# Math Puzzle

**URL:** <https://boards.straightdope.com/t/math-puzzle/494636>\
**Category:** Factual Questions\
**Created:** [April 28, 2009, 5:03pm UTC](https://boards.straightdope.com/t/math-puzzle/494636 "2009-04-28T17:03:54Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![dauerbach](https://avatars.discourse-cdn.com/v4/letter/d/c2a13f/32.png) [@dauerbach](https://boards.straightdope.com/u/dauerbach)\
**Post date:** [April 28, 2009, 5:03pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/1 "2009-04-28T17:03:54Z")

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Grandma is between 50 and 70 years old. Each of her sons has as many

sons as brothers. The total number of Grandma’s sons and grandsons is

equal to the number of years that are in her age. How old is Grandma?

Now I can figure out the answer is 56. Eight sons each with 7 sons. I think the equation would be 50\<= (x-1)(x)\<=70, but how do I solve this inequality?

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**Author:** ![muldoonthief](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/muldoonthief/32/3188_2.png) [@muldoonthief](https://boards.straightdope.com/u/muldoonthief)\
**Post date:** [April 28, 2009, 5:12pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/2 "2009-04-28T17:12:34Z")

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I get 64. Her age is the sum of sons + grandsons.

s = number of sons  
g = number of grandsons total  
z = number of grandsons per son.

z = s - 1 -\> each son has as many sons as brothers.

g = s \* z = s \* (s - 1)

g = s[sup]2[/sup] - s

age = g + s = s[sup]2[/sup] - s + s

age = s[sup]2[/sup]

Since s must be an positive integer (can’t have a fractional number of sons), and age is between 50 and 70, only s = 8 by inspection, so she must be 64.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [April 28, 2009, 5:15pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/3 "2009-04-28T17:15:13Z")

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The simplest way is just to plug in various values for x. Anything less than x=8 would give an answer less than 50, and anything greater than x=8 would give an answer greater than 70.

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**Author:** ![dauerbach](https://avatars.discourse-cdn.com/v4/letter/d/c2a13f/32.png) [@dauerbach](https://boards.straightdope.com/u/dauerbach)\
**Post date:** [April 28, 2009, 5:25pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/4 "2009-04-28T17:25:50Z")

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I think 56 because there will be one more son than the number of grandsons since it states the number of brothers is equal to the number of sons. If grandma had eight sons then each son has 7 brothers.

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**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\
**Post date:** [April 28, 2009, 5:26pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/5 "2009-04-28T17:26:14Z")

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The question as posed either leaves out an important piece of information, or makes the assumption that Grandma has no daughters, or that her daughters all have no sons.

But with that assumption, 64 is the right answer.

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**Author:** ![muldoonthief](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/muldoonthief/32/3188_2.png) [@muldoonthief](https://boards.straightdope.com/u/muldoonthief)\
**Post date:** [April 28, 2009, 5:30pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/6 "2009-04-28T17:30:29Z")

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> [@dauerbach](#):
>
> I think 56 because there will be one more son than the number of grandsons since it states the number of brothers is equal to the number of sons. If grandma had eight sons then each son has 7 brothers.

But your **OP** said her age is equal to the total number of her sons AND grandsons. 8 sons, each with 7 sons of his own, gives 8 + (8\*7) = 64.

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**Author:** ![Sinaijon](https://avatars.discourse-cdn.com/v4/letter/s/dc4da7/32.png) [@Sinaijon](https://boards.straightdope.com/u/Sinaijon)\
**Post date:** [April 28, 2009, 5:42pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/7 "2009-04-28T17:42:00Z")

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> [@muldoonthief](#):
>
> But your **OP** said her age is equal to the total number of her sons AND grandsons. 8 sons, each with 7 sons of his own, gives 8 + (8\*7) = 64.

Yep. The general equation is then

50 \< x + x(x-1) \<= 70  
50 \<= x\*x \<= 70  
7.07 \<= x \<= 8.36

So, provided that Granny isn’t King Solomon, the correct answer is the only integer value between 7.07 and 8.36.

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<div class="post-metadata">

**Author:** ![dauerbach](https://avatars.discourse-cdn.com/v4/letter/d/c2a13f/32.png) [@dauerbach](https://boards.straightdope.com/u/dauerbach)\
**Post date:** [April 28, 2009, 5:44pm UTC](https://boards.straightdope.com/t/math-puzzle/494636/8 "2009-04-28T17:44:13Z")

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Right, my mistake.