Lesson 3: Problem Solving

New Concept

This lesson focuses on translating word problems into solvable algebraic equations.

What’s next

Next, we'll break down how to apply formulas in interactive examples. You'll then get to practice writing equations with a set of problems.

Property

Many problems can be solved by using familiar formulas. To solve an equation, we undo the operations performed on the variable.When using a formula like distance equals rate times time ($d=rt$), substitute the known values for the variables and then solve for the unknown variable.

Examples

• To find the interest earned on a 2000 dollars deposit at a 4% annual rate for one year, use $I = Prt$. Substituting the values gives $I = (2000)(0.04)(1)$, so the interest is 80 dollars.

• A company has costs of 5000 dollars and wants to make a profit of 2500 dollars. To find the needed revenue, use $P = R - C$. The equation becomes $2500 = R - 5000$. Solving for $R$ shows they need a revenue of 7500 dollars.

Property

To solve some problems, we can write an equation by translating mathematical words into algebraic symbols. Look for keywords to help you: "sum" indicates addition, "product" indicates multiplication, and "is" means equals. Practice writing algebraic equations to develop the skills needed for more complex problems.

Examples

• "The product of a number and 9 is 108." We can let $n$ be the number. The equation is $9n = 108$. Solving for $n$, we find the number is 12.

• "A number decreased by 15 is 40." Let the number be $x$. The equation is $x - 15 = 40$. Adding 15 to both sides shows the number is 55.

Property

Steps for Modeling a Problem:

1. Identify the unknown quantity and choose a variable to represent it.
2. Find some quantity that can be expressed in two different ways, and write an equation.
3. Solve the equation, and answer the question in the problem.

Examples

• A recipe's ratio of flour to sugar is 5 to 2. If you use 10 cups of flour, how much sugar do you need? Let $s$ be the number of cups of sugar. The ratio can be written as $\frac{10}{s}$ and as $\frac{5}{2}$. The equation is $\frac{10}{s} = \frac{5}{2}$, so $s=4$ cups of sugar.

• A student and their backpack weigh 145 pounds together. If the student weighs 120 pounds, how much does the backpack weigh? Let $b$ be the backpack's weight. The total weight is both $120+b$ and 145. So, $120+b = 145$, which means the backpack weighs 25 pounds.

Property

1. It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
2. Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

• To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$. Defining variables just as "Sam" and "Tim" would be unclear.

• If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$. The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.