Solve the Mystery of Four Numbers with Unusual Sums | Homework Help

[loves-to-run]

Okay so there are four numbers represented by A,B,C,D. Their sums when adding three of the four equal the following sums.. 20,22,24, and 27. This is what I have discovered so far but am really stuck.

a+b+c+d=x 20= b+c+d or 20= x-a 22=x-b 24= x-c 27=x-d

Thanks in advance to anyone who can solve this, I really appreciate it!

 

[HallsofIvy]

First, you should be aware that there is a homework forum and I am going to move this thread there.

Second, the 4 3-member subsets of {A, B, C, D} are {A, B, C}, {A, C, D}, {A, B, D}, and {B, C, D}. You have 4 equations: A+ B+ C= 20, A+ C+ D= 22, A+ B+ D= 24, and B+ C+ D= 27. You should be able to solve 4 equations for the 4 unknown numbers. There is no need to introduce a fifth, "x".
Those should be easy to solve. For example, from the first equation, A= 20- B- C. Replace A in the other 3 equations and you have reduced form 4 to 3 equations. Keep doing that.

 

[Architeuthis Dux]

I would approach this problem by first organizing the given information into a clear mathematical equation, as you have done. From there, I would look for patterns and relationships between the numbers and their sums.

One possible approach is to start by rearranging the given equations to isolate each variable on one side. For example, we can rewrite the first equation as a = x - (b+c+d), and similarly for the other three equations. This allows us to see that all four variables are dependent on x, which can help us find a solution.

Next, we can look at the given sums and see if there are any patterns or relationships between them. For instance, we can see that 20 is two less than 22, which is two less than 24, and so on. This suggests that there may be a constant difference between the sums.

To test this hypothesis, we can subtract 2 from each sum and see if the resulting numbers (18, 20, 22, and 25) have any relationships with the given equations. We can see that 18 can be written as x - (b+c) and 25 can be written as x - d. This means that the missing variable in these equations is a, and we can solve for it by setting them equal to each other: x - (b+c) = x - d. This simplifies to b + c = d.

Now, we can substitute this relationship into the original equations. For example, we can rewrite 20= b+c+d as 20= (b+c) + (b+c) + d. Using the relationship we found earlier, we can rewrite this as 20 = d + (b+c) + (b+c) = 2d + (b+c). Similarly, we can rewrite the other three equations as 22 = 2x - a, 24 = 2x - b, and 27 = 2x - c.

At this point, we have four equations with four variables (a, b, c, and d) and can solve for x by setting them equal to each other. This will give us a system of equations to solve. Once we have x, we can plug it back into the original equations to solve for the remaining variables.

In conclusion, by organizing the given information and looking for patterns and relationships, we can use mathematical principles to solve this mystery of four numbers with unusual sums.