Fractions and Factors

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We know that natural numbers and integers can also be called whole numbers. Suppose you divide a bar of chocolate into two equal parts or halves, then how do you represent the value numerically? This kind of number, a fraction, is also a main type of number that we use in mathematics.

We can use factors to help us simplify fractions into their simplest form. This article explores the key concepts of fractions and factors as well as some applications.

Meaning of fractions and factors: An introduction

Let's start out by defining and introducing the concepts of fractions and factors.

Components of a fraction: Numerator and denominator

First, let's start with the definition of a fraction.

The numerical value which represents the part of any whole value or thing is known as a fraction. Fractions are known as rational numbers (by set theory). In mathematics, we say that the rational numbers are in set $ℚ .$

Fractions can be represented as $\frac{a}{b}$ , with a being known as the numerator, and b being known as the denominator. Essentially, the numerator is divided by the denominator.

Let's try to see this from a more visual perspective. Imagine 1 pizza having 8 slices.

Fractions and factors pizza slices StudySmarter Pizza with 8 slices, pixabay.com

If I take 1 slice of pizza, I have taken $\frac{1}{8}$ of the pizza. This is because we have 1 pizza, and we've divided it into 8 slices. So, we can see that the singular pizza slice (1) is the numerator, and the total number of slices (8) is the denominator.

A fraction can also be thought of as dividing a numerator by a denominator. Let's look at an example to see this in action.

I have a pie with 8 slices. I want to share it equally between 4 people. What fraction of the pie will each person receive?

Solution:

The pie has 8 slices, and we want to share it equally between 4 people. Therefore, we calculate that $8 \div 4 = 2 .$ This means each person receives 2 slices of pie.

If each person receives 2 slices, it means they receive $\frac{2}{8}$ of the pie. This is the number of slices each person receives (2) divided by the total number of slices of the pie (8), with the numerator being divided by the denominator.

Using factors for integers

Whole numbers are also known as integers. In math, they are represented as $ℤ .$ All integers contain factors.

Factors of an integer are numbers that divide exactly into that integer.

This means that if you do long division, dividing the integer by its factor, you would find no remainder.

As an example, 10 can be divided by 2 to equal 5, $10 \div 2 = 5,$ which means 2 is a factor of 10. Similarly, 10 can be divided by 5 to equal 2, $10 \div 5 = 2,$ which means 5 is also a factor of 10. So, 2 and 5 are a pair of factors of 10.

All integers are divisible by 1, so 1 is a factor of all integers. The integer itself is always also a factor of itself, as when you divide a number by itself, you get 1. Since this process leaves no remainder, we know that the number is a factor of itself.

All integers are divisible by 1 and themselves, therefore having at least two factors. Integers that are only divisible by 1 and themselves are known as prime numbers.

The one exception to the fact that all integers have at least two factors is the number 1. The number 1 doesn't count as a prime number as it is divisible by 1 and itself, but since 1 is itself, it is the only number to contain one factor.

Let's look at a small example.

List all the factors of the number 24.

Solution:

So, how many numbers is 24 divisible by? We have:

$24 \div 1 = 24, 24 \div 2 = 12, 24 \div 3 = 8, 24 \div 4 = 6, 24 \div 6 = 4, 24 \div 8 = 3, 24 \div 12 = 2, 24 \div 24 = 1$

Apart from the numbers 1, 2, 3, 4, 6, 8, 12, and 24, all other numbers when divided by 24 do not return whole numbers. This means our factors are 1, 2, 3, 4, 6, 8, 12, and 24.

Factors

Now that we understand the basic ideas and concepts behind fractions and factors, let's take a closer look at factors in particular. Understanding factors will help us later on as we learn methods for simplifying fractions into their simplest and smallest forms.

What is a prime factor decomposition?

A prime factor decomposition is simply analyzing an integer as a product of prime factors. In other words, we determine all the prime factors which, when multiplied, amount to the given integer.

A prime factor is simply a factor of an integer which is also a prime number. We can find the prime factor decomposition by drawing a factor tree. A factor tree shows us exactly how we can break down an integer into its factors, and then further break these factors down until we ultimately reach prime factors.

Let's look at a visual example of this.

Draw a factor tree for 100, and write out the prime factor decomposition of 100.

Solution:

Initially, we can break down 100 into $50 \times 2$ , which looks like this:

Fractions and Factors, prime factorization example, StudySmarter Prime factor of 100, StudySmarter Originals

Now, we can stop breaking 2 down into factors as it is a prime factor, which means it only can be divided by 1 and itself. However, 50 is not prime; therefore, we have to break it down further. We can break 50 down into $25 \times 2$ . We can add this to our factor tree like this:

Fractions and Factors, prime factorization example, StudySmarter Prime factors of 50, StudySmarter Originals

Again, 2 is prime, so we don't break this down further. However, we can break down 25 further into $5 \times 5$ . And we can add it to our factor tree like this:

Fractions and Factors, prime factorization example, StudySmarter Factor tree for 100, StudySmarter Originals

Now, as 5 is prime, we can stop there, as we cannot break any of these numbers down any further. This means that we have finished drawing our factor tree!

When writing the prime factor decomposition, we can circle all the factors we identified to be prime for easy reference.

Fractions and Factors, prime factorization example, StudySmarter Prime numbers circled after decomposition, StudySmarter Originals

When these numbers are multiplied by each other they give us 100, so our prime factor decomposition is

$2 \times 2 \times 5 \times 5$

We can make this look nicer using indices: $2^{2} \times 5^{2}$ .

What is the highest common factor?

The highest common factor (greatest common divisor), is something we can find when we use the method of prime factor decomposition on two or more different numbers. The greatest common divisor is a number which is a factor of all considered numbers. Specifically, it is the largest one possible.

There is a method to do this, which we will look at through an example.

Find the greatest common divisor of 100 and 120.

Solution:

STEP	EXAMPLE
STEP 1: Find the prime factor decomposition of both numbers.	The prime factor decomposition of 100, which we know from above is $2^{2} \times 5^{2} .$ If we use a factor tree to find the prime factor decomposition of 120 we get the following: Factor tree of 120, StudySmarter Originals Therefore, our prime factor decomposition for the number 120 is $2^{3} \times 3 \times 5 .$
STEP 2: Write these two out in power notation (so if there's just one number, write it with a power of 1).	$100 = 2^{2} \times 5^{2} 120 = 2^{3} \times 3^{1} \times 5^{1}$
STEP 3: If one of the numbers is missing a factor from another number's prime number decomposition, write that missing factor down in the prime factor decomposition to the power of 0.	100 is missing 3, so we put in a 3 to the power of 0: $100 = 2^{2} \times 5^{2} \times 3^{0}$
STEP 4: Compare the same base numbers and select the one with the lowest power.	Between $5^{1}$ and $5^{2}$ , select $5^{1}$ Between $2^{2}$ and $2^{3}$ , select $2^{2}$ Between $3^{0}$ and $3^{1}$ , select $3^{0}$
STEP 5: Multiply these selected numbers together.	$2^{2} \times 3^{0} \times 5^{1} = 20$ , so our greatest common divisor is 20.

Fractions

We have learned that fractions are made up of numerators on top and denominators on bottom. Fractions have integer values in both their numerator and denominator, but the denominator should not be zero. When a fraction has the same factors in the numerator and denominator, we can simplify the form of the fraction.

Comparing fractions and factors: How can we use factors to simplify fractions?

When we determine that a fraction can be simplified, it means that we can divide the numerator and the denominator by the same number to arrive at a simpler or smaller fraction. This can only be done if both the numerator and denominator share a factor.

If we took our earlier answer of $\frac{2}{8}$ , both 2 and 8 share the factor of 2. Therefore, if we divide both 2 and 8 by 2 we get $2 \div 2 = 1, 8 \div 2 = 4 .$ Therefore, we can simplify our fraction to $\frac{1}{4}$ .

Sometimes in exam questions, you may be asked to provide your answer in its simplest form. This means that you should simplify the fraction before giving an answer. Let's look at examples.

Simplify $\frac{56}{96} .$

Solution:

First, we need to think of the factor that 56 and 96 share. They both share 8 as a factor. Therefore, we just need to divide each of them (both the numerator and the denominator) by 8.

$\Rightarrow \frac{56 \div 8}{96 \div 8} = \frac{7}{12}$

This means that our new simplified fraction is $\frac{7}{12} .$

Simplify $\frac{5}{65} .$

Solution: Here, 5 and 65 share 5 as a factor. So, we divide both the numerator and denominator by 5.

$\Rightarrow \frac{5 \div 5}{65 \div 5} = \frac{1}{13}$

Hence, the simplified fraction is $\frac{1}{13} .$

Fractions and factors are important in a variety of applied situations. When learning other topics, we will often find that we may have to determine a common factor or simplify a fraction as part of our problem solving.

Rules in fractions

There are certain rules which apply when using basic mathematical operations on fractions. We will see the rules in fractions for the following operations:

Addition and subtraction
Multiplication
Division

Addition and Subtraction

The addition or subtraction of fractions is performed based on the type of denominator they have. We need to check if the denominators of given fractions are the same or different. Let's see the steps to perform addition or subtraction if the denominator is the same for all the fractions.

Add/subtract the numerators and keep the denominator as it is.
Reduce the fraction if possible.

$\Rightarrow \frac{a}{b} \pm \frac{c}{b} = \frac{a \pm c}{b}$

Where a, b, and c are integers.

When the denominators are not the same, then the following steps should be followed.

Make the denominator of all fractions the same. To do this, you can multiply the numerator and denominator of one fraction with the denominator of another fraction and vice versa.
After making the denominator the same, add/subtract the numerators without changing the denominator.
Simply the fraction when possible.

$\Rightarrow \frac{a}{b} \pm \frac{c}{d} = \frac{a \times d}{b \times d} \pm \frac{c \times b}{d \times b}$

Multiplication

When multiplying the fractions, the denominators need not be the same, unlike for addition/subtraction. Instead, just multiply the numerators with each other, and multiply the denominators with each other. Then, reduce the fraction to a simplified form. Remember that the fractions typically should not be mixed fractions. If it is a mixed fraction, then first convert it into proper or improper fractions.

$\Rightarrow \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Division

When dividing the fractions, we convert them into multiplication form to find the answer. So, to convert it into the multiplication form, inverse the second fraction (that is, flip the numerator and denominator) and change the division sign into the multiplication sign. Now you can perform the multiplication steps as usual.

$\Rightarrow \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Fractions and factors example

Let us see some solved examples for fractions and factors.

Find the highest common factor (HCF) of 48, 108, and 140.

Solution:

STEP	EXAMPLE
STEP 1: Find the prime factor decomposition of all the three given numbers.	Prime factor decomposition of 48 using factor tree is $2 \times 2 \times 2 \times 2 \times \times 3 = 2^{4} \times 3 .$ Factor tree of 48, StudySmarter Originals Similarly, prime factor decomposition of $108 = 2^{2} \times 3^{3}$ .Prime factor decomposition of 140 is $2^{2} \times 5 \times 7 .$
STEP 2: Write all three numbers in power notation.	$48 = 2^{4} \times 3^{1}, 108 = 2^{2} \times 3^{3}, 140 = 2^{2} \times 5^{1} \times 7^{1}$
STEP 3: Write a missing number of a factor from the other numbers' prime number decomposition to the power of 0.	$48 = 2^{4} \times 3^{1} \times 5^{0} \times 7^{0} 108 = 2^{2} \times 3^{3} \times 5^{0} \times 7^{0} 140 = 2^{2} \times 3^{0} \times 5^{1} \times 7^{1}$
STEP 4: Compare the same base numbers and select the one with the lowest power.	From $2^{2} a n d 2^{4}$ , select $2^{2}$ From $3^{0}, 3^{1}, 3^{3}$ , select $3^{0}$ From $5^{0}, 5^{1}$ , select $5^{0}$ From $7^{0}, 7^{1}$ , select $7^{0}$
STEP 5: Multiply the selected numbers.	$2^{2} \times 3^{0} \times 5^{0} \times 7^{0}$ So, HCF (or GCD) is 4 for the given three numbers.

Hailey's friend lives 25 miles away from her house. She has already traveled 11 miles. Represent the traveled distance using a fraction.

Solution: The total distance from Hailey's house to her friend's house is 25 miles. So, the denominator will be 25.

Hailey traveled 11 miles. So, the numerator will be 11.

Hence, the distance traveled in fractions will be $\frac{11}{25} .$

Solve the following fractions.

$1) \frac{6}{7} + \frac{2}{7} 2) \frac{6}{7} - \frac{1}{3} 3) \frac{2}{3} \times \frac{1}{2} 4) \frac{2}{3} \div \frac{1}{2}$

Solution:

1) $\frac{6}{7} + \frac{2}{7}$

For $\frac{6}{7}$ and $\frac{2}{7}$ , both the fractions have the same denominators. So, we can carry out addition without changing the denominator. Here, we will add the numerator and keep the denominator as it is.

$\Rightarrow \frac{6}{7} + \frac{2}{7} = \frac{6 + 2}{7} = \frac{8}{7}$

2) $\frac{6}{7} - \frac{1}{3}$

Here, both the fractions $\frac{6}{7}, \frac{1}{3}$ have different denominators. First, we will make their denominators the same and then subtract the obtained fractions.

$\Rightarrow \frac{6}{7} - \frac{1}{3} = \frac{6 \times 3}{7 \times 3} - \frac{1 \times 7}{3 \times 7} = \frac{18}{21} - \frac{7}{21} = \frac{18 - 7}{21} = \frac{11}{21}$

3) $\frac{2}{3} \times \frac{1}{2}$

For the multiplication of fractions, we multiply the numerators with each other and the denominators with each other.

$\Rightarrow \frac{2}{3} \times \frac{1}{2} = \frac{2 \times 1}{3 \times 2} = \frac{1}{3}$

4) $\frac{2}{3} \div \frac{1}{2}$

For the division of fractions, we flip the second fraction to convert the expression into one of multiplication. Then, we can multiply the fractions to obtain our answer.

$\Rightarrow \frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{2 \times 2}{3 \times 1} = \frac{4}{3}$

Fractions and factors - Key takeaways

The numerator is the top of a fraction, while the denominator is the bottom.
Factors are numbers by which other numbers divide exactly into.
Numbers with only two factors are known as prime numbers.
Prime factor decomposition is used to help us calculate the greatest common divisors.
Fractions can be simplified if the numerator and denominator share a common factor.

Frequently Asked Questions about Fractions and Factors

Fractions are rational numbers containing a numerator and a denominator. Factors are numbers that can exactly divide a number they are a factor of.

An example of a fraction would be 1/2, or a half as it is better known an example of a factor would be 3 and 5 being a pair of factors of 15 as 3 multiplied by 5 is 15.

Solving using fractions is understanding that a fraction is essentially a numerator being divided by a denominator.

The rules for add/ sub are to operate the numerator and change the denominator when necessary. For multiplication, multiply numerators and denominators respectively. And for division, convert it into multiplication form to solve it.

The component of fractions is the numerator and denominator, And the components of factors are the prime numbers in exponent form.

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Is the following fraction in its simplest form:

9/16

yes

Is the following fraction in its simplest form:

20/25

Is the following fraction in its simplest form:

4/9

yes

Is the following fraction in its simplest form:

24/82

Simplify 85/90

17/18

What is a prime number?

A prime number is a number that is only divisible by itself and 1.

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Fractions and Factors

Meaning of fractions and factors: An introduction