The difference of two positive numbers is 18 18. The larger number is one less than twice the smaller number. Find the numbers.

Question

Find the numbers.

Answer 1

Let X be the larger number and Y be the smaller number.

 

The difference of the numbers is 18 ⇒ X - Y = 18. Since both numbers are positive and 18 positive we say X - Y instead of Y - X.

 

The larger number is one less than twice the smaller ⇒ 2Y - 1 = X

Breakdown: The larger number (X)    is (=)   one less than (subtract one from the next quantity)    twice the smaller (2*Y)

 

Now we have TWO equations and TWO unknowns:

1) X - Y = 18

2) X = 2Y - 1

 

We can solve this a number of ways but since equation 2) already has X in terms of Y lets substitute.

 

X - Y = 18 ⇒ by equation 2) ⇒ (2Y - 1) - Y = 18

Which simplifies to Y - 1 = 18

Adding one to both sides of the equation yields Y = 19

 

Now we have Y we can plug it into either equation 1) or 2). since equation 2) looks simpler to me lets plug Y in there.

2) X = 2*19 - 1 = 38 - 1 = 37.

 

Ergo the numbers are 37 and 19.

You can verify this is correct by plugging in both numbers into the equations and if both sides are equal then we did it right.