# How well do we need to do in the remaining months to meet an annual percentage goal?

I am required to have a employment team meet a yearly percentage rate of 50% for a process. The team is currently short of the goal with a rate of 43%. My boss wants to know what percentage is needed each month to meet the rate at the end of the year. Is there a way I can get the percentage needed if I don't know what my future numerators or denominators are going to be?

          cases complete     total cases     monthly % complete     Total % complete
JAN             20                40                 50%                  50%

FEB             10                30                 33%                  43%

MAR             40                50                 80%                  58%

APR             20                60                 30%                  50%

MAY             20                80                 25%                  43%


I need to make sure the team is above 50% after December. I have 7 more months to reach my required rate. Is it even possible to calculate a target % rate to show me what rate is needed each month to reach 50% at the end of the year? My boss insists that there is a way. I explained that we are currently 50 cases short of meeting our goal and that we need to make up the 50 cases by the end of the year and also meet the 50% goal each month to make 5o% at the end of the year. She was not satisfied with that response. She wants a percentage. Please help.

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I'm very happy you used the homework tag - otherwise, I would have recommended that your boss hire a mathematical consultant, as industry should pay for what it needs. I'm going to add the "algebra-precalculus" tag, as policy here is not to use the homework tag all by itself. –  Gerry Myerson Apr 20 '12 at 6:45
In principle you are right. For example, suppose that for the next $7$ months you have just $1$ case a month (I know, highly unlikely, but I am arguing principle). Suppose that your clearance rate from now on is $100\%$. You will still not meet the overall $50\%$ target. However, if you make a reasonable estimate of monthly number of cases based on the past, then it is possible to come up with a percentage. –  André Nicolas Apr 20 '12 at 6:53

Your answer of "we need to make up the 50 cases by the end of the year and also meet the 50% goal each month to make 50% at the end of the year" is almost correct. You in fact need to make up 20 cases, plus half the new cases.

So if $N$ new cases come in the remaining seven months, the percentage is $\left(\frac{20}{N}+\frac{1}{2}\right)\times 100$%. If $N$ is say $140$ then this is about $64.3$% but if $N$ is 35 then it is over $100$% and so impossible.

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