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?
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.
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.
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.
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.
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.
Right, my mistake.