The difference of two positive numbers is one half of their sum. The sum of the squares of the two numbers is 90. Find the two numbers.

Question

 Solve the question and put it in an equation. 

Answer 1

Just in case you prefer answers with more words (also, the answer above states that a=10, but I calculated that is is 9, which I think is what David meant as his last step shows 3 times 3.)

 

The word "difference" implies subtraction, and the word "sum" implies addition. So, if you represent one number with a and the other with b, the difference is a-b and the sum is a+b.

 

Since the problem tells you that the difference of these numbers is equal to half of their sum, you must divide (a+b) by 2, which gives you the equation:

 

a - b = (a+b)/2

 

It's important to put (a+b) in parentheses so you don't accidentally calculate half of b, or b/2; remember, it's the full quantity of a plus b that you're taking half of.

 

The sum of the squares of the two numbers is 90, which again implies addition: a²+b²=90

 

So now we have two equations:
a - b = (a+b)/2
a²+b²=90

 

The first thing you want to do is use one equation to solve for either a or b, so you can plug that value into the other equation. Since the second one would involve a lot of square roots, I'm going to use the first one.

 

1. Multiply both sides of the equation by 2 so you can get rid of the denominator that you have on the left side. This gives you: 2(a-b) = a+b.

2. Distribute the 2 on the left side, which gives you: 2a-2b = a + b

3. Subtract a from both sides, giving you: a - 2b = b
4. Add 2b to both sides, which gives you: a = 3b

 

Now you have solved for a, and you can plug it into your second equation.

 

1. Substitute 3b for all of the As in this equation: a²+b²=90

2. This gives you (3b)²+b²=90. Don't forget you're squaring the whole quantity of 3 times b, not just the b.

3. Simplify the equation to get: 9b²+b² = 90.

4. Add together your b² terms to get: 10
5. Divide both sides by 10 to get: b²= 90/10 or 9.

6. Since b² is equal to 9, b can either equal -3 or 3. However, the problem tells you that both a and b are positive, so it must be 3. 

 

Now that you have solved for b, you can plug it back into either equation to solve for a. I'm going to use the first equation again: a - b = (a+b)/2

1. Replace all Bs with 3 since you now know that b=3. This gives you: a - 3 = (a+3)/2

2. Multiply both sides by 2 to get: 2(a-3) = a+3
3. Distribute the 2 on the left side, and you'll get 2a-6=a+3
4. Subtract an a from each side, giving you: a-6=3
5. Add 6 to both sides, which gives you: a=9

 

So, a=9 and b=3

Answer 2

a - b = (a + b)/2

a² + b² = 90

 

2a - 2b = a + b

a - 3b = 0

a = 3b

 

(3b)² + b² = 90

9b² + b² = 90

10b² = 90

b² = 9

b = 3

 

a = 3(3)

a = 10