The sum of the digits of a two-digit counting number is 10. If the digits are reversed, the new number is two-less than three times the original number. What is the original number?

Answer 1

TU is the number where T = the tens' digit and U is the one's digit. TU = 10T + U

T+U=10 or T=10-U

reverse the digits gives:

UT = 10U+T = 3(TU)-2 = 3(10T+U)-2 = 30T+3U-2

10U+T = 30T + 3U-2,

7U =29T-2

substitute 10-U for T and solve for U

7(10-T) = 29T-2

70-7T = 29T-2

36T = 72

T = 72/36 = 2

U=10-2= 8

original number was 28

82 =3(28)-2 = 84-2=82

Answer 2

A two digit number can be represented as x•10 + y. For example, 37 = 10•3 + 7.

The original number = (10x + y)

The sum of the digits is 10 so

x + y = 10

If the digits are reversed we have (10y + x) = new number.

The new number is 2 less than 3x the original number so

(10y + x) = 3(10x + y) - 2

10y + x = 30x + 3y - 2

Now solve for one variable. Let's rewrite x + y = 10 to y = (10 - x) and substitute that for y in the second equation:

10(10 - x) = 30x + 3(10 - x) - 2

100 - 10x = 30x + 30 - 3x -2

100 - 10x = 27x + 28

72 = 36x

x = 72/36 = 2

So x = 2, we can now solve for y

2 + y = 10

y = 8

Hence, the original number is 28.

The new number is 82

Checking: 28•3 = 84

So our new number 82 is 2 less than 3x the original number, 28.

Answer 3

The sum of the digits of a two-digit counting number is 10. If the digits are reversed, the new number is two-less than three times the original number. What is the original number?

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let a = first digit

let b = 2nd digit

let x = original

let y = new

------EACH ONE OF THESE 3 EQUATIONS COME DIRECTLY FROM THE WORD PROBLEM-------

a + b = 10

10a + b = x

10b + a = 3x -2

-----------YOU MUST GET THE EQUATIONS FROM THE PROBLEM FIRST BEFORE CONTINUING--------

10b + a = 3(10a + b) - 2.....substitute 2nd equation into the 3rd equation

10 b + a = 30a + 3b - 2

-29a + 7b = -2.......and......rewrite 1st equation as a = 10-b......so....

-29 (10-b) + 7b = -2......substitute the rewrite into this equation and solve for b

-290 + 29b + 7b = -2

36b = 288......b = 288/36 = 8.....so...a = 10 - 8 = 2........a=2 and b=8

original number x = 10a+b = 10(2) + 8 = 28 and then new number y = 3x-2 = 3(28) - 2 = 82