Ratios of numbers, algebra one

Question

This is a word problem that states: "The ratio of two numbers is 3:4. Their sum is to the sum of their squares as 7 is to 50. Find the numbers. (Hint: The sum of their squares would be (3x)^2 + (4x)^2.)". I am unsure of how to solve this problem, and it would be amazing if I could receive help. Thank you! 

-Jason

Answer 1

Jason,

 

What do we know? Since the ration of the numbers is 3:4 then the numbers could be denoted as 3x and 4x (we must remember that x is what we will solve for but not 'the answer').  The also know that the sum (3x + 4x) is to the sum of the squares (3x)^2 + (4x)^2 as 7 is to 50.  

 

This means that 7x/7 = 25x^2/50.

 

x=x^2/2

 

2x=x^2

 

2=x.  

 

The numbers would be 6 and 8.

Answer 2

Numbers are x and y

 

From the 1st sentence, we can write:  y = 3x/4

 

From the 2nd sentence:

SUM / SUM of squares = 7/50

So:

 

(x + y) / (x^2 + y^2) = 7/50

 

Next, substitute for y, based on the 1st equation:

(x + 3x/4) / (x^2 + (3x/4)^2) = 7/50

 

Cross-multiply:

50 (x + 3x/4) = 7 (x^2 + (3x/4)^2)

 

expand:

50x + 150x/4 = 7x^2 + 63x^2 / 16

 

87.5x = 10.94x^2

 

x = 0, or x = 8.00

then y = 3x/4 = 6.00

 

CHECK:

(6 + 8) / (36 + 64) = 14/100 = 7/50  OK!!