Problem 112
Question
Write the fraction as a terminating or repeating decimal. $$ \frac{3}{5} $$
Step-by-Step Solution
Verified Answer
The fraction \(\frac{3}{5}\) as a decimal is 0.6, which is a terminating decimal.
1Step 1: Simplify the Fraction
The fraction \(\frac{3}{5}\) is already in its simplest form, so there's no need to further simplify it.
2Step 2: Perform the Division
Divide the numerator (3) by the denominator (5). 3 ÷ 5 equals 0.6.
3Step 3: Identify the Decimal Type
The decimal 0.6 is a terminating decimal because it ends and doesn't continue or repeat any digit.
Key Concepts
Converting Fractions to DecimalsSimplifying FractionsDecimal Classification
Converting Fractions to Decimals
When you come across a fraction, such as \(\frac{3}{5}\), and you're asked to convert it into a decimal, you're engaging in one of the most fundamental aspects of arithmetic. Converting fractions to decimals is essentially about division. The top number (numerator) is being divided by the bottom number (denominator).
Here's how you would tackle the problem: divide 3 (the numerator) by 5 (the denominator). When you perform this division, you may do it on a calculator, or if you're doing it by hand, you would write it out like a long division problem. In the case of \(\frac{3}{5}\), you would end up with 0.6, which leads us to conclude that \(\frac{3}{5}\) in decimal form is 0.6.
Understanding this conversion process is vital because it allows you to move fluidly between fractions and decimals, which is necessary for complex calculations in algebra, geometry, and beyond.
Here's how you would tackle the problem: divide 3 (the numerator) by 5 (the denominator). When you perform this division, you may do it on a calculator, or if you're doing it by hand, you would write it out like a long division problem. In the case of \(\frac{3}{5}\), you would end up with 0.6, which leads us to conclude that \(\frac{3}{5}\) in decimal form is 0.6.
Understanding this conversion process is vital because it allows you to move fluidly between fractions and decimals, which is necessary for complex calculations in algebra, geometry, and beyond.
Simplifying Fractions
The process of simplifying fractions is a key skill in mathematics. It involves reducing the fraction to its simplest form where the numerator and denominator are as small as possible and can't be divided any further by a common number (except for 1).
For the fraction \(\frac{3}{5}\), you'll notice it's already simplified. There are no common factors between 3 and 5 other than 1, so it's in its simplest form. Simplification of fractions is essential for comparing sizes of different fractions and for performing arithmetic operations such as addition, subtraction, multiplication, and division of fractions.
To simplify a fraction, you find the greatest common divisor (GCD) for the numerator and denominator, then divide both by that number. Simplified fractions are also easier to convert into decimals, as the division becomes more straightforward when numbers are smaller and simpler.
For the fraction \(\frac{3}{5}\), you'll notice it's already simplified. There are no common factors between 3 and 5 other than 1, so it's in its simplest form. Simplification of fractions is essential for comparing sizes of different fractions and for performing arithmetic operations such as addition, subtraction, multiplication, and division of fractions.
To simplify a fraction, you find the greatest common divisor (GCD) for the numerator and denominator, then divide both by that number. Simplified fractions are also easier to convert into decimals, as the division becomes more straightforward when numbers are smaller and simpler.
Decimal Classification
Decimals can be classified into two main types: terminating and repeating. Terminating decimals are those that come to an end, such as 0.5, 0.75, or 0.6, which we observed when converting \(\frac{3}{5}\) to a decimal. These decimals stop after a certain number of digits after the decimal point.
On the other hand, repeating decimals do not terminate. Instead, they have a sequence of digits that keep repeating infinitely. A common example is \(\frac{1}{3}\), which equals 0.333... The dots indicate that the 3's continue indefinitely.
Recognizing whether a decimal is terminating or repeating is important for understanding the nature of the number you're working with and has implications in real-world situations, such as when measuring or in finance. For example, when \(\frac{3}{5}\) equals 0.6, we can see it's a terminating decimal because there are no digits after the 6 that continue infinitely.
On the other hand, repeating decimals do not terminate. Instead, they have a sequence of digits that keep repeating infinitely. A common example is \(\frac{1}{3}\), which equals 0.333... The dots indicate that the 3's continue indefinitely.
Recognizing whether a decimal is terminating or repeating is important for understanding the nature of the number you're working with and has implications in real-world situations, such as when measuring or in finance. For example, when \(\frac{3}{5}\) equals 0.6, we can see it's a terminating decimal because there are no digits after the 6 that continue infinitely.
Other exercises in this chapter
Problem 111
Write the fraction as a terminating or repeating decimal. $$ \frac{8}{9} $$
View solution Problem 111
Divide. Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{2}{3} \div 63 $$
View solution Problem 112
Divide. Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{5}{6} \div \frac{1}{5} $$
View solution Problem 113
Write the fraction as a terminating or repeating decimal. $$ \frac{9}{10} $$
View solution