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# Rate Problems: The Rate Equation

A software programmer does 20% of the work in 80% of the time, and 80% of the work in the remaining 20% of the time. If the code he writes is $$x$$ lines long and he was given one month (30 days) to accomplish the task, then, assuming that the programmer works at a constant rate in each of the two stages, how many lines of code did he complete in the last two weeks, in terms of $$x$$?
Incorrect. Does the programmer work at a constant rate throughout the month?
Incorrect. You probably got the basic mathematical concept right, but were careless in conversion of time units. Remember that in the GMAT, you don't get points for arriving halfway to the solution. How many weeks are there in a 30-day month? Now you probably think that you need to go back to the question and be more careful this time.
Incorrect. You calculated the work that would be done in the first two weeks, while you were asked about the last two weeks.
Incorrect.
[[snippet]]
Very good. Did you get it right on your first try?
In that case, you already know that there isn't much you needed to work out in order to choose this answer, as it was the only one larger than 80% of $$x$$.
Now let's see what we would have had to do if __Ballparking__ couldn't eliminate all four other answers, but instead only three of them, meaning that we would still have to solve the question.
Avoiding unnecessary errors could be aided by the following means: Start with converting time units: Two weeks are not half of a month. Instead of talking in days, weeks, and months, the question should be presented only in days. So, the $$x$$ lines of code are written in 30 days. 20% of the work is done in 80% of the time, or the first 24 days. 80% of the work is done in the remaining 20% of the time, or the last six days. We're asked how much work was done in the last two weeks—in other words, how much work was done in the last 14 days. How do you calculate the work that was done in the last 14 days?
I like the way you think.
Instead of calculating the work done in the last 14 days of the month, let's make the easier calculation of the work that was done in the first 16 days of the month, when the rate was still slow. When we have that value, we subtract it from the total $$x$$, and get the result of what was left for the last 14 days.
In order to make life even easier let's __Plug In__ a good number instead of $$x$$. The number of code lines written in 30 days should be divisible by 30 if we want to work with convenient numbers. We chose $$x=300$$. So if 20% of 300 lines are written in the first 24 days, how many are written per day? Work is 20% of 300, or $$\frac{300}{5} = 60$$ lines. Rate is Work over Time, or $$\frac{60}{24} = 2.5$$ lines per day
At a rate of 2.5 lines per day, how many lines are written in the first 16 days? >$$\text{Work} = \text{Time} \times \text{Rate} = 16 \times 2.5= 40$$ lines So if 40 lines were written in the first 16 days, than $$300-40=260$$ lines were written in the remaining 14 days. That is your **goal value**. If you Plug In $$x=300$$, then A will be the correct answer, $$\frac{13x}{15}$$.
This will never take you two minutes. Take a second look; don't be pushed to do the longest, most tedious calculation.
If the last 80% of the work is done in the last 20% of the month, that means that in the last two weeks, more than that time frame, we expect more than 80% of the work to be done. Did you solve the question by eliminating answers based on this criterion?
Excellent! Way to go! If you stick to this line of thinking—first __Ballparking__ for a general estimation of the answer—you will be rewarded by avoiding tedious calculations and by scoring higher.
$$\frac{13x}{15}$$
$$\frac{7x}{15}$$
$$\frac{7x}{60}$$
$$\frac{2x}{3}$$
$$\frac{x}{2}$$
Yes
No
I divided this timeframe to the first eight days where the rate is slower and then the last six days where the rate is faster.
What?! There must be an easier way.
Yes, I did.
No. I just solved the question. Still I would like to compare my solution with a detailed solution.
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