Lesson Twenty Six Problems with 2 Unknowns

- consecutive number
- digits

Problem (page 442)

The units digit of a two digit number is 2 more than the tens digit. If the digits are reversed, the new number is 39 less than twice the original number.

25

In the case above, the units digit is 3 more than the tens digit.

To solve the problem, we need to create a dictionary.
Original number is tens digit (t) + units digit (u) or tu.

The units digit is two more than the tens digit
u = 2 + t

If the digits are reversed, the new number is 39 less than twice the original number
ut = 2tu 39
u 10 + t = 2(10 t + u) 39 → 10u + t = 20t + 2u 39

With two equations, we can use algebra to solve for the variables
u = 2 + t
10u + t = 20t + 2u 39

Substitute for u
10(t + 2) + t = 20t + 2(t + 2) 39

Distribute
10t + 20 + t = 20t + 2t + 4 + -39

Simplify
11t + 20 = 22t 35

 55    =   11t 
 11         11

t = 5

If t = 5, this means the tens digit is 5. What is the units digit?
u = 2 + t
u = 7

tu = 57

Answer: The number is 57.

Problem 35

The units digit of a 2 digit number is 40% of the tens digit. If the digits are reversed, the resulting number is 27 less than the original number.

The units digit of a 2 digit number is 40% of the tens digit
u = 0.40t

If the digits are reversed, the resulting number is 27 less than the original number.
ut = tu 27, which is really
10u + t = 10t + u 27

The two equations to use are:
u = 0.40t
10u + t = 10t + u 27

Substitute
10 • (0.40 + t) = 10t + 0.40t 27 → 4t + t = 0.40t 27

Simplify
5t = 10.4t 27
-5.4t = -27
t = 5

Substitute 5 back into the equation for t
u = .4(5)
u = 2

tu = 52

Lesson Twenty Six Exercises

The Magic Square: Using each number from 1-9 only once, place each digit in a box so that the sum of every row, column, and diagonal is the same.