Finding rate, total amount or number of units

A printing machine prints one hundred and twenty posters in 4 minutes. The machine works at the same rate the whole time. How many posters does it print in 10 minutes? Let's draw it. One bar for the 4 minutes, cut into 4 equal units. Each unit is 1 minute. Those 4 minutes produce one hundred and twenty posters altogether. A rate is an amount for each one, so find the amount for 1 minute first. one hundred and twenty posters shared over 4 minutes. one hundred and twenty divided by 4 is 30. So 1 minute is 30 posters. That is the rate. 30 posters per minute. Now the 10 minutes. Same rate, so the same unit, ten times over. Every extra minute is another 30 posters. 10 times 30 is three hundred. The machine prints three hundred posters in 10 minutes. The rate times the number of units is the total.

Your turn. The printing machine prints 30 posters each minute. It must print four hundred and fifty posters. How many minutes does that take? Try it first. Pause if you like. Let's finish it together. This time we know the total and the rate, and the number of minutes is the one we want. four hundred and fifty posters, at 30 a minute. four hundred and fifty divided by 30 is 15. So it takes 15 minutes. Count the units. 15 of them, 30 posters in each. The total divided by the rate is the number of units. One thing to watch. To find a time, divide by the rate, 30 each minute. The 4 minutes and the one hundred and twenty posters belong to the first part of the question, and neither of them is the rate.

Here is a different kind of rate question. A car park charges 3 dollars for the first hour and 2 dollars for every further half hour or part of a half hour. How much does it cost to park there for 2 hours 15 minutes? Try it first. Pause if you like. The first hour is charged separately and costs 3 dollars. Subtract the first hour from the total time. Two hours fifteen minutes minus one hour. That is one hour fifteen minutes. One hour fifteen minutes is seventy-five minutes. Now count the half hours inside 75 minutes. 75 divided by 30 is 2, remainder 15. Two whole half hours, and 15 minutes more. Those 15 minutes are part of a half hour. The sign says a part is charged as a whole one. So that makes 3 half hours. 3 half hours at 2 dollars each is 6 dollars. And 6 dollars plus 3 dollars is 9 dollars. The stay costs 9 dollars. So, three things. The rate, the total, and the number of units. First find the rate, then solve the problem. And when the rate changes partway, cut the bar at the change and price each stretch. A part of a block is charged as a whole one.