What Are the Steps to Combine 2D Shapes into New Figures?
The Worksheet on Division of Rational Numbers is a practical and essential resource for students learning about rational numbers and their operations. Mastering the division of rational numbers is important for excelling in school mathematics, competitive exams, and for solving real-life mathematical problems. Practicing this concept helps strengthen problem-solving and logical reasoning skills in students preparing for exams like JEE and school assessments.
Understanding Division of Rational Numbers
A rational number is any number that can be expressed as a fraction in the form \( \frac{a}{b} \), where both a and b are integers and \( b \neq 0 \). The division of rational numbers involves dividing one rational number by another, which is achieved by multiplying the first number by the reciprocal of the second. This operation forms the basis for many topics in algebra and number theory, making it a core skill in mathematics.
How to Divide Rational Numbers: The Formula
If you have two rational numbers \( \frac{a}{b} \) and \( \frac{c}{d} \) (where \( c \neq 0 \) and \( d \neq 0 \)), then:
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)
Here, we multiply the first rational number by the reciprocal (or multiplicative inverse) of the second.
Example: Divide \( \frac{2}{3} \) by \( \frac{4}{5} \):
\( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)
Step-by-Step Worked Examples
Let’s look at a few examples to understand division of rational numbers better:
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Divide \( \frac{9}{7} \) by \( \frac{3}{14} \):
- Step 1: Write the reciprocal of the second number: \( \frac{14}{3} \)
- Step 2: Multiply: \( \frac{9}{7} \times \frac{14}{3} = \frac{9 \times 14}{7 \times 3} = \frac{126}{21} = 6 \)
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Divide \( -\frac{5}{6} \) by \( \frac{2}{3} \):
- Step 1: Reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \)
- Step 2: Multiply: \( -\frac{5}{6} \times \frac{3}{2} = -\frac{15}{12} = -\frac{5}{4} \)
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Divide \( \frac{7}{9} \) by \( -\frac{14}{27} \):
- Step 1: Reciprocal of \( -\frac{14}{27} \) is \( -\frac{27}{14} \)
- Step 2: Multiply: \( \frac{7}{9} \times -\frac{27}{14} = -\frac{189}{126} = -\frac{3}{2} \)
Practice Problems: Try for Yourself
- Find \( \frac{8}{15} \div \frac{4}{5} \)
- Calculate \( -\frac{3}{8} \div \frac{9}{16} \)
- Divide \( \frac{5}{12} \) by \( -\frac{10}{9} \)
- Solve \( \frac{7}{2} \div \frac{21}{4} \)
- Find the value of \( -\frac{11}{13} \div -\frac{22}{39} \)
- Calculate \( \frac{2}{5} \div \frac{3}{10} \)
- Divide \( \frac{1}{4} \) by \( \frac{5}{8} \)
- Find \( \frac{-16}{5} \div \frac{4}{25} \)
- Calculate \( \frac{3}{7} \div \frac{6}{21} \)
- Solve \( \frac{0}{2} \div \frac{5}{7} \) (Hint: What is zero divided by a non-zero rational number?)
Common Mistakes to Avoid
- Not taking the reciprocal of the divisor before multiplying.
- Forgetting to simplify the final answer to lowest terms.
- Dividing numerators and denominators directly (e.g., dividing 2/3 by 4/5 as \( \frac{2\div4}{3\div5} \)), which is incorrect.
- Misplacing negative signs in rational number division.
- Attempting to divide by zero, which is undefined.
Real-World Applications
The division of rational numbers is used in many real-world settings. For example, cooking measurements, financial calculations, or dividing resources equally require rational number division. Understanding how to divide rational numbers accurately helps students apply mathematics in areas like science experiments, accounting, and even in technology where precise ratios are involved. At Vedantu, we simplify such core concepts so students can excel not only in exams but also in daily life challenges.
For students wanting to deepen their understanding, check out Vedantu’s Rational Numbers page, which covers all operations with rational numbers, and explore related concepts on Fraction and Decimals.
In summary, being comfortable with the division of rational numbers sets a strong foundation for higher-level mathematics, competitive exams, and daily reasoning. Consistent practice with worksheets, like those available at Vedantu, ensures students avoid common errors and develop lasting confidence with rational number operations.
FAQs on How to Combine and Take Apart 2D Shapes
1. What is meant by combining 2D shapes?
Combining 2D shapes involves joining two or more flat figures, such as triangles, rectangles, and circles, to create a new, more complex shape, often called a composite shape. This is a key concept in geometry, helping build spatial reasoning skills.
2. What are some examples of taking apart shapes?
Taking apart, or decomposing, 2D shapes means breaking down a larger shape into smaller, simpler components. For example, you can divide a rectangle into two triangles or split a hexagon into six triangles. This skill is crucial for understanding area and perimeter calculations.
3. Why do we need to learn about combining and partitioning shapes?
Understanding how to combine and partition 2D shapes is fundamental for developing strong spatial reasoning skills. It's essential for solving problems related to area, perimeter, and more advanced geometry concepts. It also improves problem-solving abilities in various real-world situations.
4. Can I find worksheets for combining and taking apart 2D shapes?
Yes, many resources, including educational websites and textbooks, offer worksheets and PDFs for practicing combining and decomposing 2D shapes. These activities help reinforce understanding and build practical skills.
5. Which 2D shapes are most often combined to make new shapes?
Triangles, rectangles, squares, and parallelograms are frequently used to create composite 2D shapes. Understanding their properties is key to successful shape manipulation and problem-solving.
6. Can shapes be combined and separated?
Yes, absolutely! 2D shapes can be combined to form new, more complex shapes (composition), and complex shapes can be broken down into simpler ones (decomposition). This is a core concept in geometry.
7. What are 2D shapes that fit together?
Many 2D shapes can fit together to form larger shapes or tessellations. For example, squares, equilateral triangles, and hexagons are shapes that tessellate—meaning they can cover a surface completely without gaps or overlaps. This concept is related to plane geometry and spatial reasoning.
8. What are different shapes combined together?
Various 2D shapes can be combined, such as triangles and squares to make a house shape, or rectangles and circles to make a design. The possibilities are endless, depending on the shapes involved and how they are arranged.
9. What is a combination shape?
A combination shape, also known as a composite shape, is a shape formed by joining two or more basic geometric shapes. Understanding how to identify and analyze these shapes is essential for solving geometry problems.
10. How can combining shapes help in real-life problem-solving (e.g., design, architecture)?
Combining and decomposing shapes is crucial in various real-world applications. In design and architecture, understanding how shapes fit together is essential for creating functional and aesthetically pleasing structures. For example, architects use this knowledge to design buildings and interior spaces.
11. Is there more than one way to partition the same shape into different combinations of simpler shapes?
Yes, often there are multiple ways to decompose a single 2D shape into smaller, simpler shapes. This highlights the flexibility and adaptability of geometric concepts and the importance of creative problem-solving.
12. How do combining and decomposing shapes relate to fractions and measurement?
Combining and decomposing 2D shapes is closely related to fractions and measurement. For instance, partitioning a shape into equal parts helps visualize fractions, and understanding the area and perimeter of composite shapes requires measurement skills.
