SOLUTION: mathematical expression for a three-digit number whose hundred digits is half the tens digit and the tens digit is 2 more than the units digit

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Question 1122469: mathematical expression for a three-digit number whose hundred digits is half the tens digit and the tens digit is 2 more than the units digit
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


100s digit:

10s digit:

1s digit:

Value of 100s digit:

Value of 10s digit:

Value of 1s digit:

Value of entire 3 digit number is the sum of the three values above.


John

My calculator said it, I believe it, that settles it

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the hundreds digit;
then the tens digit is 2x (the tens digit is twice the hundreds digit);
then the units digit is 2x-2 (it is 2 less than the tens digit)

The value of the 3-digit number ABC is 100A+10B+C. So an expression for the 3-digit number in this problem is

100(x)+10(2x)+1(2x-2)

Simplify if required....

Note that x and 2x are both single-digit positive integers; that means x can have only the values 1, 2, 3, or 4.

Those values of x in the expression shown will produce all of the 3-digit numbers that satisfy the conditions of the problem.

120
242
364
486

Another completely different way to solve the problem is to make a list of the 3-digit numbers that satisfy the given conditions and find an algebraic expression that produces exactly that list.

Making the list is relatively easy; choose a hundreds digit; then the tens digit is twice the hundreds digit; then the units digit is 2 less than the tens digit:

1; 1*2 = 2; 2-2 - 0 --> 120
2; 2*2 = 4; 4-2 = 2 --> 242
3; 3*2 = 6; 6-2 = 4 --> 364
4; 4*2 = 8; 8-2 = 6 --> 486
5; 5*2 = 10 too big; the list is complete.

The 3-digit numbers satisfying the given conditions are

120, 242, 364, and 486.

A bit of simple algebra, seeing the first number 120 and a common difference of 122 between successive numbers in the list, yields the linear expression

122x-2

which is the simplified form of the expression shown above.