Working with Fractions Class 7 Notes
hardWorking with Fractions Class 7 Notes
Working with Fractions Class 7 Notes
Fractions
Numerator Fractions greater than 1 Mixed Fractions Fractions less than 1 Denominator Unit Fraction
Reciprocal
Reciprocal of fraction [latex]\frac {a}{b}[/latex] is [latex]\frac {b}{a}[/latex].
Multiplication and Division of Fractions
Product of fractions [latex]\frac {a}{b}[/latex] × [latex]\frac {c}{d}[/latex] = [latex]=\frac{a \times c}{b \times d}=[/latex] = [latex]\frac {ac}{bd}[/latex]
Product of two fractions is less than each fraction, if both the fractions are less than 1
Product of two fractions is greater than each fraction, if both the fractions are greater than 1
Product of two fractions is greater than one fraction and less than the other fraction, if one fraction is less than 1 and the other fraction is greater than 1
Quotient of fractions
[latex]\frac {a}{b}[/latex] ÷ [latex]\frac {c}{d}[/latex] = [latex]\frac {a}{b}[/latex] × Reciprocal of [latex]\frac {c}{d}[/latex]
= [latex]\frac {a}{b}[/latex] × [latex]\frac {d}{c}[/latex] = [latex]\frac {ad}{bc}[/latex]
If a fraction is divided by a fraction less than 1, then the quotient will be greater than the dividend
If a fraction is divided by a fraction greater than 1, then the quotient will be less than the dividend
CHAPTER AT A GLANCE
Important Terms and Their Meanings
| Term | Meaning |
| Fraction | A number that represents a part of a whole, expressed as [latex]\frac {a}{b}[/latex], where a is the numerator and b is the denominator. |
| Numerator | The top part of a fraction that indicates how many parts of the whole are being considered. |
| Denominator | The bottom part of a fraction that indicates the total number of equal parts the whole is divided into. |
| Unit Fraction | A fraction where the numerator is 1, such as [latex]\frac {1}{3}[/latex], representing one part of a whole divided into equal parts. |
| Non-Unit Fraction | A fraction where the numerator is other than 1, such as [latex]\frac {3}{4}[/latex], indicating multiple parts of the whole. |
| Mixed Number/ Mixed Fraction | A number that combines a whole number and a proper fraction, such as 1[latex]\frac {1}{4}[/latex]. |
| Equivalent Fractions | Different fractions that represent the same value, such as [latex]\frac {1}{2}[/latex] and [latex]\frac {2}{4}[/latex]. |
| Simplifying Fractions | The process of reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common factor (GCF). |
| Reciprocal | The inverse of a fraction, obtained by swapping the numerator and denominator. For example, the receprocal of [latex]\frac {3}{4}[/latex] is [latex]\frac {4}{3}[/latex]. |
| Arithmetic Operations | Basic mathematical operations that can be performed on fractions, including addition, subtraction, multiplication and division. |
Multiplication of Fractions
Understanding Multiplication of Fractions
- Multiplication of fractions involves multiplying the numerators and denominators separately. [latex]\frac {a}{b}[/latex] × [latex]\frac {c}{d}[/latex] = [latex]\frac{a \times c}{b \times d}[/latex]
- Example : To multiply [latex]\frac {2}{3}[/latex] and [latex]\frac {4}{5}[/latex], compute [latex]\frac{2 \times 4}{3 \times 5}[/latex] = [latex]\frac {8}{15}[/latex]
Multiplying Fractions with Whole Numbers
- To multiply a fraction by a whole number, multiply the numerator by the whole number.
- Example : 5 × [latex]\frac {2}{3}[/latex] = [latex]\frac {10}{3}[/latex]
Multiplying Mixed Numbers
- Convert mixed numbers to improper fractions before multiplying.
- Example: 1[latex]\frac {1}{4}[/latex] × 8 = [latex]\frac {5}{4}[/latex] × 8 = 10.
Mnemonics
- “Top times top, bottom times bottom”: This simple phrase reminds you that you multiply the numerators together and the denominators together.
- “Cross and Cancel” : Before multiplying, look for any common factors across the numerators and denominators. This can simplify your calculations and make the multiplication easier.
Division of Fractions
Steps to Divide Fractions
1. Identify the Dividend and Divisor:
- The dividend is the fraction you want to divide (the first fraction).
- The divisor is the fraction you are dividing by (the second fraction).
For example, in the expression [latex]\frac {a}{b}[/latex] ÷ [latex]\frac {c}{d}[/latex], [latex]\frac {a}{b}[/latex] is the dividend and [latex]\frac {c}{d}[/latex] is the divisor.
Find the Reciprocal of the Divisor:
The reciprocal of a fraction [latex]\frac {c}{d}[/latex] is obtained by flipping the numerator and denominator, resulting in [latex]\frac {d}{c}[/latex].
Multiply the Dividend by the Reciprocal:
Instead of dividing, you multiply the dividend by the reciprocal of the divisor. This can be expressed mathematically as :
[latex]\frac {a}{b}[/latex] ÷ [latex]\frac {c}{d}[/latex] = [latex]\frac {a}{b}[/latex] × [latex]\frac {d}{c}[/latex]
Multiply the Numerators and Denominators :
Now, multiply the numerators together and the denominators together: [latex]\frac{a \times d}{b \times c}[/latex]
Dividing Whole Numbers by Fractions
To divide a whole number by a fraction, multiply by the reciprocal of the fraction.
Example : 6 ÷ [latex]\frac {1}{4}[/latex] = 6 × 4 = 24.
Dividing Fractions by Whole Numbers
To divide a fraction by a whole number, multiply, by the reciprocal of the whole number.
Example : [latex]\frac {1}{4}[/latex] ÷ 5 = [latex]\frac {1}{4}[/latex] × [latex]\frac {1}{5}[/latex] = [latex]\frac {1}{20}[/latex].
Key Points to Remember
Reciprocal: The reciprocal of a fraction [latex]\frac {a}{b}[/latex] is [latex]\frac {b}{a}[/latex]. When you multiply a fraction by its reciprocal, the result is always 1.
Greater or Less than the Dividend : When dividing by a fraction less than 1, the quotient will be greater than the dividend. Conversely, when dividing by a fraction greater than 1, the quotient will be less than the dividend.