Find two positive numbers given the geometric mean is 42 and the sum is 85.
We’ll use an algebraic approach to answer this problem. We’ll introduce the letters 𝑎 and 𝑏 to represent the two numbers. And as we’re told these numbers are both positive, 𝑎 and 𝑏 are both greater than zero. We’ll now use the information given in the question to form some equations. We’re told that the sum of these two numbers is 85, so this gives the equation 𝑎 plus 𝑏 equals 85. Second, we’re told that the geometric mean of these two numbers is 42. Well, the geometric mean of two numbers 𝑎 and 𝑏, which must have the same sign, is defined to be equal to the square root of their product 𝑎𝑏. We can only find the geometric mean of two numbers which have the same sign because if the two numbers were of opposite signs, then their product would be negative. And the square root of a negative number gives a nonreal result.
So if the geometric mean of our two numbers is 42, then we have the equation the square root of 𝑎𝑏 equals 42. We now have a pair of simultaneous equations in 𝑎 and 𝑏, which we need to solve. We’ll begin by rearranging the first equation to give an expression for 𝑏 in terms of 𝑎. By subtracting 𝑎 from each side, we have 𝑏 is equal to 85 minus 𝑎. We can make the second equation simpler by squaring both sides. Now as both sides are strictly positive, we won’t be creating any extra solutions by doing this. So we have 𝑎𝑏 is equal to 42 squared, and 42 squared is 1764. To solve these equations, we’re going to take the expression for 𝑏 from our first equation, that’s 85 minus 𝑎, and substitute it into the second equation as this will give an equation in 𝑎 only. This gives 𝑎 multiplied by 85 minus 𝑎 is equal to 1764.
Next, we’ll distribute the parentheses on the left-hand side, giving 85𝑎 minus 𝑎 squared, and this is still equal to 1764. Finally, we’ll group all of the terms on the same side of the equation, in this case the right-hand side, so that the coefficient of 𝑎 squared is positive. And we have the equation zero equals 𝑎 squared minus 85𝑎 plus 1764. This is a quadratic equation in 𝑎. And in fact, it can be solved by factoring. This may take a bit of trial and error, and in fact, you may find it more efficient to use the quadratic formula. But it does factor as 𝑎 minus 36 multiplied by 𝑎 minus 49.
We know this factorization is correct because if we add these two numbers of negative 36 and negative 49 together, we get negative 85, which is the coefficient of 𝑎 in our quadratic equation. And if we multiply them together, we get positive 1764, which is the constant term.
Next, we recall that if a product of two factors is equal to zero, then at least one of the individual factors must be equal to zero. So either 𝑎 minus 36 equals zero or 𝑎 minus 49 equals zero. And we have two straightforward linear equations to solve for 𝑎. To solve the first equation, we add 36 to each side, giving 𝑎 equals 36. And to solve the second, we add 49 to each side, giving 𝑎 equals 49. So there are two possible values for 𝑎. In fact, these are the values of both 𝑎 and 𝑏. We didn’t define which of our two numbers 𝑎 and 𝑏 was larger. So by following this method, we’ve found two solutions. Either 𝑎 is equal to 36, in which case 𝑏 will be equal to 49, or 𝑎 is equal to 49, in which case 𝑏 is equal to 36.
Let’s just confirm this though by substituting each value of 𝑎 back into the equation for 𝑏. If 𝑎 is equal to 36, then 𝑏 is 85 minus 36, which is 49. And if 𝑎 is 49, 𝑏 is 85 minus 49, which is 36. So we found the two numbers we’re looking for. They are 36 and 49. We used the equation concerning the sum of these two numbers to calculate the second number. So let’s use the information about their geometric mean to check our answer.
The geometric mean of these two numbers is the square root of their product. That’s the square root of 36 multiplied by 49. But these numbers are both square numbers. 36 is six squared and 49 is seven squared. So we have the square root of six squared multiplied by seven squared. Using laws of radicals or surds, we can say that this is equal to the square root of six squared multiplied by the square root of seven squared. But the square root of six squared is just six, and the square root of seven squared is seven. So the geometric mean is six multiplied by seven, which is equal to 42. And this confirms that our answer is correct.
The two positive numbers then that have a sum of 85 and a geometric mean of 42 are 36 and 49.