How to Solve Digit Problems Part III

This is the third part of the tutorial series on Solving Digit Problems. In Part 1 and Part 2, we used one variable to solve digit problems. In this post, we learn how to use two variables to solve digit problems. We still use the problem in Part 2.

Problem

The sum of the digits of a 2-digit number is 9. If the digits are reversed, the new number is 45 more than the original number. Find the numbers.

Solution and Explanation

Let t = tens digit and u = units digit.

From the first sentence in the problem, we know that

t + u = 9 (1).

Also, as we have learned in the first two parts of this series, 2-digit numbers with tens digit t and ones digit u can be represented (or has value) 10t + u. For example, the number 25 with t = 2 and u = 5 has value 10(2) + 5.

So, the we can represent the original number as

10t + u.

If we reverse the digit, the specific example which is 25 becomes 52. This becomes 10(5) + 2. Hence, we can represent the reverse number as

10u + t.

Therefore, we our representation is as follows:

original number: 10t + u
new number (with digits reversed): 10u + t.

In the second sentence in the problem, it says when the digits are reversed, the new number is 45 more than the original number. That means that if we add 45 to the original number, they will be equal. That is,

original number + 45 = new number. 

Substituting the representations above, we have

10t + u + 45 = 10u + t.

We can simplify the equation by putting the variables on the right.

45 = 10u + t – (10t + u)
45 = 10u + t – 10t – u
45 = 9u – 9t (2).

Thus, we have 2 systems of equations.

t + u = 9 (1)
9u – 9t = 45 (2).

Note: We just change reverse the position of the expressions in equation (2).

We can solve this using elimination or substitution. In this solution, we use substitution.

First, we find the value of t in (1)

t + u = 9.

Subtracting u from both sides, we have
t = 9 – u.

Next, we substitute 9 – u to the value of t in (2)

9u – 9t = 45
9u – 9(9-u) = 45
9u – 81 + 9u = 45
18u – 81 = 45
18u = 45 + 81
18u = 126.

Dividing both sides by 18,

u = 7

So, the units digit is 7.

To find t, we substitute in one of the equations in (1) and (2). We substitute in (1),

t + u = 9
t + 7 = 9
t = 2

So, our number has tens digit 2 and ones digit 7. Therefore, the number is 27.

If we check, if the number is reversed, it becomes 72. Let’s see if the number with reversed digit is 45 more than the original number.

72 – 27 = 45.

Therefore, we are correct.

Having two equations in two variables is an example of systems of equation. In the process above, we solved for the value of one of the variables in (1) and substituted it in (2). We will discuss systems of equations, particularly linear equations in two variables in details in the next posts.

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