๐ Rational Number Problems: Single-Step vs. Multi-Step
Rational numbers are simply numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This includes fractions, decimals that terminate or repeat, and integers. Let's break down single-step and multi-step problems involving these numbers.
๐งฎ Single-Step Rational Number Problems
These problems require only one operation (addition, subtraction, multiplication, or division) to find the solution. They are straightforward and test your basic understanding of operations with rational numbers.
- โ Addition: Sarah has $\frac{1}{4}$ of a pizza, and John gives her $\frac{1}{8}$ of his pizza. How much pizza does Sarah have in total?
- โ Subtraction: A recipe calls for 2.5 cups of flour, but you only have 1.75 cups. How much more flour do you need?
- โ๏ธ Multiplication: A store is having a sale where everything is 0.75 of the original price. If a book originally costs $12, what is the sale price?
- โ Division: You have $\frac{3}{4}$ of a cake, and you want to divide it equally among 3 friends. How much cake does each friend get?
โ Multi-Step Rational Number Problems
These problems require two or more operations to find the solution. You need to carefully analyze the problem and break it down into smaller, manageable steps.
- โโ Addition and Subtraction: Tom earns $15.50 mowing lawns and spends $7.25 on a movie ticket and $3.50 on snacks. How much money does Tom have left?
- โ๏ธโ Multiplication and Division: A company makes 250 boxes of cookies. If each box contains 12 cookies and they want to divide the cookies equally among 5 stores, how many cookies does each store receive?
- โโ๏ธ Addition and Multiplication: You buy 3 shirts that cost $9.75 each and a pair of pants for $25.50. What is the total cost?
- โโ Subtraction and Division: A rope that is 15.75 meters long is cut into two pieces. One piece is 6.25 meters long. The other piece is then cut into 5 equal sections. How long is each of the 5 sections?
๐ Comparison Table
| Feature | Single-Step Problems | Multi-Step Problems |
|---|
| Number of Operations | One | Two or more |
| Complexity | Less complex | More complex |
| Analysis Required | Minimal analysis | Requires careful analysis and breaking down the problem |
| Solution Strategy | Direct application of a single operation | Involves multiple operations in a specific order |
| Example | $\frac{1}{2} + \frac{1}{4} = ?$ | $(\frac{1}{2} + \frac{1}{4}) \times 2 = ?$ |
๐ก Key Takeaways
- โ
Identify the Operations: Carefully read the problem to identify all the mathematical operations needed.
- โ๏ธ Break It Down: For multi-step problems, break the problem into smaller steps. Solve each step individually.
- ๐ข Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when solving multi-step problems.
- โ๏ธ Check Your Work: Always double-check your calculations to ensure accuracy.
- โ Rational Numbers: Be comfortable working with fractions, decimals, and integers.