Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ 99.05249 $$

Short answer

Rounded: 99.052, Truncated: 99.052

Step-by-step solution

- Identify the Decimal Places

First, count the number of decimal places. The number 99.05249 has five decimal places: 0, 5, 2, 4, 9.

- Rounding to Three Decimal Places

To round to three decimal places, look at the fourth decimal place (4 in this case). Since 4 is less than 5, we round down, meaning the third decimal place (2) remains the same.

- Write the Rounded Number

After rounding, the number becomes 99.052 because the 4 does not increase the 2 in the third decimal place.

- Truncating to Three Decimal Places

To truncate to three decimal places, simply remove all digits after the third decimal place. So, the fourth and fifth decimal places (4 and 9) are dropped.

- Write the Truncated Number

After truncating, the number becomes 99.052 since we drop the digits beyond the third decimal place.

Key concepts

decimal places

Decimal places refer to the digits that appear after the decimal point in a number. For example, in the number 99.05249, the digits 0, 5, 2, 4, and 9 are the decimal places. Knowing the number of decimal places is critical when rounding or truncating a number.
In our problem, we needed to work with three decimal places. This means focusing only on the first three digits after the decimal point.

rounding

Rounding is the process of decreasing or increasing a number to a certain degree of precision. To round a number to three decimal places, we must look at the fourth decimal place to decide whether to round up or down.
Here is how to round the number 99.05249:

• Identify the fourth decimal place (which is 4).
• If the fourth decimal place is 5 or greater, round up. If it’s less than 5, round down.

In our case, since the fourth decimal place is 4 (which is less than 5), we round down, and the number remains 99.052.

truncation

Truncation is simpler than rounding. It involves cutting off the number after a certain decimal place without rounding. For instance, truncating 99.05249 to three decimal places means we stop at the third decimal place and ignore any digits that come after it:

• Identify the third decimal place.
• Remove any digits that come after the third decimal place.

So, truncating 99.05249 gives us 99.052. Unlike rounding, the digits following the cutoff do not influence the digits we keep.

mathematical approximation

Mathematical approximation is useful when an exact number isn't required and a simpler, close value is sufficient. Rounding and truncation are both techniques of approximation. They help in providing a clearer, simpler representation of numbers, especially in contexts like reports, where too many decimal places can be confusing.
Approximating to three decimal places means ensuring that the represented number is easy to use and understand, while still reasonably close to the original value. Both rounding and truncation serve this purpose in different ways.