Simple math word puzzle

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Wildcard
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Simple math word puzzle

Postby Wildcard » Thu Apr 23, 2015 4:12 am UTC

Here's a little puzzle I wrote. The math is pretty straightforward on this one but the wording of the question, though unambiguous, goes beyond many high school students' "word problem translating" skills.

Here goes:

If the difference between two numbers is 20, what is the difference between their product and the product of two other numbers whose difference is 26 and whose sum is the same as the sum of the first two numbers?
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ConMan
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Re: Simple math word puzzle

Postby ConMan » Thu Apr 23, 2015 5:48 am UTC

I get 138.

Spoiler:
Let the first two numbers be a and b, so a-b = 20. Let the second two numbers be c and d, so c-d=26 and a+b=c+d. Then we want to find |ab-cd|.

Handily, (a+b)²-(a-b)²=2ab, so we can take ab-cd=1/2 {[(a+b)²-(a-b)²]-[(c+d)²-(c-d)²]} = 1/2 {(a+b)²-20² - (a+b)²+26²} = 1/2(26²-20²) = 1/2×6×46 = 3×46 = 138.
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Re: Simple math word puzzle

Postby Wildcard » Thu Apr 23, 2015 7:31 am UTC

ConMan wrote:I get 138.

Spoiler:
Let the first two numbers be a and b, so a-b = 20. Let the second two numbers be c and d, so c-d=26 and a+b=c+d. Then we want to find |ab-cd|.

Handily, (a+b)²-(a-b)²=2ab, so we can take ab-cd=1/2 {[(a+b)²-(a-b)²]-[(c+d)²-(c-d)²]} = 1/2 {(a+b)²-20² - (a+b)²+26²} = 1/2(26²-20²) = 1/2×6×46 = 3×46 = 138.

I think your algebra is off....
Spoiler:
(a+b)²-(a-b)²=2ab

I get (a+b)²-(a-b)² = (2ab)-(-2ab) = 4ab.
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Re: Simple math word puzzle

Postby Yat » Thu Apr 23, 2015 11:49 am UTC

Here's my method:

Spoiler:
The two pairs of integers have the same sum, so they have the same arithmetic mean, x.
The first two numbers are x-10 and x+10, the two other are x-13 and x+13.
The first product is x²-100, the second one is x²-169.
The difference between the products is 69.

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Re: Simple math word puzzle

Postby ingdas » Thu Apr 23, 2015 12:08 pm UTC

My experimental data tell me that the answer is
Spoiler:
69

http://dtai.cs.kuleuven.be/krr/idp-ide/ ... 070ff7ceae
Last edited by ingdas on Mon May 04, 2015 1:07 pm UTC, edited 1 time in total.

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Re: Simple math word puzzle

Postby Wildcard » Thu Apr 23, 2015 5:35 pm UTC

Yat wrote:Here's my method:

Spoiler:
The two pairs of integers have the same sum, so they have the same arithmetic mean, x.
The first two numbers are x-10 and x+10, the two other are x-13 and x+13.
The first product is x²-100, the second one is x²-169.
The difference between the products is 69.

Nicely done. That's my method also, but the high school students I've given it to (the ones who get the answer at all) follow ConMan's method and use four variables. Much longer and more fiddly that way.
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Re: Simple math word puzzle

Postby Krealr » Thu Apr 23, 2015 6:30 pm UTC

Wildcard wrote:Nicely done. That's my method also, but the high school students I've given it to (the ones who get the answer at all) follow ConMan's method and use four variables. Much longer and more fiddly that way.


That is probably because they usually see word problem of that type when they are learning systems of equations.

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Re: Simple math word puzzle

Postby SPACKlick » Thu Apr 23, 2015 7:06 pm UTC

Yat, you're not using fewer variables, you're just doing some of it in your head/from memory. I would prefer from any student to see the full solution.

Spoiler:
1. y=x+20
2. xy=wz+d
3. z=w+26
4. x+y=w+z

Putting 1 into 4
4a. 2x+20 = w+z

Putting 3 into 4a
4b. 2x+20=2w+26 ---------- -26
4c. 2x-6=2w --------------- /2
5. x-3=w

Putting 1 into 2
2a. x(x+20)=wz+d

Putting 3 into 2a
2b. x(x+20) = w(w+26)+d

Putting 5 into 2b
2c. x(x+20) = (x+23)(x-3)+d ---------------- Expand
2d. x^2+20x = x^2 +20x -69+d ------------- -(x^2 + 20x -69)
6. 69=d

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Re: Simple math word puzzle

Postby ConMan » Thu Apr 23, 2015 11:31 pm UTC

Wildcard wrote:
ConMan wrote:I get 138.

Spoiler:
Let the first two numbers be a and b, so a-b = 20. Let the second two numbers be c and d, so c-d=26 and a+b=c+d. Then we want to find |ab-cd|.

Handily, (a+b)²-(a-b)²=2ab, so we can take ab-cd=1/2 {[(a+b)²-(a-b)²]-[(c+d)²-(c-d)²]} = 1/2 {(a+b)²-20² - (a+b)²+26²} = 1/2(26²-20²) = 1/2×6×46 = 3×46 = 138.

I think your algebra is off....
Spoiler:
(a+b)²-(a-b)²=2ab

I get (a+b)²-(a-b)² = (2ab)-(-2ab) = 4ab.

Oh yeah. Stupid factor of two.
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Re: Simple math word puzzle

Postby Yat » Fri Apr 24, 2015 7:22 am UTC

SPACKlick wrote:Yat, you're not using fewer variables, you're just doing some of it in your head/from memory.
That was not my intent. I actually used one additional variable: the mean.
SPACKlick wrote: I would prefer from any student to see the full solution.

Spoiler:
1. y=x+20
2. xy=wz+d
3. z=w+26
4. x+y=w+z

Putting 1 into 4
4a. 2x+20 = w+z

Putting 3 into 4a
4b. 2x+20=2w+26 ---------- -26
4c. 2x-6=2w --------------- /2
5. x-3=w

Putting 1 into 2
2a. x(x+20)=wz+d

Putting 3 into 2a
2b. x(x+20) = w(w+26)+d

Putting 5 into 2b
2c. x(x+20) = (x+23)(x-3)+d ---------------- Expand
2d. x^2+20x = x^2 +20x -69+d ------------- -(x^2 + 20x -69)
6. 69=d
Well, I've not been a student for decades, but thanks. Now I adapt my explanations to the audience, and considering I was in the puzzles section of a webcomic forum, I think it was detailed enough.

Now, about "the full solution", this is just another method, and you couldn't really put it differently because it is just a series of operations. Of course my method could have been presented in a detailed, "classroom - proof" way, it would still illustrate the ability to look at the problem before getting head down into the bruteforce calculations. I'm not a teacher but I know this was the kind of things I was rewarded for when I was a student, and this is the kind of shortcuts that makes the problem a puzzle and not only an exercise.


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