A rate is a fraction that expresses a quantity per unit of time. For example, the rate at which a machine produces a certain product is expressed this way:
rate of production = number of units produced / time
A simple rate question might simply provide two of the three terms, and then ask you for the value of the third term. To complicate matters, the question might also require you to convert a number from one unit of measurement to another.
First Example Of Rate Problem
A printer can print pages at a rate of 15 pages per minute, how many pages can it print in 2- hours?
The correct answer is (D). Apply the following formula:
rate = no. of pages / time
The rate is given as 15 minutes, so convert the time (2; hours) to 150 minutes. Determine the number of pages by applying the formula to these numbers:
15 = no. of pages / 150
(15)(150) = no. of pages
2,250 = no. of pages
A more challenging type of rate-of-production (work) problem involves two of more workers (people or machines) working together to accomplish a task or job, In these scenarios, there’s an inverse relationship between the number of workers and the time that it takes to complete the job; in other words, the more workers, the quicker the job gets done.
A work problem might specify the rates at which certain workers work alone and ask you to determine the rate at which they work together, or vice versa. Here’s the basic formula for solving a work problem:
A/x + A/y = 1
In this formula:
- x and y represent the time needed for each of two workers, x and y, to complete the job alone.
- A represents the time it takes for both * and y to complete the job working in the aggregate (together).
So each fraction represents the portion of the job completed by a worker. The sum of the two fractions must be 1 if the job is completed.
In the real world, teamwork often creates a synergy whereby the team is more efficient than the individuals working alone. But, you can assume that no additional efficiency is gained by two or more workers working together.
One printing press can print a daily newspaper in 12 hours, while another press can print it in 18 hours. How long will the job take if both presses work simultaneously?
A. 7 hours, 12 minutes
B. 9 hours, 30 minutes
C. 10 hours, 45 minutes
D. 15 hours
E. 30 hours
The correct answer is (A). Just plug the two numbers 12 and 18 into our work formula, then solve for A:
A/12 + A/18 = 1
3A/36 + 2A/36 = 1
5A/36 = 1
5A = 36
A = 36/5 or 7.2
Both presses working simultaneously can do the job in 7.2 hours-or 7 hours, 12 minutes.
Had you needed to guess the answer, you could have easily ruled out choices (D) and (E), which both nonsensically suggest that the aggregate time it takes both presses together to produce the newspaper is longer than the time it takes either press alone. Remember: In work problems, use your common sense to narrow down answer choices!