Simplify The Following Expression. 3 11/5 Go In To 3/-9/5
Introduction
Understanding how to simplify complex algebraic expressions is an essential skill for students and professionals alike. Whether you're working on a math homework, preparing for an exam, or tackling real-world problems involving fractions and mixed numbers, mastering the process of simplifying expressions is crucial.Today, we will explore the specific problem: Simplify The Following Expression. 3 11/5 Go In To 3/-9/5. This problem involves converting a mixed number into an improper fraction and then performing division between two fractions. By carefully analyzing each step, you will learn how to simplify similar expressions efficiently and accurately.
Understanding the Components of the Expression
Before diving into the simplification process, let's clarify what the expression entails:- Mixed Number: 3 11/5
- Fraction to Divide By: 3/-9/5
Step 1: Converting the Mixed Number to an Improper Fraction
What is a Mixed Number?
A mixed number combines a whole number with a fraction, such as 3 11/5. To work with fractions algebraically, it's often easier to convert mixed numbers into improper fractions.Converting 3 11/5 to an Improper Fraction
Here's a step-by-step guide:- Multiply the whole number by the denominator of the fractional part:
- 3 × 5 = 15
- Add the numerator of the fractional part:
- 15 + 11 = 26
- Place the sum over the original denominator:
- 26/5
This conversion allows us to work with a single fraction rather than a mixed number, simplifying subsequent calculations.
Step 2: Understanding the Division Expression
The division operation is expressed as:
\[
\frac{26}{5} \div \frac{3}{- \frac{9}{5}}
\]
Note: The second fraction appears to be 3 divided by -9/5, but based on standard notation, it might be better interpreted as:
\[
\frac{26}{5} \div \frac{-9}{5}
\]
or
\[
\frac{26}{5} \div \left( \frac{3}{- \frac{9}{5}} \right)
\]
However, given the problem statement, it seems the intended expression is:
\[
\frac{26}{5} \div \frac{-9}{5}
\]
To clarify, the expression "3/-9/5" is interpreted as:
- 3 divided by (-9/5)
which simplifies to:
\[
3 \div \left( -\frac{9}{5} \right)
\]
Let's analyze both possibilities.
Scenario 1: The expression is:
\[
\text{(mixed number)} \div \text{(fraction)} = \frac{26}{5} \div \frac{-9}{5}
\]
Scenario 2: The expression involves dividing 3 by -9/5, then working with 11/5, but since the problem explicitly states "3 11/5," it's most consistent to interpret the full expression as:
\[
\left( 3 + \frac{11}{5} \right) \div \left( \frac{3}{ - \frac{9}{5} } \right)
\]
which simplifies to:
\[
\frac{26}{5} \div \left( 3 \div \left( -\frac{9}{5} \right) \right)
\]
Given the ambiguity, let's focus on the most plausible interpretation: the expression is to convert "3 11/5" into an improper fraction and divide it by "3/-9/5", which is 3 divided by -9/5.
Final assumption:
The expression is:
\[
\frac{26}{5} \div \left( \frac{3}{ - \frac{9}{5} } \right)
\]
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Note: For clarity, let's proceed with this interpretation.
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Step 3: Simplify the Divisor: 3 divided by -9/5
To simplify:
\[
3 \div \left( -\frac{9}{5} \right)
\]
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal:
\[
3 \times \left( -\frac{5}{9} \right) = -\frac{15}{9}
\]
Simplify:
\[
-\frac{15}{9} = -\frac{5}{3}
\]
Result:
The divisor simplifies to \(-\frac{5}{3}\).
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Summary so far:
- The numerator (mixed number 3 11/5) converted to improper fraction: \(\frac{26}{5}\)
- The divisor (3 divided by -9/5) simplifies to: \(-\frac{5}{3}\)
Now, the original division simplifies to:
\[
\frac{26}{5} \div \left( -\frac{5}{3} \right)
\]
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Step 4: Dividing Fractions
Dividing two fractions involves multiplying the first fraction by the reciprocal of the second:
\[
\frac{26}{5} \div \left( -\frac{5}{3} \right) = \frac{26}{5} \times \left( -\frac{3}{5} \right)
\]
Perform the multiplication:
\[
\frac{26 \times (-3)}{5 \times 5} = \frac{-78}{25}
\]
Final simplified result:
\[
\boxed{\frac{-78}{25}}
\]
This is an improper fraction, but it can be expressed as a mixed number or decimal if needed.
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Optional: Converting to a Mixed Number or Decimal
- Mixed Number:
78 ÷ 25 = 3 with a remainder of 3
Since the numerator is negative:
\[
-78/25 = - (3 + \frac{3}{25}) = -3 \frac{3}{25}
\]
- Decimal:
Divide 78 by 25:
78 ÷ 25 = 3.12
Apply the negative sign:
\(-3.12\)
---
Summary of the Simplification Process
- Converted the mixed number 3 11/5 into an improper fraction: \(\frac{26}{5}\)
- Interpreted and simplified the divisor:
- Calculated \(3 \div -\frac{9}{5}\) to get \(-\frac{5}{3}\)
- \(\frac{26}{5} \div -\frac{5}{3} = \frac{26}{5} \times -\frac{3}{5} = -\frac{78}{25}\)
- Expressed the answer as a mixed number: \(-3 \frac{3}{25}\)
Additional Tips for Simplifying Complex Fractions and Mixed Numbers
- Always convert mixed numbers to improper fractions for easier calculations.
- When dividing fractions, remember to multiply by the reciprocal.
- Simplify fractions at each step to keep numbers manageable.
- Be mindful of negative signs; they can appear in numerator, denominator, or both.
- Converting improper fractions back to mixed numbers can make the answer more understandable.
Common Mistakes to Avoid
- Misinterpreting the expression: Clarify whether the division involves the mixed number or just parts of it.
- Forgetting to convert mixed numbers: Always convert to improper fractions before operations.
- Incorrect reciprocal usage: Remember that dividing by a fraction is multiplying by its reciprocal.
- Ignoring signs: Negative signs can change the entire value; handle them carefully.
Conclusion
Simplifying the expression "3 11/5 go in to 3/-9/5" involves multiple steps, including converting mixed numbers to improper fractions, simplifying division of fractions, and handling negative signs properly. By following a systematic approach:
- Convert mixed numbers to improper fractions.
- Simplify division expressions by multiplying by reciprocals.
- Simplify fractions at each step to reduce errors.
- Express the final answer in the most understandable form, whether as an improper fraction, mixed number, or decimal.
Mastering these steps enhances your mathematical skills and prepares you for more complex algebraic and fractional problems.
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