I divide 10 units into a quantity and a remaining part, so that the sum of...?

Question

The problem states:
"I divide 10 units into a quantity and a remaining part, so that the sum of the squares of the two portions is 58 units."

After I finally figured out exactly what it meant (worded poorly) I developed 2 equations:

x = bq (quantity), r = remainder

10/b = bq + r 
= x + r
58 = (bq)^2 + r^2
= x^2 + r^2

The correct answer is 7 and 3
AKA b = 7, q = 1, x = 7, r = 3
I just can't figure out the proper substitution/manipulation to get there.
I've been staring at it forever, so it's possible my equations are poor or I just can't open my mind up to something new.
Any help is appreciated :)

notthejake · Accepted Answer

More agreement that this is poorly worded

using one variable:

the two portions can be x and 10 - x (since they must add to 10, what ever is in one portion is taken away from 10 for the other portion

and the sum of these squares is 58
x^2 + (10 - x)^2 = 58
x^2 + 100 - 20x + x^2 = 58
2x^2 - 20x + 42 = 0
x^2 - 10x + 21 = 0
(x - 7)(x - 3) = 0
x =7 or x = 3

which gives either (7 , 3) or (3 , 7)

either way, the two portions are 3 and 7

cryptogramcorner · Answer

Your making it more complicated than it is....

x + r = 10      don't get hung up on "divided". In this context it just means splitting 10 units up
                    into two quantities that add up to 10.
  
x^2  + r^2 = 58

so r  = 10 - x
and we have

x^2 + (10-x)^2 = 58

x^2 + 100 - 20x + x^2 = 58

2x^2 - 20x + 42 = 0

x^2 + -10x + 21 = 0

(x - 7)(x - 3) = 0

so x = 7 or x = 3  and since r = 10 - x
     r = 3 or r = 7

either way you look at it, the two numbers are 7 and 3

Prashant · Answer

You have interpreted it incorrectly. (By the way, I agree with you that its worded poorly!)

10 is divided into 2 parts only. "Divide" does not mean here the operation of division in mathematics. It means 10 is divided into 2 parts: One they call the "quantity" (q) and the other the "remaining part" (r).

This then gives the equations
q+r=10
q^2+r^2 = 58

Hope this helps.

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I divide 10 units into a quantity and a remaining part, so that the sum of...?

3 Answers