what is the number?

Question

in a two-place number the square of the digit in the ten's place is 2 less than the sum of the digits. if the digits are transposed the number is decreased by 45. what is the number?

Answer 1

Emilyn-

This question is similar to the example that I am going to walk through with you. The example states: A two-digit number has a value in the ten's place that can be squared. It will then be 5 more than the two digits combined. The value in the one's place is two less than the ten's place value. What is the number?

 

To solve an problem like this, you need to make an algebraic equation to represent the events that are occurring. Let's say that our number is represented by mn, where m is the ten's place value and n is the single digits value.

 

We know that our ten's place value gets squared, namely: m². 9=3+1+5   31

The problem then states that the value of m² is 5 more than the two digits combined. Therefore, m²=m+n+5.

 

In addition, we know that the value of our one's place number is two less than the ten's place value: n+2=m.

 

We can now solve the system of equations that we have by looking at the equation n+2=m first.

                Subtract two from both sides to simplify, to end up with n=m-2.

 

Looking at the other equation, we can now input the value we have for n anywhere that we see the n.

               This gives m²=m+m-2+5

 

Simplify algebraically by adding the common terms (the m's and then the number values).

             This gives m²=2m+3

 

We have a square, so we will have to move everything to the same side of the equation and factor. To do this, subtract 2m and 3 from both sides.

               This gives m²-2m+3=0

 

Factor as usual:

             (m-3)(m+1)=0

 

Set each of the values in parentheses equal to 0 to find the value of m.

                m-3=0

                m+1=0

        By solving algebraically, we find that m can either equal 3 or -1.

 

In this instance it does not make sense to have a negative value, so our solution for m will be 3.

 

We can plug this value back into the second equation we had (n=m-2) to find that n=3-1, or n=1.

 

Hopefully this helps you understand how to set up this type of problem. If you have any more specific questions, please let me know!