Question

Find two numbers if their ratio is 4:7 and their difference is 123.

Answer 1

`\large\boxed{-164\ and\ -287}`

Explanation:

`x,y-\text{the numbers}\\\\x:y=4:7\to(x)/(y)=(4)/(7)\qquad(1)\\\\x-y=123\qquad\text{add y to both sides}\\\\x=123+y\qquad(2)\\\\\text{Substitute (2) to (1)}:\\\\(123+y)/(y)=(4)/(7)\qquad\text{cross multiply}\\\\7(123+y)=4y\qquad\text{use distributive property}\\\\(7)(123)+(7)(y)=4y\\\\861+7y=4y\qquad\text{subtract 7y from both sides}\\\\861=-3y\qquad\text{divide both sides by (-3)}\\\\-287=y\to\boxed{y=-287}`

`\text{Put the value of y to (2)}\\\\x=123+(-287)\\\\x=123-287\\\\\boxed{x=-164}`

Answer 2

The two numbers are -164 and -287

They could also be 164 and 287

Explanation:

The ratio of 2 numbers 4/7

x/y = 4/7

Using cross products

7x =4y

Their difference is 123

x-y = 123

x = 123+y

Substituting in

7(123+y) =4y

Distribute

861 +7y = 4y

Subtract 7y from each side

861 +7y-7y = 4y-7y

861 = -3y

Divide each side by -3

861/-3 = -3y/-3

-287 = y

Now we need to find x

x = 123+y

x = 123+ -287

x = -164

The numbers could also be 164 and 287

The difference is 287-164 = 123

And the ratio is 164/287 = 4/7