Fractions and Decimal Conversions

Fractions: Related Terms, Types, Equivalent Fractions, Reduction

Understanding Fractions

A fraction is a mathematical notation used to represent a part of a whole or a part of a collection. It expresses a quantity that is obtained by dividing one integer by another. Fractions are a fundamental concept in arithmetic and form a critical part of the set of rational numbers ($\mathbb{Q}$), as every rational number can be written as a fraction.

When we use fractions, we are typically considering a situation where a whole unit or a group of units is divided into a certain number of equal parts, and we are looking at a specific number of those parts.

Examples:

• If a pizza is divided into 8 equal slices, and you eat 3 slices, you have eaten $\frac{3}{8}$ of the pizza.
• If there are 10 marbles in a bag, and 4 are blue, then $\frac{4}{10}$ of the marbles are blue.

Related Terms of a Fraction

A fraction is written in the form $\frac{p}{q}$, separated by a horizontal or slanted line called the fraction bar. The two integers $p$ and $q$ have specific names:

• Numerator ($p$): This is the number written above the fraction bar. It indicates how many of the equal parts of the whole (or collection) are being considered or taken. The numerator can be any integer (positive, negative, or zero).
• Denominator ($q$): This is the number written below the fraction bar. It indicates the total number of equal parts into which the whole (or each unit in the collection) is divided. The denominator must be a non-zero integer ($q \neq 0$). Division by zero is mathematically undefined.
• Fraction Bar: The horizontal line ($\frac{p}{q}$) or sometimes a slanted line ($p/q$) that separates the numerator and the denominator. It signifies the operation of division. The fraction $\frac{p}{q}$ can be read as "$p$ divided by $q$" or "$p$ over $q$".

Example: In the fraction $\frac{7}{12}$, $7$ is the numerator (representing 7 parts being considered), and $12$ is the denominator (representing that the whole is divided into 12 equal parts). The fraction $\frac{7}{12}$ represents the value obtained by dividing 7 by 12.

Types of Fractions

Fractions can be classified into different types based on the relationship between their numerator and denominator, and their structure:

1. Proper Fractions:

   In a proper fraction, the absolute value of the numerator is strictly less than the absolute value of the denominator. When dealing with positive fractions, this means the numerator is less than the denominator ($p < q$, assuming $p, q > 0$). Proper fractions represent a value that is greater than $0$ and less than $1$.

   Examples: $\frac{1}{2}, \frac{3}{4}, \frac{7}{10}, \frac{99}{100}$. On the number line, these fractions lie between 0 and 1.

2. Improper Fractions:

   In an improper fraction, the absolute value of the numerator is greater than or equal to the absolute value of the denominator. When dealing with positive fractions, this means the numerator is greater than or equal to the denominator ($p \ge q$, assuming $p, q > 0$). Improper fractions represent a value that is greater than or equal to $1$.

   Examples: $\frac{5}{2}$ ($=2.5$), $\frac{4}{4}$ ($=1$), $\frac{10}{3}$ ($\approx 3.33$), $\frac{17}{5}$ ($=3.4$). On the number line, these fractions lie at or to the right of 1 (for positive improper fractions).

3. Mixed Numbers (or Mixed Fractions):

   A mixed number is a combination of a whole number (an integer) and a proper fraction. It is written in the form $a\frac{b}{c}$, and it represents the sum of the whole number and the fraction, i.e., $a + \frac{b}{c}$. Mixed numbers are often used as an alternative way to express improper fractions.

   Examples: $1\frac{1}{2}, 3\frac{3}{4}, 5\frac{1}{10}$.

   Conversion between Mixed Numbers and Improper Fractions:

   We can convert between these two forms:

   a) Converting from Mixed Number to Improper Fraction: To convert a mixed number $a\frac{b}{c}$ to an improper fraction, multiply the whole number part ($a$) by the denominator of the fractional part ($c$), add the numerator of the fractional part ($b$) to this product. This sum becomes the numerator of the improper fraction, and the denominator remains the same ($c$).

   Formula: $a\frac{b}{c} = \frac{(a \times c) + b}{c}$

   [Mixed to Improper]

   Example 1. Convert $4\frac{3}{5}$ to an improper fraction.

   Answer:

   Here, the whole number $a=4$, the numerator $b=3$, and the denominator $c=5$.

   Using the formula:

   $\quad 4\frac{3}{5} = \frac{(4 \times 5) + 3}{5}$

   Perform the multiplication and addition in the numerator:

   $\quad = \frac{20 + 3}{5} = \frac{23}{5}$

   So, $4\frac{3}{5}$ is equivalent to the improper fraction $\mathbf{\frac{23}{5}}$.

   b) Converting from Improper Fraction to Mixed Number: To convert an improper fraction $\frac{p}{q}$ to a mixed number, divide the numerator ($p$) by the denominator ($q$). The quotient of this division becomes the whole number part of the mixed number. The remainder of the division becomes the numerator of the fractional part, and the original denominator ($q$) remains the denominator.

   If $p \div q = \text{Quotient } Q \text{ with Remainder } R$, then $\frac{p}{q} = Q \frac{R}{q}$.

   [Improper to Mixed]

   Example 2. Convert $\frac{17}{3}$ to a mixed number.

   Answer:

   Divide the numerator (17) by the denominator (3).

   $\quad 17 \div 3$

   $$ \begin{array}{r} 5 \leftarrow \text{Quotient } (Q) \\ 3{\overline{\smash{\big)}\,17}} \\ \underline{-~\phantom{(}15} \\ 2 \leftarrow \text{Remainder } (R) \end{array} $$

   The quotient is $5$, and the remainder is $2$. The original denominator is $3$.

   So, the whole number part is $5$, the numerator of the fractional part is $2$, and the denominator is $3$.

   $\quad \frac{17}{3} = 5\frac{2}{3}$

4. Unit Fractions:

   A unit fraction is a fraction where the numerator is 1 ($p=1$). These fractions represent a single part of a whole that has been divided into $q$ equal parts.

   Examples: $\frac{1}{2}, \frac{1}{4}, \frac{1}{10}, \frac{1}{100}$.

5. Like Fractions:

   A group of two or more fractions are called like fractions if they all have the same denominator. Working with like fractions (comparing, adding, subtracting) is simpler than working with unlike fractions.

   Examples: $\frac{1}{7}, \frac{3}{7}, \frac{5}{7}$ are like fractions because they all have a denominator of 7.

6. Unlike Fractions:

   A group of two or more fractions are called unlike fractions if they have different denominators. Operations like addition and subtraction require converting them to like fractions first.

   Examples: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$ are unlike fractions because their denominators (2, 3, and 4) are different.

Equivalent Fractions

Equivalent fractions are different fractions that represent the exact same value or the same proportion of a whole. They look different but occupy the same point on the number line.

You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero integer. This is because multiplying or dividing by $\frac{k}{k}$ (where $k \neq 0$) is equivalent to multiplying by 1, which does not change the value of the fraction.

Examples:

• To find equivalent fractions of $\frac{1}{3}$:

  Multiply numerator and denominator by 2: $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.

  Multiply numerator and denominator by 3: $\frac{1 \times 3}{3 \times 3} = \frac{3}{9}$.

  Multiply numerator and denominator by 10: $\frac{1 \times 10}{3 \times 10} = \frac{10}{30}$.

  So, $\frac{1}{3}, \frac{2}{6}, \frac{3}{9}, \frac{10}{30}$ are all equivalent fractions representing the same value.

• Checking for Equivalence: To check if two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, the simplest method is cross-multiplication (assuming $b \neq 0, d \neq 0$). The fractions are equivalent if and only if the cross-products are equal: $a \times d = b \times c$.

  Example 3. Are $\frac{4}{6}$ and $\frac{10}{15}$ equivalent fractions?

  Answer:

  Use cross-multiplication. Compare the product of the numerator of the first fraction and the denominator of the second ($4 \times 15$) with the product of the denominator of the first fraction and the numerator of the second ($6 \times 10$).

  Product 1: $4 \times 15 = 60$

  Product 2: $6 \times 10 = 60$

  Since the cross-products are equal ($60 = 60$), the fractions are equivalent.

  $\mathbf{\frac{4}{6} = \frac{10}{15}}$

Reduction of Fractions to Simplest Form (Lowest Terms)

A fraction is in its simplest form or lowest terms when its numerator and denominator have no common positive factors other than $1$. In other words, the Greatest Common Divisor (GCD) of the numerator and the denominator is $1$.

Reducing a fraction to its simplest form makes it easier to understand its value and perform calculations. It's the standard way to represent a fractional value unless a specific denominator is required.

To reduce a fraction $\frac{p}{q}$ to its simplest form:

1. Find the Greatest Common Divisor (GCD) of the numerator ($p$) and the denominator ($q$).
2. Divide both the numerator and the denominator by their GCD.

   Simplest form of $\frac{p}{q} = \frac{p \div \text{GCD}(p, q)}{q \div \text{GCD}(p, q)}$

   [Reduction Formula]

Answer:

Understanding these basic concepts and terminology related to fractions is crucial before performing operations with them.

Decimals: Introduction, Types (Like/Unlike)

Introduction to Decimal Numbers

A decimal number is a number that contains a decimal point, representing both a whole number part and a fractional part. It is essentially a way of writing fractions where the denominator is a power of $10$. This system extends the place value concept used for whole numbers to represent values between integers.

A decimal number is typically written with the whole number part to the left of the decimal point and the fractional part to the right of the decimal point. The decimal point (.) serves as a separator.

The place values to the left of the decimal point are powers of $10$ (Ones $10^0$, Tens $10^1$, Hundreds $10^2$, etc.). The place values to the right of the decimal point are negative powers of $10$ (Tenths $10^{-1}$, Hundredths $10^{-2}$, Thousandths $10^{-3}$, Ten-thousandths $10^{-4}$, and so on).

Example: The decimal number $45.678$ can be interpreted and written in expanded form based on its place values:

Using powers of 10:

An important property of decimals is that adding trailing zeros to the right of the last non-zero decimal digit does not change the value of the number. For example, $0.5 = 0.50 = 0.500$. This is because $0.5 = \frac{5}{10}$, $0.50 = \frac{50}{100} = \frac{5}{10}$, and $0.500 = \frac{500}{1000} = \frac{5}{10}$. This property is very useful when comparing decimals or performing addition and subtraction.

Types of Decimal Expansions

The decimal form of a number can be classified into different types based on the behaviour of the digits after the decimal point. This classification is directly related to whether the number is rational or irrational.

1. Terminating Decimals:

   These are decimal numbers that end after a finite number of digits to the right of the decimal point. This occurs when the division process of converting a fraction to a decimal results in a remainder of zero after a certain number of steps.

   Examples: $0.75$ (ends after 5), $1.2$ (ends after 2), $0.125$ (ends after 5), $5.0$ (ends after 0), $0.00$.

   Terminating decimals are always rational numbers. They can easily be written as a fraction with a denominator that is a power of $10$. For example, $0.75 = \frac{75}{100} = \frac{3}{4}$, $1.2 = \frac{12}{10} = \frac{6}{5}$, $0.125 = \frac{125}{1000} = \frac{1}{8}$. A rational number $\frac{p}{q}$ written in its simplest form has a terminating decimal expansion if and only if the prime factors of the denominator $q$ consist only of $2$ and/or $5$.

2. Non-terminating Decimals:

   These are decimal numbers that have an infinite number of digits to the right of the decimal point. The division process does not end. Non-terminating decimals are further divided into two categories:

   • Non-terminating Repeating (or Recurring) Decimals: These decimals have an infinite number of digits after the decimal point, and a specific sequence of one or more digits (a block) repeats indefinitely. The repeating block is often indicated by a bar ($\overline{\phantom{aa}}$) over the digits that repeat or by dots above the first and last digit of the repeating block (though the bar notation is more common).

     Examples: $0.333... = 0.\overline{3}$, $1.272727... = 1.\overline{27}$, $0.1666... = 0.1\overline{6}$, $3.141414... = 3.\overline{14}$.

     Non-terminating repeating decimals are always rational numbers. They can be converted back into the $\frac{p}{q}$ form (as discussed in Section I5 of the first chapter).

   • Non-terminating Non-repeating Decimals: These decimals have an infinite number of digits after the decimal point, and there is no repeating pattern or block of digits. The digits appear randomly without any cycle.

     Examples: The decimal expansions of irrational numbers like $\sqrt{2} \approx 1.41421356237...$, $\pi \approx 3.14159265358...$, $e \approx 2.71828182845...$. Also constructed examples like $0.123456789101112...$ (listing consecutive integers after the decimal point). These are always irrational numbers.

In summary, the decimal expansion of a number precisely determines its type:
- Terminating or Non-terminating Repeating Decimal $\iff$ Rational Number
- Non-terminating Non-repeating Decimal $\iff$ Irrational Number

Like and Unlike Decimals

When comparing decimal numbers or performing addition and subtraction, it's often helpful to think about the number of decimal places. This leads to the classification of like and unlike decimals.

• Like Decimals: Decimal numbers that have the same number of digits after the decimal point.

  Examples: $5.23$ and $1.78$ are like decimals (both have 2 decimal places).

  $10.125$, $0.007$, and $45.678$ are like decimals (all have 3 decimal places).

• Unlike Decimals: Decimal numbers that have a different number of digits after the decimal point.

  Examples: $3.1$ (1 decimal place) and $4.56$ (2 decimal places) are unlike decimals.

  $0.75$ (2 decimal places) and $1.2345$ (4 decimal places) are unlike decimals.

  Unlike decimals can always be converted into like decimals by adding trailing zeros to the right of the last decimal digit of the number(s) with fewer decimal places. This process is called padding with zeros and does not change the value of the decimal number.

  Example: To compare $3.1$ and $4.56$, we can write $3.1$ as $3.10$. Now $3.10$ and $4.56$ are like decimals with 2 decimal places. Their comparison is straightforward ($3.10 < 4.56$).

  Example: To subtract $0.75$ from $5.8$, we can write $5.8$ as $5.80$. Now $5.80$ and $0.75$ are like decimals, making column subtraction easier.

The concept of like and unlike decimals primarily serves as a tool for simplifying manual comparison and vertical alignment in addition and subtraction.

Conversion Between Fractions and Decimals

Fractions and decimals are two common ways to represent parts of a whole or values between integers. For rational numbers, it is always possible to convert between these two forms. Understanding these conversion processes is essential for working with numbers in different contexts.

Converting Fractions to Decimals

To convert a fraction $\frac{p}{q}$ (where $p$ is the numerator and $q$ is the denominator, $q \neq 0$) into its decimal form, perform the division of the numerator by the denominator. This division can be done using long division.

The result of this division will be either a terminating decimal or a non-terminating repeating decimal, depending on the prime factorization of the denominator $q$ (when the fraction is in simplest form).

Answer:

Answer:

Converting Decimals to Fractions

The process for converting a decimal number back to its fractional form $\frac{p}{q}$ depends on whether the decimal is terminating or repeating.

1. Converting Terminating Decimals to Fractions

A terminating decimal can be directly written as a fraction with a denominator that is a power of $10$.

Steps for Converting Terminating Decimals:

1. Write the decimal number without the decimal point. This becomes the numerator of the initial fraction.
2. Determine the place value of the last digit in the decimal. The denominator will be the corresponding power of $10$ (e.g., 10 for tenths, 100 for hundredths, 1000 for thousandths). This is equivalent to writing $1$ followed by as many zeros as there are digits after the decimal point.
3. Write the fraction with the numerator from Step 1 and the denominator from Step 2.
4. Simplify the resulting fraction to its lowest terms by dividing the numerator and the denominator by their Greatest Common Divisor (GCD).

Answer:

Answer:

2. Converting Non-terminating Repeating Decimals to Fractions

Non-terminating repeating decimals are rational numbers, and they can always be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. We use an algebraic method involving multiplication by powers of 10 to convert them.

Method:

1. Let the given decimal number be equal to a variable, say $x$. Write out the decimal expansion showing the repeating part.
2. If there is a non-repeating part after the decimal point, multiply $x$ by a power of 10 ($10^k$) to move the decimal point just before the repeating block. $k$ is the number of digits in the non-repeating part. Let this equation be (1). (If there is no non-repeating part, start from step 3 with $x$, and equation (1) is $x = \text{the decimal}$).
3. Multiply the equation from step 2 by a power of 10 ($10^m$) to move one full repeating block to the left of the decimal point. $m$ is the number of digits in the repeating block. Let this equation be (2).
4. Subtract equation (1) from equation (2). The repeating decimal parts will cancel out, leaving a simple linear equation in $x$.
5. Solve the resulting equation for $x$, expressing it as a fraction.
6. Simplify the resulting fraction to its lowest terms.

Answer:

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These conversion methods highlight the close relationship between fractions and decimal representations, particularly for rational numbers. Terminating and repeating decimals are just different ways of writing fractions, while non-terminating non-repeating decimals are the mark of irrational numbers.