Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. If they work together, they can complete the job in 3 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jimmy to complete this job on his own.

Explanation:1. Analyze the problem to identify what is given and what is asked for. We are asked for the time it takes Jimmy to complete a job. We are given a relation between Jimmy's time and his brother's time, and we are also told the time when they work together.2. Choose a variable or set of variables that will help answer the question when the value(s) of the variable(s) is(are) found. Here, it is convenient to assign a variable to Jimmy's time, as finding the value of that will answer the question. We can call that variable "j". (It is often helpful to use variables with names that help you remember what they stand for. It is easy to confuse the meanings of generic variables such as "x" and "y".)3. Write the relations given in the problem statement as an equation or set of equations. Here, we need to realize that when two people work together, their rates of doing work are added. Those rates, in terms of jobs per hour, are the reciprocal of the job completion times in hours per job.Since Harry's time to complete the job is 8 hours less than Jimmy's we can use (j-8) to represent it. Then the total job completion rate in jobs per hour is ...   1/j + 1/(j-8) = 1/34. Solve the equation using the rules of algebra. Here, we can clear fractions by multiplying the equation by the product of the denominators:   3(j -8) +3j = j(j -8)Eliminate parentheses and collect terms:   3j -24 + 3j = j^2 -8j   -24 = j^2 -14j Complete the square by adding the square of half the j coefficient.   -24 +49 = j^2 -14j +49   25 = (j -7)^2Take the positive square root. We know that j must be greater than 8+3, so time values less than 7 will not make sense in this problem.   5 = j -7Add 7.   12 = j5. Using the value(s) of the variable(s) found, formulate an answer to the question. Here, that answer is ...   It would take Jimmy 12 hours to complete this job on his own.