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AMC 10 Math Olympiad USA Math Olympiad

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

Try this beautiful problem from Algebra: Sum of whole numbers from AMC-10A, 2012. You may use sequential hints to solve the problem

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: \(7\)

Source
Suggested Reading

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 \(7\)

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