Decimal 8.125 is equal to the fraction please answer with an explanation
The correct answer and explanation is:
Sure! Let’s convert the decimal 8.125 into a fraction step by step, and then explain the reasoning behind it.
Step 1: Understand the decimal
The number 8.125 is a mixed decimal number — it has a whole number part (8) and a decimal part (0.125).
Step 2: Convert the decimal part to a fraction
Focus on the decimal part 0.125.
- 0.125 means 125 thousandths, or 1251000\frac{125}{1000}.
Step 3: Simplify the fraction 1251000\frac{125}{1000}
- Both numerator (125) and denominator (1000) can be divided by 125.
125÷1251000÷125=18\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}
So, 0.125=180.125 = \frac{1}{8}.
Step 4: Write the mixed number as a fraction
Since 8.125=8+0.1258.125 = 8 + 0.125, this becomes: 8+18=8×88+18=648+18=6588 + \frac{1}{8} = \frac{8 \times 8}{8} + \frac{1}{8} = \frac{64}{8} + \frac{1}{8} = \frac{65}{8}
Final answer:
8.125=6588.125 = \frac{65}{8}
Explanation (about 300 words):
When converting a decimal to a fraction, you want to express the decimal as a ratio of two integers. The number 8.125 has two parts: a whole number (8) and a decimal part (0.125). The decimal part is what we focus on to convert into a fraction.
Decimals like 0.125 represent fractions where the denominator is a power of 10 based on the number of digits after the decimal point. Since 0.125 has three digits after the decimal, it can be written as 1251000\frac{125}{1000}. This is because 0.125 equals 125 thousandths.
The next step is to simplify this fraction by dividing both numerator and denominator by their greatest common divisor (GCD). Here, the GCD of 125 and 1000 is 125. Dividing both by 125, you get 18\frac{1}{8}. This means 0.125 is equivalent to the fraction 18\frac{1}{8}.
Now, returning to the original number, 8.125 can be written as a sum of the whole number and the fractional part: 8+188 + \frac{1}{8}. To combine this into a single fraction, you convert the whole number 8 to a fraction with the same denominator (8): 648\frac{64}{8}. Adding 648+18=658\frac{64}{8} + \frac{1}{8} = \frac{65}{8}.
Thus, the decimal 8.125 equals the improper fraction 658\frac{65}{8}.
This process can be applied to any decimal number by expressing the decimal part as a fraction with a denominator based on powers of 10, simplifying, and then combining with the whole number if needed.