Problem 78 - Algebra 1: Concepts and Skills

Question

Write the fraction as a decimal rounded to the nearest thousandth. $$ \frac{32}{155} $$

Step-by-Step Solution

Answer

The fraction \( \frac{32}{155} \) as a decimal rounded to the nearest thousandth is approximately 0.206.

Step 1: Divide Numerator by Denominator

First, divide the numerator by the denominator using a calculator or long division. That is, compute \( \frac{32}{155} \).

Step 2: Decimal Approximation

The answer will be a decimal number. Write down this decimal as it is.

Step 3: Rounding Decimal

Lastly, round the resulting decimal to the nearest thousandth place.

Key Concepts

Decimal Approximation

Understanding decimal approximation is essential for converting fractions like \( \frac{32}{155} \) into decimals. Imagine breaking down something into 155 equal parts, and you have 32 of those parts. To find out what portion of the 'whole' you have in a way that's easy to understand and use, you turn this fraction into a decimal number. To do so, you divide the numerator (32) by the denominator (155) using either long division or a calculator.

When you perform this division, you will likely get a number with many digits after the decimal point. This is the decimal approximation of the fraction. It could go on forever if it's an irrational number or stop after a few digits if it's rational. With \( \frac{32}{155} \), you'll get a decimal that is a precise representation of the fraction.

Rounding Decimals

Once you have the decimal approximation, you might need to round it to make it easier to work with. Rounding decimals involves adjusting the number to a specific place value — in the textbook exercise, you're asked to round to the nearest thousandth. So, how do you do this? Look at the digit in the fourth place after the decimal. If this digit is five or higher, you'll increase the third decimal place by one; if it's four or less, you keep the third decimal place as is.

This process simplifies the number without losing much precision. For \( \frac{32}{155} \), when you round the decimal approximation, you transform a long, unwieldy number into a shorter version that’s still accurate enough for most practical purposes.

Division of Fractions

Division of fractions might feel tricky at first, but it's actually a straightforward process. Every time you're converting a fraction into a decimal, you're doing a division: the numerator is divided by the denominator. This is what’s happening with \( \frac{32}{155} \). The division sign (÷) turns into the line that separates the top number (numerator) from the bottom number (denominator) of a fraction.

To perform the division, you can either simplify the fraction first, if possible, or go straight into dividing the two numbers to find the decimal form. Remember, when dividing by hand (using long division), keep an eye out for repeating decimals, where a pattern of digits keeps occurring – these can often be rounded for convenience.