Question

From a rope 68 m long, pieces of equal size are cut. If the length of one piece is \[4\dfrac{1}{4}\] m, find the number of such pieces.

Answer 1

Hint:
Here, we need to find the number of equal pieces the rope is cut in. We will solve this question by assuming the number of pieces cut to be \[x\]. We will convert the length of one piece into an improper fraction. We will use the fact that the total length of the rope will be equal to the product of the length of one piece and the number of equal pieces the rope is cut in. We will then obtain an equation in terms of \[x\]. Simplifying this equation we will get the value of \[x\] and hence, the number of equal pieces.

Complete step by step solution:
Let the number of pieces cut be \[x\].
Now, it is given that the length of each piece is \[4\dfrac{1}{4}\] m.
First, we will convert this mixed fraction to improper fraction.
Therefore, we get
\[\begin{array}{l}4\dfrac{1}{4} = \dfrac{{4 \times 4 + 1}}{4}\\ \Rightarrow 4\dfrac{1}{4} = \dfrac{{17}}{4}\end{array}\]
Thus, we get the length of each piece as \[\dfrac{{17}}{4}\] m.
Now, we know that the total length of the rope will be equal to the product of the length of one piece and the number of equal pieces the rope is cut in.
Therefore, we can form the equation
\[{\text{Total length of rope}} = {\text{Length of each equal piece}} \times {\text{Number of equal pieces}}\]
Substituting the total length of the rope as 68 m, the number of pieces as \[x\], and the length of each equal piece as \[\dfrac{{17}}{4}\] m, we get
\[\begin{array}{l} \Rightarrow 68 = \dfrac{{17}}{4} \times x\\ \Rightarrow 68 = \dfrac{{17}}{4}x\end{array}\]
Multiplying both sides of the equation by 4, we get
\[\begin{array}{l} \Rightarrow 68 \times 4 = \dfrac{{17}}{4}x \times 4\\ \Rightarrow 272 = 17x\end{array}\]
Dividing both sides of the equation by 17, we get
\[\begin{array}{l} \Rightarrow \dfrac{{272}}{{17}} = \dfrac{{17x}}{{17}}\\ \Rightarrow 16 = x\end{array}\]

\[\therefore\] The number of equal pieces of length \[4\dfrac{1}{4}\] m is 16.

Note:
Here, we converted a mixed fraction to improper fraction. An improper fraction is a fraction whose numerator is larger than its denominator. A mixed fraction is a fraction in the form \[a\dfrac{b}{c}\]. Every mixed fraction \[a\dfrac{b}{c}\] can be converted to an improper fraction \[\dfrac{{c \times a + b}}{c}\].