Often, a problem will say something like "A three-digit number is exactly 80 times as great as the sum of its digits." Sometimes you can obtain the answer through guess-and-check. Usually, however, it is easier to convert the word problem into an algebra equation and then solve that equation. But how to translate English into Algebra-Language for a problem like this?
As always, begin by assigning letters to variables. The trick here is that each digit has its own variable. Observe the following example:
WRONG:
x = the three-digit number
RIGHT:
x = digit in hundreds place
y = digit in tens place
z = digit in ones place
Why is this? Because the problem involves adding together the digits. Therefore, you must view each digit as a separate number. Suppose, for example, that you want to add up the digits of 198. There is no mathematical operation you would perform on the value 198 itself; instead, you consider 1, 9, and 8 separately, then add those separate digits up.
Now that you have your variables, you need to write the equation. Let's return to our example.
You read: "A three-digit number is exactly 80 times as great as the sum of its digits."
You think: "[three-digit number] = 80 * [sum of digits]"
Now you just have to write expressions for the three-digit number and the sum of the digits. Well, it's pretty straightforward to come up with the sum of the digits: that's
x + y + z.
But what about the expression for the three-digit number? You may be tempted to write
xyz
...but that's WRONG! Remember, when you squish variables together like that, you are actually multiplying them. You don't want to multiply x, y, and z, so you need a different way to express the number.
Once again, thinking about an actual number can help us. Let's do 198 again. You know that 1 is in the hundreds place, 9 is in the tens place, and 8 is in the ones place. So the number 198 really means 1 hundred, 9 tens, and 8 ones. Algebraically, that's:
198 = 1*100 + 9*10 + 8*1.
Aha! You just take each digit and multiply it by a power of ten! So, to generalize,
[our lovely three-digit number] = 100x + 10y + z.
Now you've got all the pieces! Time to put everything together!
You read: "A three-digit number is exactly 80 times as great as the sum of its digits."
You think: "[three-digit number] = 80 * [sum of digits]"
You jot down: "100x + 10y + z = 80 * (x + y + z)"
And, with practice, this entire process will only take a few seconds! With an algebraic equation in hand, you can now find whatever the problem is asking for. And everyone lived happily ever after. The end (of the problem).
TO SUMMARIZE: