Try this beautiful problem from Algebra based on **Sum of whole numbers**.

## Sum of whole numbers – AMC-10A, 2012- Problem 8

The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?

- \(8\)
- \(9\)
- \(7\)
- \(6\)
- \(5\)

**Key Concepts**

Algebra

Equation

Sum of digits

## Check the Answer

But try the problem first…

Answer: \(7\)

AMC-10A (2012) Problem 8

Pre College Mathematics

## Try with Hints

First hint

Let us assume three numbers are \(x\),\(y\) and \(z\)

Now according to the problem

\(x+y=12\)…………………(1)

\(y+z=17\)………………..(2)

\(z+x=19\)………………….(3).

Can you find out the values of \(x\),\(y\) and \(z\)…………….?

can you finish the problem……..

Second Hint

Adding three equations we get \(2(x=y+z)=48\)

\(\Rightarrow x+y+z=24\)………….(3)

Now subtract (3) from (1),we will get \(z=12\) and similarly if we subtract (2) and (3) from (1) we will get \(x=7\) and \(y=5\)

can you finish the problem……..

Final Step

Therefore,the numbers are \(12\), \(7\), and \(5\) and middle number is \(7\)

## Other useful links

- https://www.cheenta.com/cubical-box-amc-10a-2010-problem-20/
- https://www.youtube.com/watch?v=uApjRLII6YI

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