If The Product Of Two Consecutive Numbers Is 195, What Are The Numbers?

97 and 98

Let y be one number.

Then the next odd number would be (y+2) 

According to the question, the product of these two numbers should be equal to 195.

This implies that

y(y+2) = 195 ------ (I)

Solving the left hand side of the above equation:

Y*y + y*2 = 195

y^2 + 2y = 195

Subtracting both sides by 195 would yield the following result

y^2 + 2y – 195 = 195 – 195
y^2 + 2y – 195 = 0 ------ (ii)

The above quadratic equation can be solved either by the quadratic formula or through factorization.

For this question, we will use the factorization method to solve the equation.

Step 1: Multiply the constant with the co-efficient of the squared term. Therefore,

1*(-195) = -195.

Step 2:  Split -195 into its factors. The selected factors must equal the co-efficient of y after addition. In this case, -195 would be split into -13 and 15.

Step 3: Substituting the above factors in place of the co-efficient of y in equation (ii) would give the following.

Y^2 + (15 – 13) y -195 = 0.

Y^2 + 15y – 13 y – 195 = 0.

(y^2 + 15y) – (13y – 195) = 0.

Taking common value from the above sets,

y(y + 15) -13(y + 15) = 0.

(y+15)(y-13) = 0.

This implies that y = -15 or 13.

Since the two numbers are positive, we will ignore -15.
Therefore the two numbers are 13 and 13+2 = 15.