Problem 26 - Prealgebra

Question

Perform each of the following divisions. $$6 . 6 \longdiv { 0 . 1 9 8 }$$

Step-by-Step Solution

Answer

198 divided by 6600 simplifies to \(\frac{3}{100}\), or 0.03.

Step 1: Convert Decimal Problem to Whole Numbers

To simplify the division, first convert the dividend to a whole number. Move the decimal point in 0.198 three places to the right to form the whole number 198. This means you must also move the decimal in the divisor (6.6) three places to the right. Thus, 6.6 becomes 6600.

Step 2: Set Up the Division

After converting, set up the problem as 198 divided by 6600. However, since 198 is less than 6600, it's clear that 198 divides into 6600 zero times.

Step 3: Perform the Division

Since 198 is less than 6600, the division of 198 by 6600 is a fraction less than 1. When dividing a smaller number by a larger one, the quotient is less than 1. In this case, you can compute the decimal or fraction result more easily.

Step 4: Rewrite the Division Problem

Since 198 divided by 6600 is less than one whole, express the division as a fraction: \(\frac{198}{6600}\). Simplify this fraction by dividing both the numerator and denominator by the greatest common divisor, which is 6.

Step 5: Simplify the Fraction

Simplifying \(\frac{198}{6600}\), divide both by 6: \(\frac{198 \div 6}{6600 \div 6} = \frac{33}{1100}\). This fraction simplifies further by dividing by 11: \(\frac{33 \div 11}{1100 \div 11} = \frac{3}{100}\).

Step 6: Convert to Decimal if Needed

The fraction \(\frac{3}{100}\) can be converted to a decimal by dividing 3 by 100, resulting in 0.03. This is helpful if you need to understand the division as a decimal.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It helps in making fractions easier to understand and compare. To simplify a fraction, you need to divide the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and denominator exactly without leaving a remainder.

Let's look at an example with the fraction \( \frac{198}{6600} \). To find the GCD, list the factors of both 198 and 6600. The numbers that divide both terms are potential common divisors. Here, the GCD is 6.

To simplify, divide both the numerator and the denominator by 6:

• \( 198 \div 6 = 33 \)
• \( 6600 \div 6 = 1100 \)

This gives us the simplified fraction \( \frac{33}{1100} \).

The process doesn't stop there. You can simplify \( \frac{33}{1100} \) even further by dividing by their next GCD, which is 11:

• \( 33 \div 11 = 3 \)
• \( 1100 \div 11 = 100 \)

Finally, you get \( \frac{3}{100} \), a fully simplified form. Simplifying fractions like this makes them easier to read and use in further calculations.

Converting Decimals to Fractions

Converting decimals to fractions is a handy tool in mathematics. It allows you to express and work with numbers more flexibly. To convert a decimal to a fraction, consider the place value of the last digit in the decimal.

For example, consider the decimal 0.198. This decimal can be read as "198 thousandths" because the last digit 8 is in the thousandths place. Therefore, you can write it as

• \( \frac{198}{1000} \)

To convert this into a simpler fraction, you can then simplify \( \frac{198}{1000} \) by finding the greatest common divisor (GCD) of 198 and 1000, which is 2. Divide both the numerator and the denominator by 2:

• \( 198 \div 2 = 99 \)
• \( 1000 \div 2 = 500 \)

So, \( \frac{198}{1000} \) simplifies to \( \frac{99}{500} \). Sometimes, you may need to simplify further until you can't anymore. The resulting fraction is then equivalent to the decimal.

By converting decimals to fractions, you can easily perform mathematical operations like addition and subtraction because fractions are often more intuitive to handle in equations.

Comparing Numbers

Comparing numbers is a basic yet crucial aspect of mathematics. It helps in understanding the relative values and sizes of different numbers--whether they are integers, decimals, or fractions. When comparing numbers, consider converting them into the same format, such as both integers or both decimals.

Comparing fractions is straightforward if their denominators are the same. In the case of \( \frac{3}{100} \), you can easily compare it to another fraction with 100 as the denominator. If comparing fractions with different denominators, you may want to find a common denominator to make comparison simpler. Alternatively, convert each fraction to a decimal for an even comparison.

For instance, \( \frac{3}{100} \) translates to 0.03 as a decimal. You can compare 0.03 easily with other decimals. Simply line up the decimal points and compare digit by digit:

• Is 0.03 greater than or less than 0.1? Clearly, 0.03 is less.
• Is 0.03 equal to 0.03? Yes, they are equal.

Comparing numbers can also come in handy with ordering a list of different values from smallest to largest or vice versa. Understanding these comparisons can help in various practical applications such as finances, measurements, and more.