Question

If Amanda walks at an average speed of $ 2.72 $ miles per hour, how long will it take her to walk $ 6.8 $ miles?

Answer 1

Hint: The average speed and total distance covered is given. We form a linear equation using the unitary system. We solve the given linear equation by simplifying the equation. We divide both sides of the equation with $ 2.72 $ . Then we simplify the right-hand side fraction to get the solution for $ x $ . We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1.

Complete step-by-step answer:
Let’s assume the time taken by Amanda to cover $ 6.8 $ miles with an average speed of $ 2.72 $ miles per hour will be $ x $ hours.
We know the unitary system. The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Amanda covers $ 2.72 $ miles in 1 hour which means he will cover $ 2.72x $ miles in $ x $ hours. This should be equal to $ 6.8 $ .
So, $ 2.72x=6.8 $ .
The equation $ 2.72x=6.8 $ is a linear equation of $ x $ . We first convert the decimal numbers into fractions.
\[
   2.72x=6.8 \\
  \Rightarrow \dfrac{272x}{100}=\dfrac{68}{10} \\
  \Rightarrow 272x=680 \;
\]
The equation becomes \[272x=680\]. Dividing with 272 we get
\[
   \dfrac{272x}{272}=\dfrac{680}{272} \\
  \Rightarrow x=\dfrac{680}{272} \;
\]
We need to find the simplified form of the proper fraction \[\dfrac{680}{272}\].
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
For our given fraction \[\dfrac{680}{272}\], the G.C.D of the denominator and the numerator is 136.
Now we divide both the denominator and the numerator with 4 and get $ \dfrac{{}^{680}/{}_{136}}{{}^{272}/{}_{136}}=\dfrac{5}{2} $ .
The time taken by Amanda is $ \dfrac{5}{2} $ hours.
So, the correct answer is “ $ \dfrac{5}{2} $ hours”.

Note: We also could have formed a factorisation of the equation \[272x=680\]. We take the constant 136 common out of the reformed equation \[272x-680=0\].
Therefore, $ 272x-680=136\left( 2x-5 \right)=0 $ .
The multiplication of two terms gives 0 where one of the terms is positive and non-zero. This gives that the other term has to be zero.
So, $ \left( 2x-5 \right)=0 $ which gives $ x=\dfrac{5}{2} $ as the solution.