To simplify the expression "4 divided by 4/5," we need to follow the order of operations (PEMDAS/BODMAS), which dictates that division should be performed after any operations within parentheses or any exponentiation, but in this case, we are dealing with a fraction and division.
Understanding the Expression
The expression can be written as 4 ÷ (4⁄5). To simplify this, we can convert the division by a fraction into multiplication by its reciprocal. The reciprocal of 4⁄5 is 5⁄4.
Converting Division to Multiplication
So, 4 ÷ (4⁄5) becomes 4 * (5⁄4). This simplifies the expression by allowing us to multiply instead of divide by a fraction.
| Operation | Expression | Result |
|---|---|---|
| Multiplication | 4 * (5/4) | 20/4 = 5 |
This approach helps in simplifying complex expressions by leveraging the properties of fractions and the relationship between division and multiplication.
Key Points
- To simplify "4 divided by 4/5," we convert the division into multiplication by the reciprocal of the divisor.
- The expression becomes 4 * (5/4), which simplifies to 20/4.
- The final simplified result of the expression is 5.
- Using the reciprocal to convert division by a fraction into multiplication is a fundamental technique in algebra and arithmetic.
- This method can be applied to any expression involving division by a fraction, making it a powerful tool for simplification.
Applying the Concept to Similar Problems
Understanding how to simplify expressions like “4 divided by 4⁄5” can help in tackling more complex problems that involve similar operations. By recognizing the pattern and applying the rule of converting division by a fraction into multiplication by its reciprocal, one can simplify a wide range of expressions efficiently.
For instance, if we encounter an expression like "8 divided by 3/4," we can apply the same principle: convert the division into multiplication by the reciprocal of the fraction. Thus, "8 divided by 3/4" becomes "8 * (4/3)," which simplifies to 32/3.
Real-World Applications
While expressions like these might seem abstract, they have real-world applications in various fields, including science, engineering, and economics, where calculations involving fractions and division are common. Mastering the simplification of such expressions can enhance problem-solving skills and improve accuracy in calculations.
What is the first step in simplifying an expression like "4 divided by 4/5"?
+The first step is to convert the division by a fraction into multiplication by its reciprocal. In this case, the expression becomes 4 * (5/4).
How does converting division to multiplication by the reciprocal simplify the expression?
+It simplifies the expression by allowing us to perform a straightforward multiplication operation instead of dealing with the complexities of dividing by a fraction.
What is the result of simplifying the expression "4 divided by 4/5"?
+The result of simplifying "4 divided by 4/5" is 5.
In conclusion, simplifying expressions involving division by fractions can be efficiently managed by converting the division into multiplication by the reciprocal of the fraction. This technique not only simplifies the calculation process but also enhances understanding and proficiency in handling arithmetic and algebraic expressions.
