This section of our PERT Study Guide covers word problems. The PERT will write some problems as real-world situations and others as words rather than numerical expressions. For these questions, you will have to turn the situation or sentence into an equation and solve it. Here, we walk you through this essential skill and the types of questions you may see on the exam.
Translating Words into Equations
Translating verbal expressions into mathematical equations is a crucial skill in problem-solving. This process involves identifying keywords in the problem that signal specific operations or relationships and using these to construct an accurate equation.
- Addition keywords: Look for words like “total,” “sum,” “increased by,” or “together.” For example, “The total cost of a shirt and pants is \$50” translates to $x + y = 50$.
- Subtraction keywords: Words like “difference,” “decreased by,” or “less than” indicate subtraction. For example, “The difference between a number and 7 is 3” translates to $x − 7 = 3$.
- Multiplication keywords: Keywords such as “product,” “times,” or “of” signify multiplication. For instance, “Twice a number is 10” translates to $2x = 10$.
- Division keywords: Phrases like “divided by,” “per,” or “ratio of” indicate division. For example, “A number divided by 4 is 5” translates to $\frac{x}{4} = 5$.
Example
Translate “Three times the sum of a number and 4 is equal to 18” into an equation.
First, break it into parts:
- “The sum of a number and 4” is $x + 4$
- “Three times” means $3(x + 4)$
- “Is equal to 18” becomes $= 18$
Final equation: $3(x + 4) = 18$
How to Tackle Word Problems
Solving word problems can feel challenging, but breaking them into smaller steps can make the process manageable. The first step is to carefully read the problem and identify what it is asking. Underline or highlight key information, such as quantities, units, and relationships. Then, determine the unknown quantity you need to solve for and assign it a variable, such as $x$.
Next, translate the verbal description into a mathematical equation. Look for keywords that indicate operations (e.g. “total” for addition, “difference” for subtraction). Clearly label your variables and write down the relationships between quantities. For example, if the problem involves finding the total cost of items, you might set up an equation like Total = Price × Quantity.
Finally, solve the equation step by step, checking your work as you go. After finding the solution, revisit the original problem to ensure your answer makes sense in context. If the problem involves multiple steps, consider solving one part at a time before combining results. For instance, in a distance problem, you might first calculate time and then use that to find distance.
Rate, Distance, and Time Problems
Rate, distance, and time problems are common in real-world applications and are solved using the following formula:
$d = rt$, where $d$ is distance, $r$ is rate (or speed), and $t$ is time.
Understanding how to manipulate this formula is key. Consider which variables you have and where you can move the others to isolate them.
To find distance: $d = rt$
To find rate: $r = \dfrac{d}{t}$
To find time: $t = \dfrac{d}{r}$
Example
A car travels at a speed of 60 miles per hour for 2.5 hours. How far does it travel?
$d = rt$
$d = 60 × 2.5 = 150 \text{ miles}$
Key Tip: Always start by carefully reading the problem and underlining or highlighting keywords that indicate operations or relationships. Write down what each variable represents and check if the units match (e.g. time in hours, distance in miles). Taking these steps reduces errors and clarifies your approach.