

saritalr,

Your approach is also same and correct..

[spoiler]5^13 - 3(5^13-5^12) = A

=5^13-3.5^12(5-1)

=5^13-12.5^12

=5^12(5-12)=-7.5^12[/spoiler]

selango wrote:
A=5^13-12.5^12=5^12(5-12)

=-7.5^12

Thanks Selango. Can you explain the above line - are you factoring something out?

nicolezl wrote:
In the second line, you factor out 5^12 from (5^13 - 5^12), which gives you 5^12*(5-1).

Does that help?

Very helpful! Just what I needed - thank you!

saritalr wrote:

selango wrote:
A=5^13-12.5^12=5^12(5-12)

=-7.5^12

Thanks Selango. Can you explain the above line - are you factoring something out?

A=5^13-12.5^12

Here I am taking 5^12 as common factor.

=5^12(5-12)

=5^12(-7)

Hope this clarify.