In this section, we're going to solve some application problems.
The solution to these application problems will involve the equations that we've been solving previous to this.
Written over here on the board, what's called the Blueprint for Problem Solving;
and it's an outline of the steps that we want to follow when we solve these application problems.
The first step is to read the problem and then mentally list to ourselves the known and unknown items that are given in the problem.
Step 2: We want to assign a variable to one of the unknown items in the problem,
and then translate the other information in the problem into expressions that involve that variable.
Step 3: We reread the problem and write an equation. Step 4: We solve that equation. Step 5: We write the answer in complete sentences.
And then, step 6: We reread the problem and check our solution with the original words of the problem.
So, this Blueprint for Problem Solving is written in the book. And you want to use it as a guide to solving these application problems.
Let's look at our first one: the sum of a number and 3 is 5. Find the number. Well, as we read through this problem,
we see that we have the sum of a number and 3, and that the result is 5. We want to find the number.
So, the sum, 3 + 5 those are known items. The actual number itself is an unknown item.
Step 2: We want to assign a variable to the unknown item. So, let's say, let x equal the number in question.
So, we're going to let x equal the number in question. Then, the sum of the number and 3 will be x + 3.
Now, step 3, what I want to do is reread the problem and write an equation that describes the situation.
So, I have the sum of the number and 3 is equal to 5. I've let x be equal to the number. I know that the sum of the number and 3 is x + 3.
Now, as I reread the problem, I see that the sum of the number and 3 is 5. So, that will give me the equation that will describe the situation.
The sum of the number and 3 is equal to 5. The next step is to solve this equation. So, I do that.
I think you can just see that the solution is x = 2. So, I want to write the solution in a complete sentence.
So, I'm going to say, "The number is 2," meaning that the number asked for in this problem is 2.
Now, let's take this 2 and just go back and check it with the original words of the problem. Is the sum of 2 and 3, 5? The answer is, yes;
so that solution checks with the original words of this problem.
Let's look at our next example. This time, we have three times the sum of a number and 1 is 18. Find the number.
So, I read the problem, and I see I have three times the sum (so that's going to be addition) of a number (I don't know what that is) and 1 is 18.
Find the number. So again, let's say, let's let x equal the number; and by that, I mean let's let x equal the number that's asked for in the problem.
Then, I have 3 times the sum of that number and 1. So, 3 times the sum of that number and 1 would look like this.
So, that's my second step, where I let x equal the unknown item in the problem.
And then, I translate some of the other information in the problem into expressions that involve that variable x.
Now, the next thing I want to do is step 3, which is, write the equation that describes the situation.
So, that's that 3 times the sum of a number and 1 turns out to be 18. So, here's my equation: 3(x) + 1 = 18.
So, that's step 3, where I have the equation that describes the situation. So, I want to solve the equation, now.
So, I'll multiply 3(x) and get 3x; 3(1) and get 3; that comes out to be 18; add -3 to both sides; I have 3x = 15.
Multiply both sides by 1/3, and I get x is equal to 5. So, my solution is x = 5, which I will write out as, "The number is 5."
Now, let's go back and check the number 5 in the original words of the problem: 3 times the sum of 5 and 1
well, 5 and 1 is 6; 3 times that is 18; is 18. So, certainly that solution, x = 5, checks in the original.
The next problem that we want to do involves a geometric figure: the length of a rectangle is twice its width; the perimeter is 30 meters.
Find the length and the width. Well, what I know when I read this, what's known and unknown is, well, I know that I have a rectangle.
Its length is twice its width, and its perimeter is 30 meters. What I want to find, the things I don't know, are the length and the width.
So, what I'm going to do is assign a variable to the width of this rectangle.
And one way to do it when you have a geometric figure is to actually draw a little picture of that figure. So, I'm going to draw in the rectangle;
and I'm just going to label the width with x. Then, I'll write the length in terms of that; and I know that the length is twice the width.
So, the length is going to be 2x. So, the width is x over here and over here; and the length is 2x here and 2x down here.
Now, I can write an equation that describes the situation by simply adding up all four sides to get the perimeter:
2x + 2x + x + x is the perimeter of this rectangle; and that comes out to be 30. So, 2x + 2x = 4x, + 1x = 5x, plus another 1x is 6x.
So, I have 6x = 30; so, x = 5. Now, I want to write my solution in a complete sentence.
So, I'm going to say, "The width is 5." (And what were my units here? Meters.) "The length is twice that, or 10 meters."
Now, let's check the solutions in the original words of the problem.
If I have a rectangle that's 5 meters wide and 10 meters long, then the length is twice the width, so, that part of it checks.
The perimeter would be 5 + 5, which is 10; + 10, which is 20; plus, another 10, which is 30.
So, if I add up the length all the way around the outside here 5 + 10 + 5 + 10 I am going to end up with 30.
So, these solutions do check in the original words of the problem. Problem 4 is an age problem; and it says Dale is 4 years older than Sue.
Five years ago, the sum of their ages was 64. How old are they now? So, as I read the problem, I find that I know that Dale is 4 years older than Sue;
and that five years ago, if I added up their ages, the result was 64. How old are they now? So, what I don't know is their ages now.
So, one way to do an age problem like this is with a little table. So, I think I'll put the table here, and let you take a look at it.
I have two columns right here; and I'm going to use for my two columns, the two people in the problem.
So, one will be Dale, and the other will be Sue. For my two, I'm sorry, the two rows in the table right here: one is for Dale; one is for Sue.
The two columns, I'm going to put in the two different times: one is now and the other is 5 years ago; 5 years ago.
Now, if I let x be equal to Sue's age, then I've assigned a variable to one of the unknown quantities in the problem,
and I've done it in table form, right here. So, when I put an x right here, what I mean is that, let's let Sue's age now be equal to x.
Okay, then I can write Dale's age in terms of that because I know that Dale is 4 years older than Sue.
So, whatever Sue's age is, Dale is 4 years older than that. Now, these two boxes in the table, I can fill in this way.
This is Sue's age now; 5 years ago she was 5 years younger. So, I subtract 5 from her age now to get her age 5 years ago;
likewise, the same thing with Dale's age. He's x + 4 years old now minus 5 is his age 5 years ago.
So, this comes out to be just x - 1. Now, I reread the problem, and what it says is that 5 years ago, the sum of their ages was 64.
So, 5 years ago, these are their two ages; and the sum of those two things turns out to be 64. So, x - 1 + x + 5 has to be equal to 64
that's the sum of their ages 5 years ago is 64. So, x + x = 2x; -1 + -5 = -6; is equal to 64.
Now, I'll add 6 to both sides, and I get 2x is equal to 64 + 6, which is 70. So, x = 35. x is equal to 35 is Sue's age now.
So, we've solved part of the problem. Sue's age now is 35. Dale is 4 years older than that, so Dale must be 39 years old.
So, 35 + 4 = 39. Dale is 39; Sue is 35. If I look at their ages 5 years ago, what I'll find is that those two ages add up to 64.
So, that's a look at solving an age problem that involves finding an equation, like we had in the previous section,
the solution to which gives us the solution to this application age problem.