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Scott and Greg were asked to add two whole numbers . . .

by jay armstrong
(pittsburg, ca)











































scott and greg were asked to add two whole numbers. Instead, scott subtracted the two numbers and got 10 and greg multiplied them and got 651. what was the correct sum?

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Sep 19, 2010
Word Problem
by: Staff

The question:

by Jay Armstrong
(Pittsburg, Ca)

Scott and Greg were asked to add two whole numbers.

Instead, Scott subtracted the two numbers and got 10.

Greg multiplied them and got 651.

What was the correct sum?



The answer:

1. Scott and Greg were asked to add two whole numbers.

Since Scott and Greg were asked to add two whole numbers, the final answer will be TWO POSITIVE NUMBERS which are INTEGERS. Whole numbers are the set of numbers: 0, 1, 2, 3, …

You can solve this problem using equations, or you can easily solve this problem by simple trial and error.

I’m going to solve it both ways for you.

I will start with the equation method:



2. Selecting variable names:

X = 1st number
Y = 2nd number


3. Scott subtracted the two numbers and got 10.

X – Y = 10


4. Greg multiplied them and got 651.

X * Y = 651


5. We now have two equations with two unknowns:

Equation 1: X – Y = 10
Equation 2: X * Y = 651


6. Solve the equations for X & Y. I am going to use substitution:

Equation 1: X – Y = 10

Add Y to each side of the first equation

X – Y + Y= 10 + Y

X = 10 + Y

X = Y + 10


Substitute (Y + 10) for the X in equation 2, then solve for Y.


Equation 2: X * Y = 651

X * Y = 651

(Y + 10) * Y = 651

Y^2 + 10 * Y = 651

Y^2 + 10Y = 651

Y^2 + 10Y – 651 = 651 - 651

Y^2 + 10Y – 651 = 0

Factor the equation

(-Y – 31) * (-Y + 21) = 0


At this point we have two possibilities to choose from. Solve for Y in each case:

Possibility 1: -Y – 31 = 0

Or

Possibility 2: -Y + 21 = 0


Possibility 1: -Y – 31 = 0
-Y – 31 = 0

-Y + Y – 31 = 0 + Y

– 31 = Y

Y = – 31; this solution will not work because both numbers must be whole numbers (positive integers). – 31 is a negative integer.


Possibility 2: -Y + 21 = 0

-Y + Y + 21 = 0 + Y

21 = Y

Y = 21, this solution is a positive integer – it is a whole number. 21 is the solution for the 2nd number.


Substitute 21 for Y in the 2nd equation:


Equation 2: X * Y = 651

X * Y = 651

X * 21 = 651

(X * 21)/21 = 651/21

X = 31, this solution is a positive integer – it is a whole number. 31 is the solution for the 1st number.


To summarize, the two numbers are:

X = 31
Y = 21


7. What was the correct sum?

X + Y = 31 + 21 = 52


The final answer to your original question is: the sum is 52.




To solve for X and Y by simple trial and error:

Recognize that both X and Y must be positive INTEGERS. That doesn’t leave too many possibilities.

Equation 2: X * Y = 651

X * Y = 651

Factor the number 651

651 = 3 * 7 * 31

There are only three possibilities for the two numbers: (31, 21), (93, 7), (217, 3)

Now use equation 1 to find the two numbers whose difference is 10:

Equation 1: X – Y = 10

X – Y = 10

31 – 21 = 10, this is the only combination which works

X = 31
Y = 21

The final answer is the same: 31 + 21 = 52




Thanks for writing.


Staff
www.solving-math-problems.com



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