Convert 4/7 into its decimal form and round to the nearest thousandth.
The Correct Answer and Explanation is:
To convert the fraction ( \frac{4}{7} ) into its decimal form, we divide the numerator (4) by the denominator (7). This division can be carried out using long division or a calculator.
Step-by-Step Calculation:
- Long Division Setup: We set up the division of 4 by 7. Since 4 is less than 7, we start with 0 and add a decimal point.
- Decimal Placement: We can append a zero to the 4 (making it 40) to continue dividing. So we consider how many times 7 goes into 40.
- Finding the Quotient:
- 7 goes into 40 five times (since ( 7 \times 5 = 35 )).
- We subtract 35 from 40, leaving a remainder of 5.
- Continue Division: Bring down another 0, making it 50.
- 7 goes into 50 seven times (since ( 7 \times 7 = 49 )).
- Subtract 49 from 50, leaving a remainder of 1.
- Bring Down Another 0: Bring down another 0 to get 10.
- 7 goes into 10 once (since ( 7 \times 1 = 7 )).
- Subtract 7 from 10, leaving a remainder of 3.
- Continue Bringing Down Zeros: Bring down another 0 to get 30.
- 7 goes into 30 four times (since ( 7 \times 4 = 28 )).
- Subtract 28 from 30, leaving a remainder of 2.
- Bring Down Another 0: Bring down another 0 to get 20.
- 7 goes into 20 twice (since ( 7 \times 2 = 14 )).
- Subtract 14 from 20, leaving a remainder of 6.
- Continue the Process: Bringing down a 0 to get 60.
- 7 goes into 60 eight times (since ( 7 \times 8 = 56 )).
- Subtract 56 from 60, leaving a remainder of 4, which means we are back where we started, and the decimal will start repeating.
The division continues indefinitely, producing a repeating decimal:
[
\frac{4}{7} \approx 0.571428\ldots
]
Rounding to the Nearest Thousandth:
To round ( 0.571428 ) to the nearest thousandth, we look at the fourth decimal place. The first three decimal places are 571, and the fourth decimal place is 4. Since 4 is less than 5, we do not round up. Therefore, the rounded value is:
[
\frac{4}{7} \approx 0.571
]
Conclusion:
In decimal form, ( \frac{4}{7} ) equals approximately ( 0.571 ) when rounded to the nearest thousandth. This process illustrates not only how to convert a fraction to a decimal but also how to round it appropriately for precision in various mathematical and practical applications.