Problem 81 - Algebra 1: Concepts and Skills

Question

Write the fraction as a decimal rounded to the nearest thousandth. $$ -\frac{63}{89} $$

Step-by-Step Solution

Answer

The fraction \(-\frac{63}{89}\) as a decimal rounded to the nearest thousandth is -0.708.

Step 1: Division

Divide the numerator by the denominator. So, perform the division: -63 ÷ 89.

Step 2: Conversion to Decimal

The result will be a decimal number. Write it down.

Step 3: Rounding

Since we want the division to be rounded to the nearest thousandth place, look at the digit in the fourth place after the decimal point. If it's 5 or more, round up the third digit. If it's less than 5, leave the third digit as it is. Write down your final answer.

Key Concepts

Decimal Rounding

Understanding how to round decimals is a fundamental skill in mathematics and it's often required in various fields to simplify numbers. Rounding to the nearest thousandth means looking at the fourth digit after the decimal point. If this digit is 5 or higher, you increase the third digit by one. If it's less than 5, the third digit stays the same. This concept is essential when you want to approximate values while retaining a certain level of precision.

For example, if you have the decimal 0.12345, and you need to round to the nearest thousandth, you would determine the rounding based on the fourth digit (which is '5' in this case). Since '5' is the midpoint in our number system, it becomes the threshold for rounding up. Thus, 0.12345 would be rounded to 0.123.

Division of Fractions

When converting a fraction to a decimal, division is the key operation that will take you there. You divide the numerator (the top number) by the denominator (the bottom number). This division can be done using long division or a calculator. For instance, with the fraction \( -\frac{63}{89} \), you divide -63 by 89.

Understanding Division

Think of division as a way to find out how many times the denominator can fit into the numerator. In our case, since the fraction is negative, we know that the resulting decimal will also be negative. This core concept of division is pivotal in numerous mathematical applications, beyond just converting fractions to decimals.

Decimal Conversion

Decimal conversion involves turning fractions into decimals. It is done through division, where the fraction's numerator is divided by its denominator. The result of this division is a decimal that can sometimes be finite or infinite (repeating).

Finite vs. Infinite Decimals

Some fractions, when divided, result in a finite number of digits after the decimal point (non-repeating), while others have an infinite number of digits that repeat in a pattern. It is crucial to understand whether the decimal will terminate or repeat to properly round the number or to use it in further calculations successfully.

Fraction Arithmetic

Fraction arithmetic encompasses addition, subtraction, multiplication, and division of fractions. It is important to understand that fractions represent a part of a whole and arithmetic operations on fractions follow specific rules.

Basic Rules

• Addition/Subtraction: To add or subtract fractions, they must have the same denominator. If they don't, find a common denominator, then add or subtract the numerators.
• Multiplication: Multiply the numerators to get the new numerator and the denominators to get the new denominator.
• Division: Multiply by the reciprocal of the fraction that is the divisor.

This arithmetic is not just limited to numbers. It also applies to algebra, where you work with variables as parts of the fractions.