Question

A machine can produce 27 inches of ribbon every 3 minutes. How many feet of ribbon can the machine make in one hour?

Answer 1

Hint: We are given that a machine work to produce 27 inches of ribbon every 3 minutes, we have to look how much feet ribbon produce after one hour, to do so, we will be using firstly the unitary method to first find the ribbon produce in 1 minute and then use it further to find the actual produce in an hour and then we need dimensional analysis to find the given term in inches to get the amount of equitant in feet.

Complete step by step solution:
We are given that machines produce 27 inches of ribbon every 3 minutes, we are asked to find how many feet ribbon produce in one hour.
To do so, we will learn about unitary methods.
Unitary method is a technique in which we find the value of 1 unit (single unit) then we multiply by ‘n’ to find the value of ‘n’ units.
Now as we have that machine produce 27 inches ribbon in every 3 minutes so it means –
$3\text{minute}=27\text{inches}$
So, we will find ribbon produce in a minute. So, we divide both sides by ‘3’.
So, $ \dfrac{3\text{minute}}{3}=\dfrac{27\text{inches}}{3}$
We get –
$1\text{minute}=9\text{inches}$
Mean in 1 minute 9 inches of ribbon is produced.
Now, we will need to find the ribbon produced if the same machine continues working for an hour.
Before we move forward, we will learn about dimensional analysis.
Dimensional analysis is a technique which helps us in establishing the relation between different quantities.
In our case, we will use the fact that 1 hour is the same as 60 minutes.
So, as we have machines, we continue working for an hour.
Mean machine working for 60 minutes.
In one minute the machine produced 9 inches.
So for 60 minutes –
We multiplied both sides by 60.
We get –
In 60 minutes the machine produces $9\times 60$ inch ribbon.
So, a 60 inch machine produces 540 inches of ribbon.
Hence in 1 hour the machine produces 540 inches of ribbon as we have to find the answer into feet.
So, we will find a relation between inches and feet.
We know that 12 inches is the same as 1 feet.
So, we will find the value of 1 inch.
As $12\text{inch}=1\text{feet}$
So, by using unitary method,
$1\text{inch}=\dfrac{1}{12}\text{feet}$ .
Now, for 540 inches,
We multiply previous equation by 540 on both sides, so we get –
$\Rightarrow 540\text{inches}=\dfrac{1}{12}\times 540\text{feet}$ .
By simplifying, we get –
$=45\text{feet}$
So, we get –
In one hour the machine will produce 45 feet of ribbon.

Note: Remember $\dfrac{1}{a}\times b=\dfrac{b}{a}$ , when we solve our calculation in only fraction, we need to simplify and then solve, for example $\dfrac{2}{16}\times 20$ , here we cancel, 4 on both numerator as well as denominator $\dfrac{2}{16}\times 20=\dfrac{10}{4}$ , we get $\dfrac{10}{4}$ , we can also cancel ‘2’ on both numerator, so $\dfrac{10}{4}=\dfrac{5}{2}$ , now as nothing remain common, now we divide 5 by 2 we get 2.5.
So, $\dfrac{2}{16}\times 20=2.5$ .
Doing simplification makes calculation easier.