Express each of the following as a rational number: (a) 4.125
The correct answer and explanation is:
Correct Answer:
(a) 4.125=3384.125 = \dfrac{33}{8}
Explanation (300 words):
A rational number is any number that can be expressed as a fraction ab\dfrac{a}{b}, where aa and bb are integers and b≠0b \neq 0. In other words, any decimal that terminates or repeats is a rational number because it can be written as a ratio of two integers.
Now let’s convert the decimal 4.125 to a fraction:
Step 1: Identify the decimal type
4.125 is a terminating decimal because it ends after a few digits. This type of decimal can easily be written as a fraction.
Step 2: Convert to fraction
The decimal part 0.125 has three digits after the decimal point, so we multiply it by 1000 to eliminate the decimal: 4.125=412510004.125 = \frac{4125}{1000}
Step 3: Simplify the fraction
Now simplify 41251000\frac{4125}{1000} by dividing both the numerator and denominator by their greatest common divisor (GCD).
The GCD of 4125 and 1000 is 125. 4125÷1251000÷125=338\frac{4125 \div 125}{1000 \div 125} = \frac{33}{8}
Step 4: Final Answer
4.125=3384.125 = \frac{33}{8}
This confirms that 4.125 is a rational number because it can be written as a simple fraction.
Summary:
- Rational numbers include all decimals that end or repeat.
- 4.125 is a terminating decimal.
- It simplifies to 338\dfrac{33}{8}, confirming it’s rational.
Therefore, 4.125 expressed as a rational number is 338\dfrac{33}{8}.