Understanding Rounding Rules and Finding the Minimal Possible Value

When dealing with decimal numbers, rounding is a common practice to simplify the representation of numbers while ensuring they remain close to their actual values. One instance of rounding is to round a number to the nearest tenth. This involves looking at the second digit after the decimal point (hundredths place) to decide whether to round up or down.

Standard Rounding Rules

In order to round a number to the nearest tenth, we follow the standard rounding rules:

When the hundredth digit is 5 or greater, we round up. When the hundredth digit is 4 or less, we round down. In the case where the hundredth digit is exactly 5, the convention is to round to the nearest even number to avoid bias.

Lets take an example: If a number has three digits after its decimal point and upon rounding it to the nearest tenth, the result is 8.4, we need to determine the minimal possible value of the original number.

Determining the Minimal Possible Value

Given that the rounded value is 8.4, the original number must be between 8.350 and 8.450. However, the exact minimal value that would round to 8.4 when following the rounding rules is 8.350. This is because 8.350 is exactly halfway between 8.3 and 8.4, and following the rounding rule of rounding up when the digit is 5, 8.350 would round up to 8.4.

Let's break down the steps in rounding 8.350:

Identify the hundredth digit (second digit after the decimal): In this case, it is 5. Determine whether to round up or down: Since the hundredth digit is 5, we round up according to the standard rounding rules. Apply the rounding: 8.350 rounds up to 8.4.

Example of Rounding Process

Consider another example where we have a number 8.449. The rounded value to the nearest tenth is 8.4. To apply the rounding rules:

Identify the hundredth digit (second digit after the decimal): In this case, it is 4. Determine whether to round up or down: Since the hundredth digit is 4, we round down according to the standard rounding rules. Apply the rounding: 8.449 rounds down to 8.4.

Conclusion

Understanding and applying the standard rounding rules is crucial for accurate data representation and analysis. By following the rounding rules, we can ensure that our numbers are appropriately simplified without introducing significant errors. For the problem at hand, the minimal possible value of the number with three digits after the decimal point that, when rounded to the nearest tenth, results in 8.4 is 8.350.