Question

If it takes 42 minutes to load \[3\dfrac{1}{2}\] trucks, how many minutes will it take to load \[6\dfrac{1}{2}\] trucks?

Answer 1

Hint: Convert the number of trucks which is given in mixed fraction into improper fraction. Use the relation: - \[a\dfrac{b}{c}=\dfrac{\left( a\times c \right)+b}{c}\] for the conversion. Now, apply the unitary method to determine the time it will take to load one truck. Multiply both the sides with \[\dfrac{13}{2}\] to get the required time.

Complete step-by-step solution:
Here, it is given to us that the time taken to load \[3\dfrac{1}{2}\] trucks is 42 minutes and we are asked to determine the time in minutes that will be taken to load \[6\dfrac{1}{2}\] trucks.
First, let us convert the number of trucks which is given in mixed fraction into improper fraction using the relation: - \[a\dfrac{b}{c}=\dfrac{\left( a\times c \right)+b}{c}\]. So, we have,
\[\Rightarrow 3\dfrac{1}{2}\] trucks = \[\dfrac{\left( 3\times 2 \right)+1}{2}=\dfrac{7}{2}\] trucks
\[\Rightarrow 6\dfrac{1}{2}\] trucks = \[\dfrac{\left( 6\times 2 \right)+1}{2}=\dfrac{13}{2}\] trucks
\[\because \] Time taken to load \[3\dfrac{1}{2}\] trucks = 42 minutes
\[\Rightarrow \] Time taken to load \[\dfrac{7}{2}\] trucks = 42 minutes
Now, applying the unitary method to determine the time it will take to load 1 truck, we have,
\[\Rightarrow \] Time taken to load 1 truck = \[\dfrac{42}{\dfrac{7}{2}}\] minutes
\[\Rightarrow \] Time taken to load 1 truck = \[\dfrac{42\times 2}{7}=12\] minutes
Now, multiplying both the sides with \[\dfrac{13}{2}\], we get,
\[\Rightarrow \] Time taken to load\[\dfrac{13}{2}\] trucks = \[12\times \dfrac{13}{2}\] minutes
\[\Rightarrow \] Time taken to load\[\dfrac{13}{2}\] trucks = 78 minutes
\[\Rightarrow \] Time taken to load \[6\dfrac{1}{2}\] trucks = 78 minutes
Hence, it will take a total of 78 minutes to load \[6\dfrac{1}{2}\] trucks. Therefore, 78 minutes is our answer.

Note: One may note that we have applied the basic unitary method to solve the question. Actually, this question is from the topic ‘ratio and proportion’. There can be another method also to solve the question. What we can do is we will assume the time taken to load \[6\dfrac{1}{2}\] trucks as ‘x’ minutes. Now, we will write the given information in proportion form as: - 42 minutes : \[\dfrac{7}{2}\] trucks :: x minutes : \[\dfrac{13}{2}\] trucks, which is equal to the expression: - \[\dfrac{42}{\dfrac{7}{2}}=\dfrac{x}{\dfrac{13}{2}}\]. Hence, by cross – multiplication we will simplify the expression and get the value of x. But one thing to remember is that you must convert the mixed fraction into the improper fraction for the calculations.