Hi, there--
The Problem:
Three times the difference of eight and twice a number is five times the number.
Solution:
It's not clear what you are being asked to do here. It might be that you only need to translate
the sentence into an algebraic equation. You may also need to solve it. I'll walk you through
both steps.
Let's break this sentence into pieces and translate as we go. First, we define a variable:
Let n be "a number."
We translate the phrase "twice a number" as "2 times n," or 2n.
We translate the phrase "five times the number" as 5 times n, or 5n.
"The difference of eight and twice a number" means subtraction, so we have 2n - 8.
"Three times the difference" is 3 times (2n - 8), or 3(2n - 8).
When we translate sentences into algebra, the word "is" means equality, so we use an
equals sign. Let's put this all together,
3(2n - 8) = 5n.
Now, let's solve for n:
Use the distributive property to clear the parentheses.
6n - 24 = 5n
Combine like terms by subtract 6n from both sides of the equation. (6n-6n=0)
6n -24 - 6n = 5n - 6n
-24 = -n
Divide both sides of the equation by -1. ((-n)/(-1)=n)
(-24)/(-1) = (-n)/(-1)
24 = n
Rewrite this equation with in "n=" form.
n = 24
The number is 24. Check your answer in the original English sentence.
Twice the number is 2 * 24 = 48
The difference between 8 and 48 is 48 - 8 = 40
Three times forty is 3 * 40 = 120.
Five times 24 is 5 * 24 = 120.
Both halves of the sentence equal 120, so 24 is the correct number.
Good luck!
Mrs Figgy
