Two positive integers are in the ratio 1:3. If their sum is added to their product, the result is 224. Find the integers.

Question

This is a quadratic word problem. 

Answer 1

x = 3y

 

(x+y) + xy = 224

 

plug in value of x in secod equation

 

(3y + y) + 3yy = 224

 

3y² + 4y - 224 = 0

 

y = (-4 + √(16 + 2688))/6 = (-4 + √(2704))/6 = (-4 + 52)/6 = 8 or- 9.3

 

plug in value of y to get x

 

Answer 2

 

x and y are the integers

x/y=1/3

x+y+xy=224

if x/y=1/3 then y=3x (by cross-multiplying)

substitute

x+(3x)+x(3x)=224

4x+3x²=224

3x²+4x-224=0

224=2*2*2*2*2*7

3*2*2*2*2*2*7=3*8*28 and 3*8=24, 28-24=4 (coefficient of the middle term)

(3x+28)(x-8)=0

x-8=0

x=8

y=3x=24

check: 8/24=1/3 and 8+24+8*24=224, 32+192=224, 224=224