The product of two numbers is 50. If you divide the larger number by the smaller number you get 2. What are the numbers?

x = larger number

y = smaller number

xy = 50
x/y = 2 or x = 2y

Substitute, x = 2y in xy = 50
2y * y = 50
2y^2 = 50
y^2 = 25
y = +- 5

To find x,
Substitute y = +- 5 in either equation
xy = 50
5x = 50
x = 10

-5x = 50
x = -10

Numbers are 10, 5 or -10, -5

The factors of 50 are:

10 and 5
25 and 2

Which of these pairs meets the other criterion?

To find the two numbers, let's assign variables. Let's say the smaller number is x and the larger number is y.

We know that the product of the two numbers is 50, so we can write the equation:
x * y = 50

We are also given that if you divide the larger number by the smaller number, you get 2. So we can write the second equation as:
y / x = 2

Now we have a system of two equations. To solve for x and y, we can use either substitution or elimination method.

Method 1: Substitution
From the second equation, we can rearrange it to solve for y:
y = 2x

Now we substitute this value of y into the first equation:
x * (2x) = 50
2x^2 = 50
Divide both sides by 2:
x^2 = 25
Take the square root of both sides (considering both positive and negative roots):
x = ±5

Substituting x = 5 back into the second equation, we get:
y = 2 * 5
y = 10

So, the two numbers are 5 and 10 (or -5 and -10).

Method 2: Elimination
Rearrange the second equation to solve for y:
y = 2x

Substitute this value of y into the first equation:
x * (2x) = 50
2x^2 = 50
x^2 = 25
x = ±5

Plugging x = 5 back into the second equation, we find:
y = 2 * 5
y = 10

Therefore, the two numbers are 5 and 10 (or -5 and -10).