Find Two Numbers

[harpazo]

Two numbers have a product of 8 and the sum of 9 ? What are the two numbers?

Solution

I am interested in the set up.

Let a and b = the two numbers

a•b = 8
a + b = 9

Correct set up?

 

[Otis]

Two numbers have a product of 8 …
a•b = 8 …

… [Two numbers have a] sum of 9 …
a + b = 9 …

Correct set up?

Yes. When we multiply numbers, the result is called their 'product'. When we add numbers, the result is called their 'sum'.

?

 

[HallsofIvy]

Now, how do you go about solving those equations?

 

[harpazo]

Now, how do you go about solving those equations?

My concern is always the set up.
I will be back to show my work.

 

[harpazo]

Now, how do you go about solving those equations?

Here it goes:

a•b = 8...Equation A
a + b = 9...Equation B

Solve B for either a or b.

a + b = 9

a = 9 - b

Plug into A.

(9 - b)b = 8

9b - b^2 = 8

b^2 - 9b + 8 = 0

Solving for b, I got b = 1 and b = 8.

Now I plug the two values found for b into EITHER A or B to find a.

I will use ab = 8.

Let b = 1.

a(1) = 8

a = 8

Let b = 8.

a(8) = 8

a = 8/8

a = 1

Answer: The two numbers are 1 and 8.

Yes?

 

[Steven G]

My concern is always the set up.
I will be back to show my work.

There is no need to ask us if you set it up right. Just solve it and see if it is correct. Seriously, that is how you learn math. Of course 8*1=8 and 8+1=9. I am sure that you know that. Please try it for yourself.

 

[harpazo]

There is no need to ask us if you set it up right. Just solve it and see if it is correct. Seriously, that is how you learn math. Of course 8*1=8 and 8+1=9. I am sure that you know that. Please try it for yourself.

I posted the complete solution.

 

[Steven G]

Here it goes:

a•b = 8...Equation A
a + b = 9...Equation B

Solve B for either a or b.

a + b = 9

a = 9 - b

Plug into A.

(9 - b)b = 8

9b - b^2 = 8

b^2 - 9b + 8 = 0

Solving for b, I got b = 1 and b = 8.

Now I plug the two values found for b into EITHER A or B to find a.

I will use ab = 8.

Let b = 1.

a(1) = 8

a = 8

Let b = 8.

a(8) = 8

a = 8/8

a = 1

Answer: The two numbers are 1 and 8.

Yes?

Algebra is good and thinking is even better (and more fun). You should always spend a couple of seconds to see if the answer is obvious. Two numbers that multiply out to 8 and add up to 9 can be done in your head if you try. Seeing things is a mathematical growth. It really is! Students and teachers always tell me that I am wrong about having students see the answer without doing any work. Thinking is always good. Hopefully the other helpers will comment, if not ask them.

 

[Steven G]

I posted the complete solution.

True, but after you asked if you had the correct setup.

 

[Otis]

… Seeing things is a mathematical growth … having students see the answer without doing [work on paper] …

Yes, indeed, whenever numbers are easily manipulated mentally. Of course, in exercises like the one in this thread, students would need to have memorized the multiplication table, before gaining the number sense needed to "see" those patterns.

?

 

[Steven G]

I would hope that by the time someone is in 9th grade algebra or an adult in their 50's would have mastered the times table by then.

 

[Otis]

I would hope …

Hope is a thin rope, when we're hanging on it.

?

 

[harpazo]

Algebra is good and thinking is even better (and more fun). You should always spend a couple of seconds to see if the answer is obvious. Two numbers that multiply out to 8 and add up to 9 can be done in your head if you try. Seeing things is a mathematical growth. It really is! Students and teachers always tell me that I am wrong about having students see the answer without doing any work. Thinking is always good. Hopefully the other helpers will comment, if not ask them.

The main reason I enjoy algebra (and math in general) is the THINKING part. As a 55 year old man, math helps to keep my memory ability alive and well. I suppose Sudoku puzzles do the same thing.

 

[harpazo]

Yes, indeed, whenever numbers are easily manipulated mentally. Of course, in exercises like the one in this thread, students would need to have memorized the multiplication table, before gaining the number sense needed to "see" those patterns.

?

Yes, thinking works when you are in a store and need to quickly do some math in your head to prevent the cashier person from pulling a fast one and stealing your money. However, the THINKING game cannot be applied to Calculus problems. In that case, math work on paper must be applied.