1. Review of Real Numbers

Simplifying Fractions

1. Review of Real Numbers

Simplifying Fractions: Videos & Practice Problems

Topic summary

Fractions represent parts of a whole, expressed as $\frac{a}{b}$ , where $a$ is the numerator and $b$ the denominator, with $b ≠ 0$ . Equivalent fractions, like one half, two fourths, and three sixths, show the same value by multiplying numerator and denominator by the same constant. Simplifying fractions involves factoring numerator and denominator to cancel the greatest common factor, reducing fractions to lowest terms, essential for understanding polynomial terms and coefficients in algebra.

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Intro to Fractions

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Intro to Fractions Video Summary

Fractions are fundamental in mathematics, representing parts of a whole. A fraction consists of three components: the numerator, the denominator, and the fraction bar. The numerator, located above the fraction bar, indicates how many parts are being considered, while the denominator, below the bar, shows the total number of equal parts the whole is divided into. Essentially, a fraction $ \frac{a}{b} $ expresses the division of $a$ by $b$, linking fractions directly to the concept of division. It is crucial to remember that the denominator $b$ cannot be zero, as division by zero is undefined.

Visualizing fractions often involves dividing shapes, such as circles, into equal parts based on the denominator. For example, the fraction $ \frac{1}{2} $ (one half) is represented by dividing a circle into two equal sections and shading one part. Similarly, $ \frac{2}{4} $ (two fourths) divides the circle into four equal parts with two shaded, and $ \frac{3}{6} $ (three sixths) divides it into six parts with three shaded. These visual models help in understanding the size and value of fractions.

Interestingly, fractions like $ \frac{1}{2} $, $ \frac{2}{4} $, and $ \frac{3}{6} $ are equivalent fractions, meaning they represent the same portion of a whole despite having different numerators and denominators. Equivalent fractions are generated by multiplying both the numerator and denominator of a fraction by the same nonzero constant. For instance, multiplying the numerator and denominator of $ \frac{1}{2} $ by 2 yields $ \frac{2}{4} $, and multiplying both by 3 results in $ \frac{3}{6} $. This property is essential for simplifying fractions and comparing their values.

Understanding fractions as division, their visual representation, and the concept of equivalent fractions lays a strong foundation for further study in algebra and other areas of mathematics. Mastery of these basics enables one to confidently work with fractions in various real-life contexts, from cooking measurements to financial calculations.

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Types of Fractions

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Types of Fractions Video Summary

Fractions can be categorized into three main types: proper fractions, improper fractions, and mixed numbers, each with distinct characteristics and visual representations. Proper fractions are those where the numerator is less than the denominator, meaning their value is always less than one. For example, the fraction $\frac{1}{2}$ has a numerator of 1 and a denominator of 2, so it represents one part out of two equal parts. Visually, this can be shown by shading one half of a shape divided into two equal sections. Another example is $\frac{4}{6}$, where four parts out of six are shaded, still less than a whole.

Improper fractions occur when the numerator is greater than or equal to the denominator, resulting in a value that is greater than or equal to one. For instance, $\frac{3}{2}$ has a numerator of 3 and a denominator of 2. Since 3 is greater than 2, this fraction represents more than one whole. To visualize this, you might shade one whole shape divided into two parts completely, and then shade one additional half of another shape. Another example is $\frac{2}{2}$, where the numerator equals the denominator, representing exactly one whole. This can be shown by shading all parts of a single shape divided into two equal sections.

Mixed numbers combine a whole number with a proper fraction, such as \$1 \frac{1}{2}$ or $2 \frac{3}{8}$. The whole number indicates how many complete shapes are fully shaded, while the proper fraction shows the additional part of another shape that is shaded. For example, $1 \frac{1}{2}$ means one whole shape is fully shaded plus half of another shape. Similarly, $2 \frac{3}{8}$ means two whole shapes are fully shaded, and three out of eight parts of a third shape are shaded.

Understanding these types of fractions is essential for visualizing and working with fractional values, especially when relating them to real-world contexts like dividing a pizza. Proper fractions represent parts less than a whole, improper fractions represent one or more wholes, and mixed numbers combine whole units with fractional parts. This foundational knowledge prepares you for more advanced operations involving fractions and mixed numbers.

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Intro to Fractions Example 1

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Intro to Fractions Example 1 Video Summary

Fractions can be visually represented by dividing shapes into equal parts, where the total number of these parts corresponds to the denominator, and the number of shaded parts corresponds to the numerator. For example, if a rectangle is divided into 10 equal sections and 7 of these are shaded, the fraction represented is seven tenths, written as $\frac{7}{10}$. This means 7 out of 10 equal parts are considered.

In another case, if a shape is divided into 5 equal parts but none are shaded, the fraction is $\frac{0}{5}$, representing zero parts out of five. This illustrates how fractions can express the absence of a quantity as well.

When a circle is divided into 12 equal parts with 7 shaded, the fraction is $\frac{7}{12}$, showing seven twelfths of the whole. This highlights that fractions apply to various shapes, not just rectangles.

Fractions can also represent quantities greater than one. For instance, two rectangles each divided into 5 equal parts with a total of 8 parts shaded can be expressed as the improper fraction $\frac{8}{5}$. This fraction indicates more than one whole unit. Alternatively, this can be written as a mixed number, combining whole units and a fraction: one whole plus $\frac{3}{5}$, or \$1 \frac{3}{5}$. This dual representation helps in understanding and comparing quantities that exceed a single whole.

Understanding fractions through visual models strengthens comprehension of numerators and denominators, improper fractions, and mixed numbers, which are fundamental concepts in mathematics. Recognizing how to interpret and convert between these forms enhances problem-solving skills and numerical literacy.

Problem

From the choices, select the fraction equivalent to the given fraction.
(A) $\frac{3}{5}$

$\frac{9}{15}$

$\frac{6}{20}$

$\frac{15}{5}$

Problem

From the choices, select the fraction equivalent to the given fraction.
(B) $\frac{11}{4}$

$\frac{55}{15}$

$\frac{33}{12}$

$\frac{66}{18}$

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Simplify Fractions (Write Fractions in Lowest Terms)

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Simplify Fractions (Write Fractions in Lowest Terms) Video Summary

Simplifying fractions involves reversing the process of creating equivalent fractions by reducing a fraction to its lowest terms. Equivalent fractions are formed by multiplying both the numerator and denominator by the same nonzero constant, which keeps the value of the fraction unchanged. For example, multiplying the fraction one-half by 2/2 results in two-fourths, and multiplying by 3/3 results in three-sixths, all representing the same value.

To simplify a fraction, you factor both the numerator and denominator and then divide out their greatest common factor (GCF). A fraction is in lowest terms when the numerator and denominator share no common factors other than 1. This process effectively cancels out the common factors, reducing the fraction to its simplest form.

Mathematically, if a fraction is expressed as $\frac{a \times c}{b \times c}$, it can be rewritten as $\frac{a}{b} \times \frac{c}{c}$. Since $\frac{c}{c} = 1$, the fraction simplifies to $\frac{a}{b}$. For instance, the fraction $\frac{3}{6}$ simplifies to $\frac{1}{2}$ by dividing numerator and denominator by their GCF, which is 3.

Consider the fraction $\frac{4}{6}$. Both 4 and 6 are divisible by 2, so factoring gives \$4 = 2 \times 2$ and $6 = 3 \times 2$. This allows the fraction to be expressed as $\frac{2 \times 2}{3 \times 2} = \frac{2}{3} \times \frac{2}{2}$. Since $\frac{2}{2} = 1$, the fraction simplifies to $\frac{2}{3}\(. This is the fraction in lowest terms because 2 and 3 share no common factors.

When the greatest common factor is not immediately obvious, prime factorization can be used to identify common factors. For example, simplifying \)\frac{80}{60}$ involves recognizing that both numbers are divisible by 10, so dividing numerator and denominator by 10 yields $\frac{8}{6}$. Since 8 and 6 are both even, dividing by 2 further simplifies the fraction to $\frac{4}{3}\(, which is in lowest terms.

In cases where no common factors exist, such as the fraction \)\frac{5}{4}$, prime factorization confirms that the fraction is already in simplest form. Five is a prime number, and four factors into $2 \times 2$, so no common factors can be canceled.

Understanding how to simplify fractions by factoring and canceling common factors is essential for working efficiently with fractions in various mathematical contexts. This skill builds on the concept of equivalent fractions and is foundational for more advanced topics involving ratios, proportions, and algebraic expressions.

Problem

Simplify the following fraction to lowest terms.
(A) $6 over 15$

$\frac{3}{5}$

$2 over 15$

$\frac{2}{5}$

$\frac13$

Problem

Simplify the following fraction to lowest terms.
(B) $288 over 24$

$144 over 24$

$14$

$18$

$12$

Problem

Simplify the following fraction to lowest terms.
(C) $28 over 56$

$2$

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

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Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This is important because it makes the fraction easier to understand and work with, especially in algebra. To simplify, you find the greatest common factor (GCF) of the numerator and denominator and divide both by that number. For example, simplifying $\frac{4}{6}$ involves dividing numerator and denominator by 2, the GCF, resulting in $\frac{2}{3}$ . Simplified fractions are essential for clear communication and accurate calculations in math and real-life applications.

To find the greatest common factor (GCF) of two numbers, you can list their prime factors and identify the largest factor they share. For example, to simplify $\frac{80}{60}$ , factor 80 as 2 × 2 × 2 × 2 × 5 and 60 as 2 × 2 × 3 × 5. The common prime factors are two 2's and one 5, so the GCF is 2 × 2 × 5 = 20. Dividing numerator and denominator by 20 simplifies the fraction to $\frac{4}{3}$ . Using the GCF ensures the fraction is reduced to its simplest form efficiently.

Equivalent fractions represent the same part of a whole but have different numerators and denominators. They are related to simplifying fractions because simplifying is the reverse process of creating equivalent fractions. For example, multiplying numerator and denominator of $\frac{1}{2}$ by 2 gives $\frac{2}{4}$ , an equivalent fraction. Simplifying $\frac{2}{4}$ by dividing numerator and denominator by 2 returns it to $\frac{1}{2}$ . Understanding this relationship helps in recognizing and simplifying fractions effectively.

The denominator of a fraction cannot be zero because division by zero is undefined in mathematics. A fraction $\frac{a}{b}$ represents $a ÷ b$ , and dividing by zero has no meaning or value. Visually, if you try to divide a shape into zero parts, it is impossible to represent. This restriction ensures fractions are valid and meaningful in calculations and algebraic expressions.

When the greatest common factor (GCF) is not obvious, you can factor the numerator and denominator into their prime factors to identify common factors. For example, to simplify $\frac{18}{24}$ , factor 18 as 2 × 3 × 3 and 24 as 2 × 2 × 2 × 3. The common factors are 2 and 3, so the GCF is 6. Dividing numerator and denominator by 6 simplifies the fraction to $\frac{3}{4}$ . Alternatively, you can divide by any common factor step-by-step until no common factors remain.