The difference of two numbers is 67. The larger number is three less than six times the smaller number. Find the numbers

Question

The difference of two numbers is 67. The larger number is three less than six times the smaller number. Find the numbers



Answer 1

The first step is to take those sentences and translate them into equations.

"The difference of two numbers is 67."

We don't know what the two numbers are, so let's call them x₁ and x₂.

The word "difference" means subtraction, and "is 67" is another way to say "equals 67".

Now, put it all together:

x₁ - x₂ = 67

*Note: it doesn't matter whether you do x₁ - x₂ or x₂ - x₁, but whichever number comes first will be the larger number, since the difference is +67 (a positive number). So if we do x₁ - x₂, that means that x₁ is larger than x_2.

x₁ > x₂

"The larger number is three less than six times the smaller number."

The larger number = x₁

The word "is" means equals (=)

"three less than" means we are subtracting 3 from something

"six times" means we are multiplying by 6

The smaller number = x₂

Now, put it all together:

x₁ = 6x₂ - 3

So now, we have two equations and two unknowns:

x₁ - x₂ = 67 equation (1)

x₁ = 6x₂ - 3 equation (2)

We can solve by substitution - equation (2) is already solved for x₁, so we can plug in 6x₂ - 3 into equation (1) in place of x₁.

(6x₂ - 3) - x₂ = 67

Now, we solve for x₂.

5x₂ - 3 = 67

5x₂ = 70

x₂ = 14

Now that we know x₂, we can solve for x₁ by plugging in 14 into equation (2) in place of x₂.

x₁ = 6(14) - 3 = 81

Note: after you solve for x₂, you can plug 14 into either equation (1) or equation (2). You will get the same answer regardless of which equation you choose.

As a final check of our answer, we should make sure that x₁ > x₂

81 > 14

Answer 2

This is just a matter of converting English into algebra.

Let x be the larger number and y be the smaller.

x-y=67 --> The difference of two numbers is 67.

x=6y-3 --> The larger number is three less than six times the smaller number.

x=67+y=6y-3

add 3 to both sides

70+y=6y

Subtract y from both sides

70=5y

Divide both sides by 5 and swap left and right sides

y = 70/5 =14

x = 67 + (14) = 81

Answer 3

y - x = 67; rewrite as y = x + 67 this also implies y is the larger number which is relevant for the other part.

y is larger (as y - x > 0) so the second portion describes y = 6x-3

y = x+67 and y = 6x-3, so

x+67 = 6x-3

70 = 5x

x = 14

y = x+67 = 14 + 67 = 81

so x = 14, y = 81 are the two numbers.