Fractions and Decimals

1. Introduction

Fractions and decimals are two common ways of representing non-integer numbers. Understanding how to convert between fractions and decimals, as well as performing operations with them, is essential for many areas of mathematics. In these notes, we cover the basic definitions, conversion methods, and operations, along with 50 examples (with solutions) to illustrate the concepts.

2. Basic Definitions

Fractions

A fraction represents a part of a whole and is written in the form:

where a is the numerator and b is the denominator.

Decimals

A decimal is another way of expressing fractions, particularly those with denominators that are powers of 10. For example:

represents the fraction 75100, which can be simplified to 34.

3. Converting Between Fractions and Decimals

Fraction to Decimal

To convert a fraction to a decimal, divide the numerator by the denominator.

Decimal to Fraction

To convert a decimal to a fraction, write the decimal over the appropriate power of 10 and then simplify.

4. Operations with Fractions and Decimals

Operations with Fractions

Addition/Subtraction: Find a common denominator, convert fractions, and then add/subtract.

Multiplication: Multiply the numerators and denominators:

Division: Multiply by the reciprocal:

Operations with Decimals

Operations with decimals are similar to whole numbers, but it is important to align the decimal points for addition and subtraction. Multiplication and division follow similar rules to whole numbers, with adjustments for the decimal places.

5. 50 Examples with Solutions

1. Converting Fraction to Decimal Convert 12 to a decimal.
   Solution: 12 = 1 ÷ 2 = 0.5
2. Converting Fraction to Decimal Convert 34 to a decimal.
   Solution: 34 = 3 ÷ 4 = 0.75
3. Converting Fraction to Decimal Convert 14 to a decimal.
   Solution: 14 = 1 ÷ 4 = 0.25
4. Converting Fraction to Decimal Convert 25 to a decimal.
   Solution: 25 = 2 ÷ 5 = 0.4
5. Converting Fraction to Decimal Convert 35 to a decimal.
   Solution: 35 = 3 ÷ 5 = 0.6
6. Converting Fraction to Decimal Convert 18 to a decimal.
   Solution: 18 = 1 ÷ 8 = 0.125
7. Converting Fraction to Decimal Convert 38 to a decimal.
   Solution: 38 = 3 ÷ 8 = 0.375
8. Converting Fraction to Decimal Convert 58 to a decimal.
   Solution: 58 = 5 ÷ 8 = 0.625
9. Converting Fraction to Decimal Convert 78 to a decimal.
   Solution: 78 = 7 ÷ 8 = 0.875
10. Converting Fraction to Decimal Convert 110 to a decimal.
    Solution: 110 = 1 ÷ 10 = 0.1
11. Converting Fraction to Decimal Convert 310 to a decimal.
    Solution: 310 = 3 ÷ 10 = 0.3
12. Converting Fraction to Decimal Convert 710 to a decimal.
    Solution: 710 = 7 ÷ 10 = 0.7
13. Adding Fractions Compute 12 + 14.
    Solution: Find a common denominator: 12 = 24, so 24 + 14 = 34. In decimal, 34 = 0.75.
14. Subtracting Fractions Compute 34 - 12.
    Solution: 34 - 24 = 14 = 0.25.
15. Multiplying Fractions Compute 12 × 23.
    Solution: 1 × 22 × 3 = 26 = 13 ≈ 0.333.
16. Dividing Fractions Compute 34 ÷ 12.
    Solution: Multiply by the reciprocal: 34 × 21 = 64 = 32 = 1.5.
17. Simplifying a Fraction Simplify 812.
    Solution: Divide numerator and denominator by 4: 8÷412÷4 = 23 ≈ 0.667.
18. Converting a Repeating Decimal to Fraction Express 0.3 as a fraction.
    Solution: 0.3 = 13.
19. Converting a Repeating Decimal to Fraction Express 0.6 as a fraction.
    Solution: 0.6 = 23.
20. Adding Fractions with Different Denominators Compute 13 + 14.
    Solution: Common denominator is 12: 412 + 312 = 712 ≈ 0.5833.
21. Multiplying Mixed Numbers Compute 112 × 213.
    Solution: Convert to improper fractions: 32 × 73 = 216 = 3.5.
22. Converting a Decimal to a Fraction Convert 0.125 to a fraction.
    Solution: 0.125 = 1251000 = 18.
23. Multiplying Decimals Compute 0.25 × 0.4.
    Solution: 0.25 × 0.4 = 0.1.
24. Converting a Decimal to a Simplified Fraction Convert 0.625 to a fraction.
    Solution: 0.625 = 6251000 = 58.

6. Summary

In these notes, we reviewed the definitions of fractions and decimals, learned how to convert between the two, and performed various operations such as addition, subtraction, multiplication, and division. The examples provided illustrate a wide range of conversions and operations, ensuring a thorough understanding of fractions and decimals.