Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ \frac{81}{5} $$

Short answer

Rounded: 16.200, Truncated: 16.200.

Step-by-step solution

Calculate the Decimal Equivalent

First, find the decimal equivalent of \(\frac{81}{5}\). Perform the division: \(\frac{81}{5} = 16.2\).

Identify Decimal Places

Observe that the result 16.2 has only one decimal place. For rounding or truncating, additional zeros should be added to show three decimal places. Therefore, write it as 16.200.

Rounding to Three Decimal Places

Since 16.200 already shows three decimal places and no further digits to consider for rounding, the rounded value remains 16.200.

Truncating to Three Decimal Places

For truncating, keep the number up to the third decimal place directly. As there are no additional digits, the truncated value remains 16.200.

Key concepts

Rounding

Rounding is a method used to reduce the digits in a number while keeping its value close to the original. The process depends on the digit immediately following the last desired decimal place. If this digit is 5 or greater, round up. Otherwise, round down.

For example, to round the number 12.3456 to three decimal places, observe the fourth decimal place (number 6). Since it is greater than 5, round up the third decimal place (number 5), resulting in 12.346.

In the exercise, rounding the number 16.200 to three decimal places is straightforward since it already has exactly three decimal places. Thus, the rounded value is 16.200.

Truncating

Truncating, or chopping off, is a simple method to limit the number of digits right after the decimal point. Unlike rounding, truncating does not consider the value of the removed digits but simply cuts off any digits after the desired decimal place.

For instance, truncating 12.3456 to three decimal places involves removing all digits after the third place without changing the remaining digits. Hence, the truncated value is 12.345.

In the given exercise, truncating 16.200 to three decimal places is equally simple. The number is already expressed with three decimal places, so the truncated value remains 16.200.

Decimal Places

Decimal places refer to the number of digits to the right of the decimal point in a number. They are crucial for precision in various mathematical calculations.

When a problem specifies rounding or truncating to a certain number of decimal places, ensure the result has exactly that number of decimal digits.

In the exercise, the given fraction \(\frac{81}{5}\) equals 16.2. The challenge involves displaying this number with three decimal places. To do this, additional zeros can be appended to create 16.200, ensuring clarity for both rounding and truncating functions.

Understanding decimal places facilitates a deeper comprehension of how numbers are handled in different contexts, maintaining precision and accuracy.