An 81 inch length piece of ribbon is to be cut into three pieces.

Question

An 81 inch length of ribbon is to be cut into three pieces. The longest piece is to be 34 inches longer thaan the shortest piece, and the thirdd piece is to be half the length of the longest piece. Find the length of ech piece of ribbon.

Answer 1

short, middle, long pieces of ribbon, totaling 81 inches.

long = short + 34

long = 2 * middle

short + middle + long = 81

this is called a "system of equations" with 3 variables (short, middle, long)

to solve a system of equations with 3 variables, you need 3 equations.  We have 3 equations, so we should be good.

in math terms, we're going to use the "substitution method"

so here's our equations:

long = short + 34

long = 2 * middle

short + middle + long = 81

since long = short + 34, we'll replace long with short + 34 in the other two equations like this:

short + 34 = 2 * middle

short + middle + short + 34 = 81

let's simplify

short + 34 = 2 * middle

2 * short + middle + 34 = 81

let's get the 34s to the other side of both equations by subtracting 34 from both sides like this:

short = 2 * middle - 34

2 * short + middle = 81 - 34

simplify

short = 2 * middle - 34

2 * short + middle = 47

in the bottom equation let's get middle to the other side:

short = 2 * middle - 34

2 * short = 47 - middle

in the bottom equation let's get short by itself.

short = 2 * middle - 34

short = (47 - middle) / 2

since short = short, we can say this:

2 * middle - 34 = (47 - middle) / 2

4 * middle - 68 = 47 - middle

5 * middle - 68 = 47

5 * middle = 115

middle = 23

but short = 2 * middle - 34 so:

short = 2 * 23 - 34

short = 46 - 34

short = 12

but short + middle + long = 81 so:

12 + 23 + long = 81

long + 35 = 81

long = 46

Answer:  short = 12, middle = 23, long = 46