Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ \frac{5}{9} $$

Short answer

(a) 0.556 (b) 0.555

Step-by-step solution

Convert Fraction to Decimal

First, convert the fraction \(\frac{5}{9}\) into a decimal by performing the division. \(\frac{5}{9} \approx 0.555555...\).

Round to Three Decimal Places

To round \(0.555555...\) to three decimal places, look at the fourth decimal place. Since it is a 5, which is greater than or equal to 5, round up the third decimal place: \(0.556\).

Truncate to Three Decimal Places

To truncate \(0.555555...\) to three decimal places, simply cut off all digits after the third decimal place, without rounding: \(0.555\).

Key concepts

Fraction to Decimal Conversion

When we convert a fraction to a decimal, we basically perform a division of the numerator by the denominator. For example, to convert the fraction \(\frac{5}{9}\) into a decimal, we divide 5 by 9, resulting in \(\frac{5}{9} \approx 0.555555...\). This decimal is a repeating decimal, which means that the pattern '55555...' continues indefinitely. Converting fractions to decimals is helpful in making the number easier to work with, especially when needing precision in calculations. This process is often the first step in various mathematical problems, allowing us to then apply other operations like rounding or truncating.

Rounding Numbers

Rounding numbers helps to simplify them, by reducing the number of digits but keeping the value close to what it was. To round a number to three decimal places, you need to look at the fourth decimal place: \(0.555555...\). For example:

• If the fourth decimal is 5 or more, you round the third decimal up.
• In this case, because the fourth decimal is 5, you round up the third decimal place, changing it from 5 to 6.
• So, \(0.555555...\) rounded to three decimal places becomes \(0.556\).

This process makes it easier to work with numbers by providing a value that is 'close enough' to the original, for practical purposes like calculations or reporting.

Truncating Numbers

Truncating numbers involves cutting off the digits beyond a certain point without rounding. It is a method used when you want to keep the digits exactly as they are up to a certain place. To truncate \(0.555555...\) to three decimal places:

• You simply leave the first three digits after the decimal point as they are
• Cut off any digits that come after: \(0.555555...\) truncated to three decimal places is \(0.555\).

This method is often used in computing and programming because it is faster and requires less processing power than rounding, providing an exact value up to a specified point.