Problem 99 - Algebra 1: Concepts and Skills

Question

Write the decimal as a fraction in simplest form. $$ 0.82 $$

Step-by-Step Solution

Answer

The decimal 0.82 as a fraction in simplest form is \(\frac{41}{50}\).

Step 1: Identifying the Decimal Digits

Here we have a decimal number 0.82. The number after the decimal point is 82, and it's in the hundredths place. Therefore, it can be written as the fraction \(\frac{82}{100}\).

Step 2: Simplifying the Fraction

We look for the greatest number that can divide both 82 and 100. In this case, that's 2, so we divide both numbers by 2. This simplifies the fraction to \(\frac{41}{50}\).

Step 3: Checking if Further Simplification is Possible

In this case, 41 and 50 don't have any common divisor other than 1. So \(\frac{41}{50}\) is the fraction in its simplest form.

Key Concepts

Decimal to Fraction Simplification

Converting a decimal to a fraction isn't as complex as it may seem at first glance. To simplify a decimal to a fraction, you must identify the decimal place value of the last digit. For example, in the decimal 0.82, the '2' is in the hundredths place, because it is two places to the right of the decimal point. Hence, you initially express 0.82 as \( \frac{82}{100} \).

But we must aim for the simplest form to make the fraction easier to work with in mathematical operations. This leads us to the process of simplifying the fraction by finding the largest common divisor, sometimes known as the Greatest Common Divisor (GCD). For 82 and 100, the GCD is 2. So, when both the numerator and denominator are divided by 2, we obtain \( \frac{41}{50} \), which is the fraction in its simplest form.

Identifying Decimal Places

Understanding decimal places is crucial for converting decimals to fractions. The location of a digit after the decimal point indicates its 'place.' So, the first digit after the decimal point is the tenths place, the second is hundredths, the third is thousandths, and so on. This naming pattern helps us determine the denominator of the fraction when converting.

For instance, with our example 0.82, the digit '8' is in the tenths place, and '2' is in the hundredths place. Hence, we write it over 100, because the last digit, '2', determines the place value. If you remember this concept, identifying decimal places becomes a natural step in the process of converting decimals to a fraction.

Simplifying Fractions

The goal of simplifying fractions is to reduce them to their simplest form where the numerator and denominator are as small as possible. This process is essential to make calculations easier and result interpretation clearer. To simplify a fraction, you must look for the highest number that divides evenly into both the numerator and denominator.

Once you find this number, you divide both the top and bottom of the fraction by it. It's important to check if you can simplify further by attempting to find another common divisor greater than 1. If no such number exists, the fraction is considered to be in simplest form, like our \( \frac{41}{50} \) example, which cannot be simplified any further since 41 and 50 share no common divisors other than 1.