A 64-inch board is cut into three pieces so that the second piece is twice as long as the first piece, and the third piece is 5 times as long as the first piece. Find the length of the longest piece.

Explanation:

x is some positive real number, and the units of each are in inches.

Add up the smaller pieces and we should get 64 inches back again

x+2x+5x = 64

8x = 64

x = 64/8

x = 8

The first piece is 8 inches long.

2x = 2*8 = 16 is the length of the second piece (in inches)

The third piece is 40 inches because 5x = 5*8 = 40

Summary:

first = 8 inches

second = 16 inches

third = 40 inches

Check: 8+16+40 = 64, which confirms the answer

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!