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Hannah has an older sister Lucy. It is known that the sum of the squares of their ages
give a number equal to 30 times the sum of their ages. Given also that the sum
of Hannah's age and 2 times Lucy's age is 90. Find Lucy's present age
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Let x = Lucy's age.
Then Hannah's age is 90-2x years.
Next you have this equation
x^2 + (90-2x)^2 = 30*(x + (90-2x)).
Simplify step by step and find x
x^2 + 8100 - 360x + 4x^2 = 30*(90-x)
5x^2 - 360x + 8100 = 2700 - 30x
5x^2 - 330x + 5400 = 0
x^2 - 66x + 1080 = 0.
Solve this quadratic equation by any way you can. Its roots are 30 and 36.
Now let analyze them.
If x= 30 is Lucy's age, then Hannah's age is 90-2x = 90-2*30 = 30,
and Lucy can not be the older sister.
If x= 36 is Lucy's age, then Hannah's age is 90-2x = 90-2*36 = 18 years,
and it satisfies the condition requirements.
ANSWER. Lucy is 36 years old.
Solved.
