Updated 20190816
Throughout science, there inevitably comes a time when it is necessary to use some "known relationship" to figure out what happened or what might happen. In physics and chemistry, especially, this is the case, but sciences like sociology use "known relationships," such as population growth models, to explain or predict certain happenings. Business science looks at factors such as supply and demand or production cost vs sales price as means of maximizing profit.
In many, many cases, the "known relationship" is expressed as a mathematical equation. The relationship between the rate something happens and the amount of something produced is very common, so this discussion will use a couple of rate problems as its example.
In most general terms, the amount of something produced is equal to the rate of production times the amount of time production took place. For instance, in a displacement problem, rate is how far something moves in a given time and motion of an object is produced. In a general form a rate problem might look like this:
O = R•t
where O is the output, R is the rate, and t is the time.
So what is this process of solving science problem that will use rate problems as its example?
There are three steps to solving ANY science problem (or word problem of any kind, for that matter).
The following two examples will be used as the steps are discussed:
EXAMPLE 1: A baker's oven will hold only 1 pan of cookies, and each pain has space for 12 cookies. The baking time on cookies (including putting the dough on the pan) is 20 minutes. Therefore, the rate cookies are baked is 12 cookies per 20 minutes. How many cookies can be baked in 80 minutes?
EXAMPLE 2: A car travels at an average rate of 20 MPH for 3 hours. How far does it go.
The first step is to write down what is asked for. What are you supposed to find?
Example 1:
LOOKING FOR "How many cookies"Example 2:
LOOKING FOR "How far does it go"
A student familiar with physics motion problems would see Example 2 as a "distance" problem and would use the distance equation variables. In such a case, R would be velocity, v, t would be time, t, and LOOKING FOR would be distance, d (or in some cases, s).The second step is to write down what's given and find an equation that relates to the question.
Example 1:
Given:
Rate of baking (R) = 12 cookies/20minutes
time (t) = 80 minutes
Intuition and general math instruction will lead you to a formula where…
Number of Cookies = R • t
Example 2:
Rate of motion (v) = 20 MPH
time (t) = 3 hours
Intuition or a quick Google search will reveal a formula (the distance formula) that related velocity and time of motion.
d = vt
From time to time, it is necessary to rearrange an equation so that it yields the answer you are trying to find. For example, if you trying to find time in Example 2 above, you will need to isolate the t variable. MANY TIMES it is easier to do the algebra BEFORE you plug in the numbers and units!
d = vt (To find t divide both sides by v.)
d/v = vt/v (The v on the right side cancels.)
d/v = tOnce you have an equation in the form you need, you are ready for the next and final step.
The third step is to plug in what was given and solve. Assuming the second step was completed, the following solutions would emerge:
Example 1:
Cookies = R•t
Cookies = 12 Cookie/20 min • 80 min
Cookies = 960 Cookie•min / 20 min
Cookies = 48 Cookie
Example 2:
d = vt
d = 20 mi/hr * 3hr
d = 60 (mi•hr)/hr
d = 60 mi
For additional examples and a video explanation, check this out:



Example 3 (All together)
A car ball rolls at a constant rate of 20 m/s. How far will it roll in 8 seconds?
Using df as final distance rolled, v as the velocity, and t as time…
Find df where
v = 20 m/s
t = 8 s
df = vt
df = (20 m/s) (8 s)
df = 160 m
SUMMARY
There are three steps to solving problems.
1: Write down what is asked for.
2: Write down what is given and find a relevant equation.
2b: Rearrange the equation so that it yields the answer you want.
3. Plug in and solve.
The same thing expressed as five steps:
1 Write down what is being asked for.
2. Write down what is given.
3. Identify a relevant equation.
4. Plug in.
5. Solve.
1: Write down what is asked for.
ReplyDelete2: Write down what is given and find a relevant equation.
2b: Rearrange the equation so that it yields the answer you want.
3. Plug in and solve.