Question

Approximate each number (a) rounded and (b) truncated to three decimal places. $$ \frac{3}{7} $$

Short answer

\(\frac{3}{7} \approx 0.429\) when rounded and \(\frac{3}{7} \approx 0.428\) when truncated to three decimal places.

Step-by-step solution

- Calculate the Decimal Equivalent

First, convert the fraction \(\frac{3}{7}\) into a decimal. Perform the division 3 ÷ 7.

- Determine the Decimal Form

Using long division or a calculator, you find that \(\frac{3}{7} \approx 0.428571428571\ldots\). This is the repeating decimal form of the fraction.

- Round to Three Decimal Places

To round to three decimal places, look at the fourth decimal place. For \(0.428571428571\ldots\), the fourth decimal place is 5. Since it is 5 or greater, round the third decimal place up from 8 to 9. Thus, \(\frac{3}{7} \approx 0.429\) rounded to three decimal places.

- Truncate to Three Decimal Places

To truncate to three decimal places, simply cut off the decimal after the third place without rounding. So, \(\frac{3}{7}\ \approx 0.428\) truncated to three decimal places.

Key concepts

Rounding Decimals

Rounding decimals involves determining which digit to adjust at a specific decimal place based on the digits that follow. When rounding to three decimal places, you look at the fourth decimal digit:

For example, let's consider the fraction \(\frac{3}{7}\). When converted to a decimal, it is approximately 0.428571428571...

To round to three decimal places, focus on the three decimal digits first:

• 0.428
  
  Now, look at the fourth decimal place, which is 5 in this instance. In rounding rules:
  • If the digit is 5 or greater, increase the last considered digit by one.
  • If it is less than 5, keep it the same.
  
  
  Applying this rule to \(\frac{3}{7}\) gives:
  • 0.428 with the fourth digit 5 ⇒ Round up ⇒ 0.429
  
  
  This is the rounded form of \(\frac{3}{7}\) to three decimal places.

Truncating Decimals

Truncating decimals is straightforward. Unlike rounding, you do not adjust the digits; you just cut off extra digits beyond a certain place.

Consider the fraction \(\frac{3}{7}\) converted to its decimal form, which extends infinitely: 0.428571428571...

To truncate it to three decimal places:

• Simply keep only the digits up to the third place.
• Ignore any digits that follow.

Thus, for \(\frac{3}{7}\), truncating to three decimal places results in:

0.428

Notice there's no rounding involved.

Long Division

Long division is a step-by-step method for dividing numbers to find exact quotients or decimal representations. It helps to convert fractions into decimals without a calculator.

Performing long division for \(\frac{3}{7}\) yields:

• 3 divided by 7 gives 0 remainder 3 (first step).
  
• Bring down 3 to get 30.
• 30 divided by 7 gives 4 remainder 2.
  
• Bring down 2 to get 20.
  
• 20 divided by 7 gives 2 remainder 6.
  
• This process continues cyclically.

Through long division, you pinpoint that \(\frac{3}{7}\) is approximately 0.428571428571..., a repeating decimal.

This efficiency shows why long division is valuable for precise calculations and understanding repeating patterns in decimals.