Answer-
He purchased a rope of 48 ft.
Solution-
Let us assume the length of the rope he had originally purchased was x ft.
He used one-fourth of the x to make a bow, so the rope he used was
[tex]=\dfrac{x}{4}\ ft[/tex]
The amount he left with was,
[tex]=(x-\dfrac{x}{4})\ ft[/tex]
Then he used one-third of the remaining to tie up a roll of carpet, so the rope he used was
[tex]=(x-\dfrac{x}{4})\dfrac{1}{3}\ ft[/tex]
Then he left with 24 ft of rope.
∵ The total amount = Amount used + Amount left
Hence,
[tex]\Rightarrow \dfrac{x}{4}+(x-\dfrac{x}{4})\dfrac{1}{3}+24=x[/tex]
[tex]\Rightarrow \dfrac{x}{4}+\dfrac{x}{3}-\dfrac{x}{12}+24=x[/tex]
[tex]\Rightarrow \dfrac{x}{4}+\dfrac{x}{3}-\dfrac{x}{12}-x=-24[/tex]
[tex]\Rightarrow \dfrac{3x+4x-x-12x}{12}=-24[/tex]
[tex]\Rightarrow \dfrac{-6}{12}x=-24[/tex]
[tex]\Rightarrow \dfrac{1}{2}x=24[/tex]
[tex]\Rightarrow x=24\times 2=48[/tex]
Therefore, he purchased a rope of 48 ft.
