Machine A produces 129 meters of cloth in three hours. Machine B produces 121 meters of the same cloth in five and a half hours. Which machine has the faster production rate?
First, let’s list what we know. We’ve been given information about machine A. It produces 129 meters of cloth in three hours. If we want to write this as a rate, we take the 129 meters and write that as a fraction out of three hours. We can do the same thing for machine B, which is 121 meters in five and a half hours.
Now, we have two rates: 129 meters per three hours and 121 meters per five and a half hours. However, to compare rates accurately, we need to have a common denominator, and that common denominator is usually one. When we use rates with the denominator of one, they’re called unit rates. We want to ask the question “how many meters can machine A and B produce in one hour?” And because they’re rates, they are proportional.
To get from three hours to one hour, we divide by three. And since rates are proportional, if we divide by three in the denominator, we need to divide by three in the numerator. 129 divided by three is 43. Machine A produces 43 meters of cloth per hour.
We need to follow a similar procedure for machine B. To go from five and a half hours to one hour, we divide by 5.5. And since these are proportional, we need to do the same thing in the numerator. 121 divided by 5.5 equals 22. Machine B produces 22 meters of cloth per hour.
Now that we have the unit rate, it’s very easy to see which machine has the faster production rate. 43 meters per hour is far more than 22 meters per hour. Therefore, machine A has the faster production rate.