Question

a two digit number has the following properties. If you add the digits together and multiply the result by 10, you will get 9 more than the reverse number? FInd all the possibilities

Answer 1

Answer:

10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Step-by-step explanation:

A two-digit number can be written as:

a*10 + b

Where a and b are single-digit numbers.

a is the tens digit

b is the units digit.

the reverse number is:

b*10 + a

We know that:

"If you add the digits together and multiply the result by 10, you will get 9 more than the reverse number"

Then:

(a + b)*10 = b*10 + a + 9

We now need to solve this for a and b, where the other restriction that we have is that a and b can be any whole number of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Then:

(a + b)*10 = b*10 + a + 9

a*10 + b*10 = b*10 + a + 9

subtracting b*10 in both sides, we get:

a*10 = a + 9

solving this for a, we get:

a*10 - a = 9

a*(10 - 1) = 9

a*9 = 9

a = 9/9

a = 1

and notice that we do not have any restriction for b.  So b can be any number of the set.

for example, if b = 2

a*10 + b = 12

now let's test the property:

10*(1 + 2) = 2*10 + 1 + 9

30 = 20 + 10 = 30

now if b = 4, we have:

a*10 + b = 1*10 + 4 = 14

10*(1 + 4) = 4*10 + 1 + 9

50 = 50

So we can see that for any value of b, this will work.

So the only restriction that we have, is that a must be equal to 1.

Then the numbers are:

10, 11, 12, 13, 14, 15, 16, 17, 18, 19