The sum of two numbers is 50. The difference of the two numbers is 4. Find the two numbers

Answer 1

Before looking at my solution, see if you can solve the problem yourself. Start by trying to formalize the word problem into an algebra equation (there are two). Then try to solve the two equations you come up with on your own. Then look at my solution. If you get stuck on any part and give up, that's completely OK. Learning how to do this takes practice and seeing other people do it definitely helps. Scroll down for the answer

I'm going to formalize the problem line by line. (To formalize is to convert the problem from words into symbols and numbers. It's tricky to do properly but, again, becomes easier with practice)

The sum of two numbers is 50.

a + b = 50 (Equation 1)

The difference of two numbers is 4.

a - b = 4 (Equation 2)

So now we have two equations that are derived from the given statements. We're now ready to do some algebra to figure out what a and b are. To clarify, we need to find an a and a b that satisfies both equations.

Let's start with the first equation. We can try to solve for either a or b first. It doesn't really matter, you'll end up with the same answer. I'm going to try to solve the first equation in terms of b(which just means that we want to get to where b is the only thing on one side of the = sign).

a + b = 50 (Equation 1)

b = 50 - a (subtract a from both sides)

Now we can try to solve the problem with the second equation by substituting our definition of b we derived from Equation 1 into Equation 2

a - b = 4(Equation 2)

a - (50 - a) = 4 (Substitution, replacing b with what we derived from Equation 1)

a - 50 + a = 4 (Remove the parenthesis, multiply negative one to both 50 and - a)

2a - 50 = 4 (Combine like-terms, by adding the a's together)

Now let's solve for a.

2a = 54 (Add 50 to both sides)

a = 27 (Divide both sides by 2)

Now that we know a. We could do a lot of work to get b. But remember the original word problem? It says that "The difference of the two numbers is 4". 25 is half of 50. So 27 must be the bigger number. So the other number must be 27-4. This is 23 (who knew that Michael Jordan would show up in our math problem?)

Now to prove it.

27 + 23 = 50 (Equation 1)

27 - 23 = 4 (Equation 2)

Now you can do this by brute force. You can try guessing numbers that will match both equations and you might be able to guess fairly quickly. But this strategy only works here because this problem is simple and in the real world, guessing is usually not the best strategy. Especially if efficiency is imperative.

Also, I solved this problem with pencil and paper in under 20 seconds. People who practice this stuff all the time can do it much faster than that. But learning how to do this takes time. If you don't practice enough, this could take... 2 minutes.. 10 minutes... or 30 minutes... or never if you find this subject to be infuriating or not worth your time(something I used to think when I was a kid btw).

It's ok if it takes a long time to do these problems. That means there's a lot of opportunity to learn by doing them. Even if you don't plan on entering the STEM fields, being able to develop mathematical maturity and logic skills is indispensable. And I would happily help anyone who wants to try. Please feel free to ask any questions.

Thanks for reading my post!

Answer 2

x+y=50

x-y=4

2x=54

x=27

y=23