# Understanding Decimal and Fractional Numbers: A Guide with Practice Problems
In mathematics, numbers can be represented in various formats, two common forms being decimal and fractional. Decimals and fractions are essential for expressing values that are not whole numbers, and they each have unique characteristics as well as applications in different areas of maths and life.
## Decimal Numbers
Decimal numbers are a part of the base-ten system, which is also known as the decimal system. A decimal number consists of a whole number part and a fractional part, separated by a decimal point (in the US, a dot is used as the decimal point). The name “decimal” comes from the Latin word ‘decimus’, meaning ‘tenth’, reflecting the fact that each position to the right of the decimal point represents a power of one-tenth.
For example, the decimal 23.45 has a whole number part of 23 and a fractional part of 45 hundredths. This is equivalent to \( 23 + \frac{45}{100} \) in fractional form.
## Fractional Numbers
Fractions represent a portion or ratio of a whole. A fraction consists of two numbers – a numerator, which is the top number indicating how many parts are taken, and a denominator, which is the bottom number indicating the total number of equal parts the whole is divided into.
For example, the fraction \( \frac{3}{4} \) indicates 3 parts out of a total of 4 equal parts. In decimal form, \( \frac{3}{4} \) is equivalent to 0.75.
## Converting Between Decimals and Fractions
To convert a fraction to a decimal, divide the numerator by the denominator. Conversely, to convert a decimal to a fraction, start by expressing the decimal part as a fraction of a power of ten (depending on the number of decimal places) and then simplify if necessary.
Let’s dive into some practical problems to better understand decimals and fractions.
### Problems and Solutions
#### Decimal Problems
1. Write the decimal 0.75 as a fraction.
**Solution:** As a fraction of a power of ten, 0.75 is \( \frac{75}{100} \). Simplify to \( \frac{3}{4} \).
2. Add the decimals 1.2 and 3.45.
**Solution:** 1.2 + 3.45 = 4.65
3. Subtract 0.89 from 2.
**Solution:** 2 – 0.89 = 1.11
4. Multiply 0.5 by 3.6.
**Solution:** 0.5 × 3.6 = 1.8
5. Divide 4.8 by 1.2.
**Solution:** 4.8 ÷ 1.2 = 4
#### Fraction Problems
6. Simplify the fraction \( \frac{8}{20} \).
**Solution:** \( \frac{8}{20} = \frac{2}{5} \) (by dividing both numerator and denominator by 4).
7. Add the fractions \( \frac{1}{4} \) and \( \frac{1}{3} \).
**Solution:** To add, find a common denominator: \( \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \).
8. Subtract \( \frac{1}{5} \) from \( \frac{3}{4} \).
**Solution:** Convert to a common denominator: \( \frac{15}{20} – \frac{4}{20} = \frac{11}{20} \).
9. Multiply \( \frac{2}{3} \) by \( \frac{1}{2} \).
**Solution:** \( \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} \), which simplifies to \( \frac{1}{3} \).
10. Divide \( \frac{3}{5} \) by \( \frac{2}{3} \).
**Solution:** \( \frac{3}{5} ÷ \frac{2}{3} = \frac{3}{5} \times \frac{3}{2} = \frac{9}{10} \).
#### Conversion Problems
11. Convert 0.125 to a fraction.
**Solution:** \( 0.125 = \frac{125}{1000} = \frac{1}{8} \) after simplification.
12. Convert \( \frac{5}{8} \) to a decimal.
**Solution:** Divide 5 by 8 to get 0.625.
13. Express 2.08 as a fraction.
**Solution:** \( 2.08 = 2 \frac{8}{100} = 2 \frac{2}{25} \).
14. Convert \( \frac{9}{10} \) to a decimal.
**Solution:** Divide 9 by 10 to get 0.9.
15. Convert 3.33 (with 3 repeating) to a fraction.
**Solution:** Let \( x = 3.333… \). Multiplying by 10, we get \( 10x = 33.333… \). Subtracting the original equation from this, \( 9x = 30 \) and therefore \( x = \frac{30}{9} = \frac{10}{3} \).
#### Mixed Problems
16. Add 2.5 to \( \frac{3}{4} \).
**Solution:** First, convert \( \frac{3}{4} \) to a decimal which is 0.75. Then, 2.5 + 0.75 = 3.25.
17. Subtract \( \frac{5}{6} \) from 1.5.
**Solution:** Convert \( \frac{5}{6} \) to a decimal which is approximately 0.833. Then, 1.5 – 0.833 ≈ 0.667.
18. Multiply 0.25 by \( \frac{6}{5} \).
**Solution:** Convert 0.25 to \( \frac{1}{4} \), then \( \frac{1}{4} \times \frac{6}{5} = \frac{6}{20} \), which simplifies to \( \frac{3}{10} \).
19. Divide \( \frac{2}{7} \) by 0.5.
**Solution:** Convert 0.5 to \( \frac{1}{2} \), then \( \frac{2}{7} ÷ \frac{1}{2} = \frac{2}{7} \times \frac{2}{1} = \frac{4}{7} \).
20. 1.75 added to \( \frac{2}{3} \).
**Solution:** Convert \( \frac{2}{3} \) to a decimal which is approximately 0.667, then 1.75 + 0.667 ≈ 2.417.
Through these problems and solutions, you can see how an understanding of both decimals and fractions is crucial in handling various mathematical tasks. Remember, practice is key to mastering these concepts and honing your number sense.
