Question

Find two positive numbers so that twice their sum equals their product and one number is 3 times the other number. Enter the smaller number first.

Answer 1

x and y

2(x+y)=xy
x is 3 times y
x=3y

easy, sub 3y for x

2(3y+y)=(3y)y
2(4y)=3y²
8y=3y²
divide both sides by y
8=3y
divdie both sides by 3
8/3=y
sub back
x=3y
x=3(8/3)
x=8

the smaller number is 8/3
larger is 8

Answer 2

Answer: 8/3 and 8

Explanation: Suppose that the two numbers are represented by P and Q.

Twice their sum equals their product: 2(P + Q) = PQ

One number is 3 times the other: P = 3Q

To solve it, first, substitute P in the first equation:

2(P+Q) = PQ

2(3Q + Q) = 3Q.Q

2(4Q) = 3Q²

8Q = 3Q²

3Q² - 8Q = 0

Q(3Q - 8) = 0

Q = 0

3Q - 8 = 0

Q = 8/3

As the question said the numbers are positive and since 0 isn't positive or negative, to find P, use Q=8/3

P = 3Q

P = 3.
`(8)/(3)`

P = 8

The positive numbers are 8/3 and 8.