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Lesson 108: Solving Word Problems with Algebra (Part 1 of 2)
In this lesson, we'll learn how to solve word problems involving some unknown value. The trick is correctly translating the word problem into symbolic form.
Word problems are always very tricky for students, and rightly so. You're given a little story of sorts, and you need to correctly translate that story into symbols, in some cases determining what parts are completely irrelevant. It just takes lots of practice.
In much earlier lessons, we learned words that mean different operations. That will help us in this lesson. For example, if we see "the sum of a number and 7 is 15", we can translate that into x + 7 = 15. We don't know what the number is, so we'll call it x. We know that "is" means "equals," so we know where our equals sign goes. Then we'll solve the problem as we've learned how.
Often the problem is more tricky, though. Try this one: Seven more than double a certain number is 29. How can we translate this into symbols? The first step is really just to keep cool. Students typically panic when they see word problems, and often either skip them, or do them wrong. Let's just slowly see what we have.
We see "seven more". That means we'll be adding 7 to something, so let's just write "+ 7". Now we see "double a certain number." We don't know what that number is, so we'll call it x. Doubling means to multiply it by 2, so we'll write 2x. Now, don't forget that we're dealing with 7 more than that, so we have to write 2x + 7. Then we see that the result is supposed to be 29, so we can write 2x + 7 = 29, and then solve it as we've learned.
Here is a tricky one: Three times the sum of a number and 5 is 21. Be very careful here, and don't rush into the problem. We can't just write 3x, like you might be tempted to. What is it that we are really multiplying by 3? It's the sum of a number and 5. We can represent that as x + 5. Now we need to multiply that sum by 3. How can we show that? We have to put the sum in parentheses, to show that the sum is what is being multiplied. We then have 3(x+5) = 21. Now we can use the distributive property to eliminate the parentheses, and solve as we've learned.
Let's try this one. How can we represent "five less than one-third of m is 13"? Don't rush and immediately start with 5. That would be wrong. We can certainly start by writing our unknown value, m. What is really the first thing being done to it? We're taking 1/3 of it. That means dividing it by 3. We now have m / 3. Now we want 5 less than the result of that operation. What we have is (m / 3) - 5 = 13. We learned how to solve this. First we'll add 5 to each side of the equation, and then we'll multiply each side by 3.
The more word problems you do, the easier they will become. After solving a word problem, try to make up your own variants of it, and see how sometimes changing or adding one small word can totally change the problem into a completely new one. You'll have more practice in the next lesson.
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