the sum of the digits of a two-digit number is 12. if the digits are reversed, the new number is 18 less than the original number. find the original number

Question

digit problems. use a system of equations to solve each problem.

Answer 1

The system of equations is as follows:

 

x+y=12

(10x+y)-(10y+x) = 18

 

From here, there are any number of possible ways to solve this system. I'll use the subtraction method, but first let's simplify equation 2:

 

x+y=12

9x-9y = 18

 

multiply both sides of equation one by 9

 

9x+9y=108

9x-9y=18

 

subtract equation two from equation one:

 

18y = 90

 

divide by 18:

 

y = 5

 

use the original first equation to find x now that we know y:

 

x + 5 = 12

x = 7

 

Now let's check that 75 actually answers the question:

 

5+7 = 12

75-57 = 18

 

So the original number is 75!

 

Answer 2

Let u be the units digit t be the tens digit

t+u=12

The original number has value of u+10t

The number with digits reversed has value t+10u

 

The equations are now t   +   u   = 12

and                            t    + 10u = u+10t-18

 

t+u=12

9t-9u=18

 

t+u=12

t-u=2

 

add and get 2t=14 or t=7

subtract and get 2u=10 or u=5

original number is 75

number with digits reversed is 57, 18 less than the original number

 

Answer 3

Hi Katie,

 

So we have a two-digit number, x.  Thus x = 10a + b, for some integers a and b.  Note that we can write any two digit number this way (i.e. 36 = 10(3) + 6).  

 

We are given that the sum of the digits of x is 12.  This means that a + b = 12.  

When we reverse the digits, we have a new number, y, where y = 10b + a.  (Note the switch).  We are told that

y = x - 18.  To find x, note that x = y + 18.  

 

We have the system of equations a + b = 12; x = y + 18.  We want to use the substitution method. I'll solve for a;

a = 12 - b.

 

From x = y + 18, we have 10a + b = 10b + a + 18.

 

Now substitute a with 12 - b:  10(12 - b) + b = 10b + 12 - b + 18

Simplify:  120 - 9b = 9b +30

Collect like terms:  18b = -90

Divide both sides by 18:  b = 5

 

Now that we know what b is, we can find a:  a = 12 - b = 12 - 5 = 7.  

Since x = 10a + b, x = 10(7) + 5 = 75.

 

To double check, note  that this means y = 57, and 75 - 57 = 18.

 

I think the key to this type of problem is the observation that two-digit numbers can be written in the form 10a + b, for some integers a and b.  Similarly, three-digit numbers are of the form 100a + 10b + c, and so on.  

 

Try working through this problem again, but solve and substitute with b instead of a!  

 

Let me know if I need to clarify anything.

Answer 4

Let's define the two digit number as xy.  

x + y = 12     (A)

yx = xy - 18  (B)

 

We could take several problem solving strategies at this point.  But let's take advantage of the fact that we have two equations and use substitution.  x = 12 - y

Rearrange B to get: xy - yx = 18

For the digits in the tens place you will need to multiply by 10.

Also, recall that addition is commutative: 12 + 10 = 10 + 2 + 10.

10 * (12 - y) + y - 10y - (12 - y) = 18

120 - 10y + y - 10y -12 + y = 18

108 - 18y = 18

y = 5

x = 7

 

Now we double check: 5 + 7 = 12, YES

57 = 75 - 18, YES

The original number is 75.