So in this first set of videos in this module, we're gonna focus on the algebraic approach to word problems. I wanna frame this by saying, don't get locked into the mindset that algebra is the only way to do word problems. This is not the only way. In fact, for some word problems it might not be the best way to do the word problem. Show Transcript
We're covering algebra first because it's the most widely applicable. It applies to the widest variety, the largest number of word problems. So that's why we're beginning it first, just have in the back of your mind that algebra is not the only way to do this. Okay, having said that, we're gonna do algebra first. And of course if we're gonna do algebra, we need variables.
To solve a word problem algebraically, we need to introduce variables. The very first thing I will say is: be thoughtful in the choice of your letter for the variable. It's true that, in algebra, the math we do does not depend on the letter at all. It's the same math, it doesn't matter what letter we pick for the variable. Nevertheless, you will help yourself by choosing a meaningful letter.
It will be helpful for you to choose a variable that, for example, begins with the same letter of the name of the person or the thing for which it stands. Don't pick plain old x. See, everyone, when they start doing algebra, they think it's a law they have to use x for algebra and x is not necessarily the best choice. Picking a non-descript variable, such as x, may confuse you later in the problem when you find you have a number, and you don't know what it is that you have found.
And this is the problem people run into on word problems all the time. They solve, they get a number, but then this number, what does it have to do with the words? What number have they actually found? That can be a frustrating thing, especially when you're working under time pressure.
For example, this is the beginning of a sample problem. Each week, Sarah makes $500 more than Mary, and Helen makes $700 more than Mary. If the sum of their weekly salaries is blah blah blah, so you can imagine that problem continuing. If I pick x for the salary of one of the women, I may solve for it correctly, but when I get that number x, how will I know whether it answers the question?
How will I know was that x standing for Mary's salary, or Sarah's salary, or Helen's salary? In other words, it's very easy to get confused. You'd avoid all that confusion if you'd just used S, M, and H, those would be very good names for the weekly salaries of Sarah, Mary, and Helen.
Some other considerations in choosing variables. Sometimes, it can be helpful to assign a variable for the smallest value, and then express everything in terms of that. For example in that last problem, it seemed that one of the women had the smallest salary. It might make sense just to assign one variable for that and express everything else in terms of that variable.
Sometimes, it can be helpful to assign a variable to the target value, what the question is asking. And of course, the advantage of this is as soon as you've solved for the value of that variable, you have the answer to your question. So some people prefer doing that. If all the quantities in the problem relate to a single quantity, make this conceptually central quantity the variable, and I'll show an example of this in a moment.
Right here, this is a good example, this problem. I'll say pause the video, think about this question, and then we'll talk about how to solve it. Okay, so Sam's car was fined for parking when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Very nice of them.
Joe paid $3 more than a quarter of the fine and Peter paid $3 less than a third of the fine, leaving Sam $4 less than half the fine to complete the payment. How much did Sam pay? Wow, well first of all, notice that everything there is expressed in terms of the fine.
So in other words, the fine, that's the conceptually central concept in every variable. How much each individual paid is related to some fraction of the fine. So it makes the most sense to say that our single variable is gonna be the fine itself, call it F for the fine. And then we can say right away that Joe's share was (F/4) + 3.
Peter's share is (F/3)- 3, and Sam's share was (F/2)- 4. And of course if we add all these up, they should add up to the fine. So we add them and then, turns out that is very easy to clear all the fractions by multiplying by 12. We multiply by 12, we do a little math, and we get that the fine is 48.
Now, we haven't answered the question yet. That's the price of the fine. So now we have to answer, how much did Sam pay? Well, half of 48 is 24, so that means Sam paid $4 less than that. Sam paid $20. In summary, do not use generic variables.
Choose variables that will help you keep track of the numbers. Sometimes it helps to choose the smallest value as the variable, especially if everything else is related to that smallest value. Sometimes it helps to choose the target value as the variable, and if all the quantities are related to one quantity, choose that quantity as the variable.
And I'd suggest as you go through, as you're practicing with word problems, solve a word problem one way and then go back and assign variables another way. And solve it again using a different combination of variables. Just experiment for yourself so you develop a feel. What does it feel like to assign variables in different ways and different problems? And that's how you build intuition for what works best for you.