Riddle Me This?????

[mbrule]

83 = x + y + (10x + y)


x = -(2/11)y + 83/11

I don't disagree with your formula, "83 = X + Y" I see that. But, where did you get the additional (10x + Y). That looks like an assumption that Sara's brother and sister have ten years between them??

Matt

 

[sawdust_maker]

Any mathematicians got an answer to this one.


When Sara asked her grandfather how old he was, he said "The 2 digits in my age are the ages of your brother and sister. If you add their ages to mine it equals 83." How young is Sara's grandfather?

If you know the answer can you explain how you got it.

This was a bonus question on my son's math test, he didn't answer it, but has bugged him all day to know what the answer is.

Thanks

Just writer out what the problem states.

Each sibling has a one digit age. Call the ages a and b.

The age of the grandfather is (10*a + b).

If we add the age of the grandfather to those of the two children, then we get 83.

(10*a + b) + a + b = 83

So

11*a + 2*b = 83

This is known as a diophantine equation. We wish to find integer solutions to the equation, in this case, where a and b are one digit integers. We can most simply do this by inspection. Note that 2*b is always an even number, when b is an integer. For the sum of 11*a + 2*b to be an odd number, we must have a as an odd integer.

So, what is the largest value of a, such that 11*a is less than 83? Clearly 11*7 = 77. If a = 7, then we must have b = (83 - 77)/2 = 3.

A quick check will verify that the other possible (odd, single digit) possible values for a, i.e., 1, 3, 5, or 9, all yield solutions that do not yield b in the range of single digit positive numbers. Those solutions would have b as {36,25,14,-8} respectively.

So no other solution can exist, at least not for one digit positive integers.
The childrens ages are 7 and 3, and the grandfather is a grandfatherly age of 73.

John

 

[LMTC]

What grade is your son in currently? Post-grad mathematics at a major university? I agree with sawdust_maker as to the nature and resolution of the problem, but good grief.....a diophantine equation problem in a math test at any level HS or below is, IMHO, just silly. Maybe my schooling is too far in the past, but I held some serious math credentials and I don't know that I could have done that in HS.

For those with too much time who still have an interest, the wikipedia article on diophantine equations is a decent read.

 

[mbrule]

Each sibling has a one digit age. Call the ages a and b.

The age of the grandfather is (10*a + b).

If we add the age of the grandfather to those of the two children, then we get 83.

(10*a + b) + a + b = 83
John

Maybe I need more coffee, but, where do you get the "10"? By my math, from the problem given:

The age of the grandfather is 83 - a - b

?????????????????

Matt

 

[LMTC]

Maybe I need more coffee, but, where do you get the "10"? By my math, from the problem given:

The age of the grandfather is 83 - a - b

?????????????????

Matt

Each digit, a & b, is a child's age. The first digit in grandpa's age, "a", has to be by a factor of 10, i.e., child is 7, grandpa is 70 something. Whatever one child is, grandpa is 10x that plus the other child's age.

 

[AlanB]

Well, it is funny, I was doing these with my son last week, he is 11 and in 5th grade.

I am terrible at math using letters, but, it is just not needed in this case.

My thought is that they are teaching estimating and solving. At least that is what I was using as an excuse with my kid, and at least too me it is a pretty practical skill, you need to be able to "guess" or "estimate" numbers fairly quickly to make some decisions in life, and that is a skill most adults learned by doing, and this is their learning time.

For me it hit home with Dean because he was struggling with the double digit division, and where I look at a problem like 87695 divided by 24, and start rounding things, doing guesses in my head, and then long handing out the division or multiplication, Dean would sit and start at 24 times 1 equals 24 and do it out long division. Probably every adult that looked at that problem above new "instantly" that it was 3 for the first number but it had to be done with experience, and they gain experience by doing the problems above.

(In the FWIW column, my wife and daughter both figure it on the theorom and start throwing letters into the discussion, I look at the problem and said, should be about 70 and started testing different numbers)

I don't think Diophantine equation is mentioned in my sons textbook.

Oh, and I was wondering where the 10 came from the same as Mat, but I know that 83-A-B does not equal the equation, there has to be a balancing one on the other side.

 

[Robert_in_NY]

Nope it's not a silly question and is a excellent way to demonstrate how to perform mathematical functions to students. Actually the teacher did give them all information needed. The correct answer as was already given is 73, simple math.

Don't take what I wrote very seriously

 

[mbrule]

Each digit, a & b, is a child's age. The first digit in grandpa's age, "a", has to be by a factor of 10, i.e., child is 7, grandpa is 70 something. Whatever one child is, grandpa is 10x that plus the other child's age.

I guess that makes sense, wow, my algebra is weak. Good algebra problem. It looks like one cannot write a simple equation or set of equations to solve this problem, but must work through the logic stated by Sawdust maker as well.

Anyway, thanks all for the input, it's fun to stimulate the brain occasionally! Now maybe I will do some work!!!

Matt

 

[silverdollar6]

What state doed the grandfather live in?

 

[Steve_Miller]

I used to love these questions in math at school, don't ask me why as I don't have a clue any more. Makes me feel stupid, but I'm guessing I'm not the only feeling that way.

Steve
Nova Scotia