This page contains a number of worked word problems. Of course, no list could be complete, but there are certain pattern that occur over and over.
Solution: Let's say your previous wage was dollars per hour. You are now making 10 percent more. In other words, you are making % of your old wage. Hence
Solution: The discount applies to the list price of . The sales tax applies to the discounted price. % of is . The discounted price therefore is (Equivalently you could argue that you buy the pants for % of the list price, which is ) You still need to pay % sales tax on the price of . Since % percent of is your total price is
Solution: Let's call the growth rate . Every year the population gets multiplied with . The problem tells us that
Solution: The main difficulty here is to understand the meaning of the phrase 5% vinegar. Think of ordinary vinegar as a mixture of water and pure (100%) vinegar. 5% vinegar means that the ratio of the pure vinegar in the mixture, and the total amount of fluid is 0.05 (or 5%). In this example the total amount of the fluid is 3 gallons, and the total amount of pure vinegar is gallons (i.e., 5% of the 1 gallon and 20% of the two gallons). The ratio of pure vinegar to total amount of fluid is, therefore,
Solution: The trick for this kind of problem is to think of the job as the basic unit. You want to do 1 job, i.e., paint that room. You can do 1/2 of that job in one hour, and your brother can do 1/3 of the job in one hour. Painting together the two of you will do of that job in hours. Thus we need to solve the equation
Solution: The solution to all such problems proceeds by drawing a picture, and then applying the Pythagorean Theorem. In this case we get the following Figure which shows the mountain in a fetching green color:
Denoting the unknown height by we get the equation
Solution: There are quite a few problems like this in the homework sets because they provide excellent exercises in applying radicals. As pointed out on the scaling page, multiplying each linear dimension with a factor multiplies every area with the square of that factor, and the volume (and weight) with the cube of that factor. Everything flows from that basic fact! In this case we multiply linear dimensions with 3, and so we multiply the area (and hence the price of the gold leaf) with and the volume (and hence the weight) of the statue with . You pay $1,800 for the gold leaf, and your original statue weights 54 tons.
You now begin to wonder how much it will cost to reinforce your floor, or whether you should perhaps remove the floor altogether and place your statue in the basement. But it is a beautiful statue!
In some of the homework problems in this course you are given (explicitly or implicitly) a factor that multiplies the area or the volume. As explained on the scaling page, if you multiply the areas with you multiply the lengths with and the volume with , and if you multiply the volume with you multiply the lengths with and the areas with .
You throw a rock and release it at a height of feet. Its initial vertical velocity is feet per second (going up) and its horizontal (forward) velocity is feet per second. How far does the rock fly until it hits the ground, and what is the maximum height that it reaches?
Solution: According to the discussion of throwing rocks the height of the rock at the time is given in general by In our case feet per second squared (since we are on earth), feet per second (the initial upward motion), and feet, the height at which we release the rock. The height of the rock at time , therefore, is given by To find out at what time the rock hits the ground (where the height is zero) we need to solve the quadratic equation which gives seconds as the time of impact. During that time the rock covers a distance that's obtained by multiplying the flight time with the horizontal velocity, which gives the distance feet. At the time that the rock reaches its maximum height its vertical velocity, which in general is given by is zero. In our case and hence the time at which the rock reaches the maximum height is given by seconds. At that time the rock is a height feet.