Question

The larger of two numbers is equal to five times the smaller. If twice the larger is

added to two times the smaller, the sum is 40. Find the numbers.

Answer 1

Explanation:

`\left \{ {{y=5x} \atop {2y+2x=40}} \right.`

`\left \{ {{y=5x} \atop {y+x=20}} \right.`

`\left \{ {{y=5x} \atop {5x+x=20}} \right.`

`\left \{ {{y=5x} \atop {6x=20}} \right.`

`\left \{ {{y=5x} \atop {x=(20)/(6)=(10)/(3)=3(1)/(3)  }} \right.`

`\left \{ {{y=5*(10)/(3)=(50)/(3)=16(2)/(3)  } \atop {x=3(1)/(3) }} \right.`

Answer 2

Let x be the smaller and y be the larger numbers.

"The larger of two numbers is equal to five times the smaller" means that

`y=5x`

"Twice the larger plus to two times the smaller" means

`2y+2x`

And we can plug y=5x here to get

`2y+2x=2(5x)+2x=10x+2x=12x`

This must equal 40, so we have

`12x=40 \iff x=(40)/(12)=(10)/(3)`

And the larger number will be five times the smaller, i.e.

`y=5\cdot (10)/(3) = (50)/(3)`