Question

The number of scarves of length half a metre that can be made from \[y\] metres of cloth is:
A) \[2y\]
B) \[\dfrac{y}{2}\]
C) \[y + 2\]
D) \[y + \dfrac{1}{2}\]

Answer 1

Hint:
We will assume that \[x\] number of scarves can be made. We will formulate a linear equation according to the information given in the question. We will solve that linear equation to find the answer.
Formulas used:
1) We can obtain the reciprocal of a fraction by interchanging its numerator and denominator. For example, the reciprocal of the fraction \[\dfrac{a}{b}\] is the fraction \[\dfrac{b}{a}\] .
2) When we divide 2 fractions, we can take the reciprocal of the 2nd fraction and change the division sign into multiplication:
\[ \Rightarrow \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}\]

Complete step by step solution:
We know that the length of each scarf is half a metre i.e. 0.5 m. We will express the length in the form of a fraction. We will remove the decimal and multiply by 10 in the denominator:
\[ \Rightarrow 0.5 = \dfrac{5}{{10}}\]
We will factorize the denominator and cancel out the common factors (if there are any):
\[\begin{array}{l} \Rightarrow 0.5 = \dfrac{5}{{5 \times 2}}\\ \Rightarrow 0.5 = \dfrac{1}{2}\end{array}\]
We will assume that \[x\] scarves can be made. So, the total length of cloth for the scarves will be:
\[ \Rightarrow \dfrac{1}{2} \times x\]
But we also know that the total cloth length is \[y\] metres. So, we will equate the 2:
\[ \Rightarrow \dfrac{1}{2} \times x = y\]
To find \[x\] , we will divide \[y\] by \[\dfrac{1}{2}\]:
   \[ \Rightarrow x = y \div \dfrac{1}{2}{\rm{ }}\left( 1 \right)\]
To solve the above equation, we will find the reciprocal of $\dfrac{1}{2}$. We will substitute 1 for \[a\] and 2 for $b$ in the formula for reciprocal of a fraction:
$\Rightarrow \dfrac{2}{1}=2$

The reciprocal of $\dfrac{1}{2}$ is 2.
We will substitute y for \[a\] , 1 for \[b\] , 2 for \[d\] and 1 for \[c\] in the formula for division of 2 fractions and we will simplify equation (1):
\[\begin{array}{l} \Rightarrow x = \dfrac{y}{1} \times \dfrac{2}{1}\\ \Rightarrow x = 2y\end{array}\]
We can see that \[x\] is equal to \[2y\] . So, the number of half a metre scarves that can be made out of \[y\] metres of cloth is \[2y\] .
\[\therefore \] Option A is the correct option.

Note:
We can also solve the question by looking at the options. We need to find \[x\] such that it satisfies equation (1). We will start by substituting the 1st option in place of \[x\] .
The L.H.S. will be:
\[2y\]
The R.H.S will be:
\[\begin{array}{l} \Rightarrow y \div \dfrac{1}{2} = y \times 2\\ \Rightarrow {\rm{ }} = 2y\end{array}\]
As the L.H.S and R.H.S are equal, option A is the correct option. If the L.H.S. and R.H.S. were not equal, then we would have to check the next options and see which of them satisfies the equation.