There are two types of fractional equations that a first year algebra course covers. The first type consists of equations that have fractional coefficients, such as:
The second type has variables in the denominator:
I introduce equations with fractional coefficients well before this unit, showing how to eliminate the denominator by multiplying all terms by a common denominator. It is also taught once more prior to this lesson. It paves the way for the second type, which is the subject of this lesson.
In essence, both types are solved in the same way: all terms are multiplied by a common denominator. In the second type, however, that denominator consists of variables. Students have had experience with such denominators from their work with adding and subtracting algebraic fractions. Nevertheless, it is new information and the connection between the two is often missing. Also, with adding or subtracting fractions, the denominator is not eliminated, whereas with equations it is.
Some students will retain a common denominator; for example, taking the equation given above, they will treat it the same as if they are adding fractions:
I will then suggest they eliminate the denominator 9n by multiplying both sides by the same, resulting in 15 = 5n. It is then a matter of showing that this same result can be obtained by multiplying each term by 9n.
I devote two days for this topic. After presenting the method of multiplying through by a common denominator on the first day, I then show how certain problems lend themselves to cross-multiplication. For example:
This can be solved by multiplying each side by 2x, which is effectively the same as cross-multiplying.
This also serves as a derivation for cross-multiplication. (I presented an informal derivation of cross-multiplication here. The cross multiplication method can be used for problems that have more than one term on each side:
I give examples where cross-multiplication is beneficial as well as when multiplying each term by the LCD is a better way to go.
Warm-Ups.
Problem 1 is a “hiding in plain sight” problem. If x represents one of the numbers, then 46 – x represents the other. Prompts might be: “If 10 is one of the numbers, how do you find the other?” The student then extends the pattern to x being one of the numbers. Problem 2 is subtraction of fractions. Problem 3 require students to see that both sides are multiplied by the denominator. Suggested prompt: “What if the problem were x/8 = 2? How do you eliminate the 8?” Problem 4 is the same as 1/3(x) -1/5(x) = 4 and requires that all terms be multiplied by the LCD, which is 15. Problem 5 requires students to see (a-b) as a single number. Prompt might include “What if it were x² - 25?” “What’s the square root of x²?” “What’s the square root of (a-b)²?”
Day 1
Multiplying by LCD. I use Problem 4 as the intro to today’s lesson. “Yesterday, we learned about equations with fractional coefficients and Problem 4 is one of those type of problems.” Since the previous lesson follows the one on adding and subtracting algebraic fractions, students may be in the habit of keeping the common denominator.
“I notice some of you when solving this, kept the common denominator.”
“Then what did we say we should do to eliminate the denominator?” Students recall that both sides are multiplied by 15.
“You can do it this way, but it’s building in an extra step. You can just multiply each term by 15 to begin with, which is how we started doing this long ago when we first learned about equations with fractional coefficients. Today we’ll learn about solving equations with fractions where the denominator has variables in the denominator.” I write on the board.
I have them write down the steps as we work the problem.
Step 1: Find the LCD. In this case it is
Step 2: Multiply each term of the equation by the LCD and cancel denominators.
Step 3: Solve the equation.
Step 4: Check for excluded values.
The original equation is used to identify excluded values which are Since the solution to the equation is -4, it stands. (Some students may ask what would happen if -4 were an excluded value. I tell them that then there would be no solution—and we will see examples of this in the next lesson.)
Examples. These are part of the homework assignment so we kill two birds at once by getting them going on the homework via making the first few examples. I work through the first one with them and others as needed.
“Let’s cancel and see what we get.”
“Now solve it.” They will get x = 4. Excluded values are 2 and -2.
Homework. Problems consist of fractional equations, including some fractional coefficient types from the previous lessons. The problems should be relatively straightforward and also include two or three addition/subtraction of fractions that are not equations. The reason is that students occasionally get combining fractions confused with equations and try to eliminate the denominator.
Day 2
Any homework problems that students found difficult should be gone over so that students understand how they are solved.
Solving by Cross Multiplication. I start out with a problem that was in their homework preferably, or something similar to it.
“In solving this, you multiplied by the LCD which is…” They tell me it’s 2x. “Go ahead and solve it.” They do so and obtain x = 10.
“Now when you did that, you multiplied both sides by 2x, and ended up with something like this.”
“Fairly simple. This problem, in fact looks a lot like the proportion problems you did in seventh grade. And when you did those problems you did something called cross multiplying. Which you were still doing, and didn’t realize it. Because watch:”
“For a lot of fractional equation problems you can just cross multiply. Let’s look at another one.”
“I want this half of the room to multiply both sides by the LCD which is…” They tell me it’s 12n. “And this half to cross multiply.”
I do this, monitoring their notebooks and pick two students representative of each side to write the solved problem on the board. Side by side, we have the multiply by LCD version and the cross multiply version:
“Which one was easier, or are they both about the same?”
Generally, they say it’s about the same, though they note that the cross multiplication method resulted in larger numbers. “OK, so sometimes it might be easier, but you now have a choice. Let’s look at this one:
“We could multiply all terms by 8x. Or we could combine the fractions on the left hand side. So do that; add 3/4x + 1/x and keep the right hand side the same. Show me what you get.” The result is :
“Now you can solve it by cross-multiplication.”
They do so and get x = 2. Some students will say they solved it by inspection, seeing that 4 times 2 is 8. There’s general agreement that this method is a bit easier than multiplying each term by 8x.
“I just want to show you things you can do that may make things easier. There are times, though when multiplying by the LCD is probably the easiest. To wit and for example,” I say writing the following on the board:
“Does this look like the type of problem that would lend itself to cross multiplication?”
There’s general agreement that it doesn’t.
“What’s the LCD? Discuss it and put it on your mini-whiteboard.” Some students will get confused and not see that 3t is a multiple of t. The general consensus will be:
“Now you could multiply each term by this chain of factors. Or you could do it an easier way. Which would you prefer?”
They will pick the easier way, although a few will say the longer way just to get attention. The Lord of the Flies mode will prevail and I proceed with the short-cut.
“Looking at the first term, the denominator is 3t – 2. If we multiplied by 3t(3t-2), we would cancel the (3t – 2) and end up multiplying 4 by 3t. We can actually skip the cancel part. We know that the denominator is missing 3t to form the LCD, so we know we can just multiply 4 by 3t and be done with it. Similarly, what would we multiply the 7 by in the second term?”
They discuss this a bit, and decide it will be 3t – 2. “And what about the third term?” The answer comes back 3(3t – 2). “What we have then is this.”
“Multiply it out, combine terms and solve. Do it in your notebooks.” I walk around and check:
I usually get the response “Can we do it the long way on something like this?” to which I respond “Of course. I’m just showing you various things you can do. Let’s do some easier ones.”
Examples. Again, the examples are taken from the homework. Students can elect to use whatever method they wish and I assure them that I won’t require them to do a short-cut on a test or quiz if they do not wish to do so.
Homework. Problems should include mostly those that can easily be solved by cross multiplication but should include some that are more complex, including those that have one of the solutions excluded.
“I want this half of the room to multiply both sides by the LCD which is…” They tell me it’s 12n. “And this half to cross multiply.”
I do this, monitoring their notebooks and pick two students representative of each side to write the solved problem on the board. Side by side, we have the multiply by LCD version and the cross multiply version: