Word Problem on Proportions: Problem type 3 (SLICE: Exp. & Rat. Expressions)

 

The problems in this objective are a bit tricky because you can’t work them all with one set “formula”.  The problems deal with quantities that vary with respect to each other.  If one quantity goes up when the other goes down, we call this indirect or inverse variation.  If this relationship holds, whatever factor you increase one quantity by (multiply), the other will decrease by that same factor (divide).  An example of this type of variation might involve getting work done.  If you increase the number of workers or machines, it will take less time to do the task. Notice that to make quantities bigger you multiply, to make them smaller you divide.

 The other type of variation is direct…when one quantity goes up, the other goes up also…or they both go down such as the case may be.  In this case if you increase one by a factor (multiply) you increase the other by the same factor (multiply). 

 Below are examples of each type:

 Example 1:

 5 workers can complete a given task in 6 days.  If there are 2 workers, how many days would it take them to finish the same task?

 Solution:

Notice that this is inverse variation because increasing the number of workers should decrease the amount of time it takes them to do a job.

Next we want to figure out how long it will take 1 worker to do the job.  Since we are decreasing the number of workers by a factor of 5 (5 workers divided by 5) we need to increase the amount of time by a factor of 5 (6 days times 5 = 30 days).  So it takes 1 worker 30 days to do the job.  This should make sense.  The less help you have the longer it takes to get the job done.

 Now, the problem asks how long it will take 2 workers.  Working from the knowledge that it takes 1 worker 30 days, if we double the number of workers (multiply by 2) it should take fewer days (divide 30 by 2 to get 15 days).

 Thus it should take 2 workers 15 days to get the task done.  Notice this answer makes sense with the given information.  It should take 2 workers longer than 5 workers but less time than 1 worker.

 Example 2:  

At an oil change shop, each mechanic can change the oil in 3 cars in ½ hour.  How long would it take 5 mechanics to change the oil in 30 cars?

 This one is a bit more complicated because we have 3 things to be concerned with…mechanics, cars, and time. 

 We know 1 mechanic changes the oil in 3 cars in half an hour.  We have 1 mechanic.  We also need to get it down to a time for 1 car. This is a direct relationship.  If I decrease the number of cars, it should take less time.  If he does 3 cars in ½ hour (30 minutes) , he should be able to do 1 (3 divided by 3) car in 10 minutes (30 divided by 3).  We need the time in hours so we express 10 minutes as 1/6 of an hour.  Notice also that ½ divided by 3 is 1/6, but I thought using the minutes would make the relationship easier to see.

 Now, to the question…How long would it take 5 mechanics to change the oil in 30 cars?  First we need to note that 5 mechanics could change the oil in 5 cars in 1/6 hour…the time doesn’t change…each mechanic is working on his own car and should get finished in 10 minutes (1/6 hour).  Another way of looking at this is that a garage with 5 mechanics on duty should be able to process 5 cars every 10 minutes (1/6 hour).

How long will it take them to do 30 cars?  That is 6 times as many cars so it should take 6 times as long or 6 time 1/6 = 1 hour. 

Strategy:

Figure out how long it takes for 1 then think about the relationship.  If you need a larger number (direct variation), multiply.  If you need a smaller number (inverse variation), divide.

 Example 3:

 If 2 farmers can harvest 10 bushels of tomatoes in 2 hours, how many farmers are needed to harvest 80 bushels of tomatoes in 1 hour?

 This time we need to figure out how many farmers and the time needs to be 1 hour.  We need to find the number of farmers to harvest the 10 bushels in 1 hour.   Notice the relationship is inverse…If you decrease the time, it will take more farmers to get the 10 bushels harvested.  Since we are cutting the time in half (2 hours divided by 2), we would need to increase the farmers by a factor of 2 (2 times 2 farmers = 4 farmers).  Thus 4 farmers can harvest 10 bushels in 1 hour.  

We need to harvest 8 times that amount in the same time ( 1 hour).  There is a direct relationship between harvest and farmers.  If I want to harvest more in the same amount of time, I will need more farmers.  Since the number of bushels is 8 times as much as 4 farmers can do, it will take 8 times as many farmers or 8 times 4 = 32 farmers to harvest 80 bushels in 1 hour.

 Example 4:

 If 10 plants require 20 milliliters (ml) of plant food every 3 days, how many ml of plant food would 50 plants require in 9 days?

 This is a direct variation problem…more plants take more food.

If 10 plants require 20 ml of plant food every 3 days, 50 plants should require 5 times as much, or 100 ml of plant food every 3 days.  Notice this time I skipped figuring the rate per plant because it is easy to see you have 5 times as many plants ( 5 times 10 = 50) so you would need 5 times as much plant food in that 3 day period.  However, you could also figure that if 10 plants require 20 ml in 3 days, 1 (10 divided by 10=1) plant would require 2 ml (20 divided by 10=2) in 3 days, and 50 plants would require 50 times that or 100 ml in 3 days.

 Now the problem asks about 9 days.  Since there are three 3-day periods in the 9 days, we need to feed the plants that 100 ml 3 times so it would take 300 ml to do the job.