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Comparing fractions

Introduction

In our daily lives, we commonly deal with fractions. Most of the time, we are asked to compare which part is smaller or larger, or if both parts are equally split. In this article, we look back at the definition of fractions and their types, then explore methods on how we can compare them.

What is a Fraction?

To motivate us into knowing about fractions, we think about a popular analogy to it. Imagine we have a box of pizza that is sliced into equal parts:

Each slice represents a part of the whole pizza. If we count the number of slices that make up the pizza, we can express the slice we will take as a mathematical quantity. This is what we will introduce as a fraction.

When we refer to fractions, we think about parts of a whole. Mathematically speaking, it is a quantity that is expressed as a quotient or a ratio between two numbers. Going back to the box of pizza we used as an analogy, if we take the number of slices we eat over the total number of slices in the box, we can express this ratio as follows:

$\frac{no.\: of\: pizza\: slices\: eaten}{no.\: of\: slices\: in\: the\: box}$

Suppose a friend ate three slices of pizza. We can then say that our friend had three-eighths of the pizza:

$\frac{no.\: of\: pizza\: slices\: eaten}{no.\: of\: slices\: in\: the\: box}=\frac{3}{8}$

We now identify the parts of a fraction. The number above is called the numerator, and the number under it is called the denominator. The numerator and denominator are separated by the line we call the fraction bar.

$\frac{3}{8}$    ←numerator     ←denominator

Types of Fractions

Fractions can be classified under three types, depending on how the numerator and denominator are related:

Proper Fractions

Proper Fractions are fractions that have a numerator that is less than its denominator:

numerator<denominator

Some examples we can think of are $\frac{1}{3}$, $\frac{4}{7}$, and $\frac{5}{8}$:

We can think of proper fractions as “proper” since the number of parts is less than the whole. In a mathematical sense, the ratio formed by proper fractions is less than 1:

$\frac{numerator}{denominator}$<1

This happens because dividing a smaller number by a bigger number results in a quotient that is less than one.

Improper Fractions

On the other hand, Improper Fractions are fractions that have a numerator that is greater than its denominator:

numerator>the denominator

Some examples we can produce are $\frac{3}{2}$, $\frac{5}{4}$, and $\frac{9}{8}$.

We can think of improper fractions as “improper” since the number of parts is more than the whole. Mathematically speaking, the ratio formed by improper fractions is greater than 1:

$\frac{numerator}{denominator}$>1

Contrary to the case of proper fractions, dividing a larger number by a small number results in a quotient that is greater than one.

Mixed Numbers

A Mixed Number can be thought of as a combination of a whole and a fraction. This is another way of writing an improper fraction, taking the parts that make a whole and then expressing the remaining parts as a proper fraction.

For example, we consider the improper fraction $\frac{11}{8}$. By visualization, we have 11 equal parts that will be grouped into 8 parts per whole:

If we put together 8 parts to form one whole, then what remains will be a proper fraction of $\frac{3}{8}$. In this regard, we can express the improper fraction as the number of wholes and the fractional part:

$\frac{11}{8}=\frac{8}{8}+\frac{3}{8}=1\frac{3}{8}$

Conveniently, we have discovered this method to convert improper fractions into mixed numbers.

Like Fractions

We can also relate two (or more) fractions based on the number on their denominators. Like Fractions are fractions that have the same denominator. We can think of like fractions as parts of the same whole. For example, we have $\frac{1}{5}$, $\frac{2}{5}$, $\frac{3}{5}$, and $\frac{4}{5}$ as like fractions since their denominators are all equal to 5.

Unlike Fractions

What if two fractions are not like fractions? In this case, we have unlike fractions. Unlike Fractions are fractions that have different denominators. We can think of unlike fractions as parts of different wholes. For example, $\frac{1}{4}$ and $\frac{2}{8}$ are unlike fractions since their denominators are not equal.

What are Equivalent Fractions?

Before we begin comparing fractions, another relevant concept that we must know is how fractions can be equivalent. Equivalent fractions are fractions that bear the same value, even if they have different numerators and denominators. To help us understand how equivalent fractions work, let us consider two unlike fractions $\frac{1}{4}$ and $\frac{2}{8}$:

We see that for the same circle split into equal parts, the area covered by one part over four equal parts is the same as the area covered by two parts over eight equal parts. Hence, we can conclude that both $\frac{1}{4}$ and $\frac{2}{8}$ are equivalent fractions.

For the previous example, we have easily shown that they are equivalent by visualizing the fractions. But how can we determine whether two of such fractions are equivalent for fractions that will involve larger numbers?

Two fractions are said to be equivalent if they have a common factor that we can divide or multiply to both the numerator and the denominator of one fraction to arrive at the other fraction.

Again, let us take $\frac{1}{4}$ and $\frac{2}{8}$ as an example. We can multiply a factor of 2 to both the numerator and denominator of $\frac{1}{4}$ and $\frac{2}{8}$:

$\frac{1×2}{4×2}=$\frac{2}{8}$

Similarly, we can divide a factor of 2 to both numerator and denominator of $\frac{2}{8}$ to obtain $\frac{1}{4}$:

$\frac{2÷2}{8÷2}=$\frac{1}{4}$

Hence, aside from the visualization we made earlier, we further prove that both $\frac{1}{4}$ and $\frac{2}{8}$ are equivalent fractions.

Comparing Fractions

Now that we have learned the underlying concepts regarding fractions, we can now proceed to our main topic which revolves around a comparison between two fractions. We note that the techniques discussed below are generally applicable to two fractions.

Intuitive Approach

Like what we have been doing with the pizza analogy, we use an Intuitive Approach that allows us to think of fractions visually. For this method, we can use any shape that we wish to use but for convenience, we will be using rectangles to represent parts of a whole:

Suppose we wish to compare two fractions $\frac{6}{8}$ and $\frac{2}{9}$. We represent the first fraction by splitting the rectangle into 8 equal parts, then shading 6 parts of it:

We also represent the second fraction by splitting the rectangle into 9 equal parts, then shading 2 parts of it:

If we compare both figures to each other, we see that $\frac{6}{8}$ covers more area than $\frac{2}{9}$.

Therefore, we conclude that the first fraction $\frac{6}{8}$ is greater than the second fraction $\frac{2}{9}$:

$\frac{6}{8}$>$\frac{2}{9}$

Again, we note that this approach can be convenient for fractions that are easy to imagine. As we consider larger denominators, the splitting might not be as effective as it is for smaller denominators. Hence, we consider other techniques.

Decimal Approach

Another method we can use to compare fractions is using the Decimal Approach. In this approach, we solve for the exact values of each fraction using either a calculator or by doing long division and then compare these decimal numbers.

For example, we want to know which fraction is larger: $\frac{15}{24}$ or $\frac{30}{40}$. By computing the exact values of both fractions, we get the following decimal numbers:

$\frac{15}{24}$=0.625

$\frac{30}{40}$=0.75

We note that both fractions are proper fractions whose numerators are smaller than their denominators. Hence, we expect that their exact values will be less than one.

Then, we compare the decimal values of both fractions. Looking at the tenths place of both decimals, we note that 0.7 is greater than 0.6, thus we conclude that 0.75>0.625 and that $\frac{30}{40}$ is greater than $\frac{15}{24}$:

$\frac{30}{40}$>$\frac{15}{24}$

Cross Multiplication

If we wish to work with multiplying numbers instead of getting their quotient, we can use Cross Multiplication to compare two fractions. To perform cross multiplication, we multiply the numerator of one fraction to the denominator of the other fraction, and then compare which cross product is larger and subsequently determine which fraction is larger.

We can use the previous example $\frac{15}{24}$ and $\frac{30}{40}$ and verify which fraction is larger.

First, we take the cross product of the numerator 15 and the denominator 40 then write the result to represent the first fraction:

Next, we also take the cross product of the numerator 30 and the denominator 30 then write the result to represent the second fraction:

Finally, we compare the two products obtained from cross multiplication:

600<720

Since 720 is larger than 600, then we conclude that $\frac{30}{40}$ is greater than $\frac{15}{24}$. This is consistent with the conclusion we made using the decimal approach.

Comparing Like Fractions

Suppose we have two like fractions. Other than the general techniques we have already discovered, there is a faster way to compare these fractions without going through much trouble.

When comparing two fractions that have the same denominator, we can treat the comparison between fractions as a comparison between their numerators. The relationship between the numerators of both fractions also applies to the relationship between the two fractions.

Let us consider an example and compare the following fractions $\frac{2}{7}$ and $\frac{6}{7}$. Since both fractions are like fractions with the same denominator 7, we compare the numerators 2 and 6.

2<6

Since the numerator of the first fraction is less than the numerator of the second fraction, we conclude that the first fraction is also less than the second fraction:

$\frac{2}{7}$<$\frac{6}{7}$

Intuitively, we can agree to this conclusion since from a pizza of seven slices, eating two slices out of the whole is less than eating six slices out of the same box!

Comparing Unlike Fractions

How about the case of two unlike fractions? We can use a similar approach to comparing like fractions, but we add extra steps to convert unlike fractions into like fractions. We do this by either finding a common denominator for both fractions or the least common denominator.

Finding the Common Denominator

The Common Denominator between two fractions can be obtained by taking the product of both denominators. We then rewrite both fractions as like fractions by multiplying both the numerator and denominator of one fraction by the denominator of the other fraction.

As an example, we compare the fractions $\frac{22}{45}$ and $\frac{6}{9}$. Their common denominator is given by:

45×9=405

We then rewrite the first fraction $\frac{22}{45}$ by multiplying both numerator and denominator by the denominator of the second fraction 9:

$\frac{22×9}{45×9}$=$\frac{198}{405}$

Next, we rewrite the second fraction $\frac{6}{9}$ by multiplying both numerator and denominator by the denominator of the second fraction 45:

$\frac{6×45}{9×45}$=$\frac{270}{405}$

Now that we have two like fractions, we can compare their numerators to determine which fraction is larger:

270>198           ∴        $\frac{270}{405}$>$\frac{198}{405}$

Since $\frac{270}{405}$ is greater than $\frac{198}{405}$, then we conclude that their equivalent fractions $\frac{6}{9}$ is greater than $\frac{22}{45}$:

$\frac{6}{9}$>$\frac{22}{45}$

Finding the Least Common Denominator

Alternatively, the Least Common Denominator (LCD) between two fractions can be obtained by taking the Least Common Multiple of both denominators. We then rewrite the equivalent fractions such that we get two like fractions. This way, we can compare them as like fractions.

We consider the two fractions in the previous example $\frac{22}{45}$ and $\frac{6}{9}$ whose denominators are 45 and 9, respectively. If we take the least common multiple of both denominators, 45, then we can rewrite both fractions to have the same denominator.

Since $\frac{22}{45}$ is already expressed in terms of the least common denominator, we only have to get the equivalent fraction of the second fraction $\frac{6}{9}$:

$\frac{6×5}{9×5}$=$\frac{30}{45}$

Thus, we can compare both like fractions using their numerators:

30>22           ∴        $\frac{30}{45}$>$\frac{22}{45}$

Since $\frac{30}{45}$ is greater than $\frac{22}{45}$, then we conclude that their equivalent fractions $\frac{6}{9}$ is greater than $\frac{22}{45}$:

$\frac{6}{9}$>$\frac{22}{45}$

We note that we have arrived at the same conclusion when we used the Common Denominator earlier.

Problem-Solving Examples

We can now proceed to solve sample problems to apply what we have learned so far. Each problem tackles the different approaches and techniques discussed and gives us a challenge on how to work through the information given to us.

Comparing Fractions Using Intuitive Approach

Sample Problem 1:

Timothy made a chocolate bar to share with five of his friends. During recess, another friend joined them, and Timothy decided to share the chocolate among all seven of them (including himself).

However, Timothy doesn’t want to get less than the share he was supposed to get sharing with five friends. He thinks that he should split the chocolate into eight parts, get two parts for himself, then share the remaining parts with his friends. Will he get more chocolate in this way?

Solution:

We first consider the chocolate bar split into six pieces, and take one piece for Timothy as shown in the figure below:

Next, we consider the chocolate bar split into eight pieces, and take two pieces for Timothy as shown in the figure below:

Comparing the combined size of the chocolate Timothy will receive in either case, we conclude that he will get more chocolate by splitting the bar into eight pieces and taking two pieces for himself.

Sample Problem 2:

On her birthday, Melissa received three round cakes of the same size: a chocolate cake from eleven of her relatives in the morning, a strawberry cake from her eight workmates in the afternoon, and a blueberry cake from ten of her cousins in the evening.

If she decided to share each cake with the group who gave it to her, which cake did she have the most?

Solution:

We first consider the chocolate cake she received in the morning, where she got a slice out of eleven parts. We compare this with the strawberry cake she had in the afternoon, where she got a slice out of eight parts.

By visualizing the slices of cake she had as parts of a whole, we can conclude that she had more strawberry cake than the chocolate cake:

Then, we compare the strawberry cake she had with the blueberry cake she had in the evening, where she got a slice out of ten parts.

Again, by visualizing the slices of cake, we can conclude that she had more strawberry cake than the blueberry cake:

Therefore, we conclude that out of all three cakes Melissa had on her birthday, she had the most share in the strawberry cake.

Comparing Fractions Using Decimal Approach

Sample Problem 3:

Suppose we have two improper fractions $\frac{52}{11}$ and $\frac{47}{8}$. Using the decimal approach, determine which fraction is larger.

Solution:

For this example, we use long division to show the process of getting the decimal values of each fraction. We begin by taking the quotient between the numerator and denominator of the first fraction $\frac{52}{11}$:

Hence, we approximate the decimal value up to the thousandths place by:

$\frac{52}{11}$≈4.727

Similarly, we take the quotient between the numerator and denominator of the second fraction $\frac{47}{8}$:

Hence, the exact decimal value of the fraction is given by:

$\frac{47}{8}$=5.875

Finally, we compare the two decimal numbers representing their equivalent fractions:

4.727<5.875

Since the decimal value of the second fraction is larger than the decimal value of the first fraction, then we conclude that the second fraction $\frac{47}{8}$ is larger than the first fraction $\frac{52}{11}$.

Comparing Fractions Using Cross Multiplication

Sample Problem 4:

Suppose we have two proper fractions $\frac{7}{23}$ and $\frac{16}{35}$. Which fraction is smaller? Show your computation using cross multiplication.

Solution:

We first consider the cross product of the numerator of the first fraction, 7, and the denominator of the second fraction, 35, then write it as a number representing the first fraction:

Next, we consider the cross product of the numerator of the second fraction, 16, and the denominator of the second fraction, 23, then write it as a number representing the second fraction:

Comparing the two cross products, we can conclude that the first cross product is smaller than the other cross product:

245<368

Therefore, we conclude that the first fraction $\frac{7}{23}$ is smaller than the second fraction $\frac{16}{35}$.

Sample Problem 5:

If we are given two proper fractions $\frac{8}{24}$ and $\frac{12}{36}$, which fraction is larger? Using cross multiplication, show your computations.

Solution:

We first consider the cross product of the numerator of the first fraction, 8, and the denominator of the second fraction, 36, then write it as a number representing the first fraction:

Next, we consider the cross product of the numerator of the second fraction, 16, and the denominator of the second fraction, 23, then write it as a number representing the second fraction:

Comparing the two cross products, we can see that both cross products are equal:

288=288

Therefore, we conclude that neither fraction is smaller than the other since both fractions are equivalent:

$\frac{8}{24}$=$\frac{12}{36}$

Comparing Like Fractions

Sample Problem 6:

Which fraction is larger, $\frac{3}{17}$ or $\frac{10}{17}$?

Solution:

Since both fractions are like fractions of the same denominator 17, we simply compare the numerators of both fractions:

3<10

Since the numerator of the second fraction is larger than the numerator of the first fraction, we then conclude that the second fraction is larger than the first fraction:

$\frac{10}{17}$>$\frac{3}{17}$

Comparing Unlike Fractions

Sample Problem 7:

We are given two fractions $\frac{3}{10}$ and $\frac{16}{30}$. We are then asked the following questions:

  1. What is the common denominator of the two fractions?
  1. What are the equivalent fractions of each fraction in terms of the common denominator?
  1. Which fraction is smaller, $\frac{3}{10}$ or $\frac{16}{30}$?

Solution:

  1. To get the common denominator of the given fractions, we multiply their denominators to get the product:

10×30=300

Hence, the common denominator of both fractions is 300.

  1. From the first fraction $\frac{3}{10}$, we multiply its denominator by 30 to arrive at the common denominator 300. To get its equivalent fraction, we also multiply the numerator by the same number:

$\frac{3×30}{10×30}$=$\frac{90}{300}$

Therefore, the equivalent fraction of the first fraction is given by $\frac{90}{300}$.

Then, from the second fraction $\frac{16}{30}$ we multiply its denominator by 10 to arrive at the common denominator 300. To get its equivalent fraction, we also multiply the numerator by the same number:

$\frac{16×10}{30×10}$=$\frac{160}{300}$

Therefore, the equivalent fraction of the second fraction is given by $\frac{160}{300}$.

  1. By comparing the equivalent like fractions $\frac{90}{300}$ and $\frac{160}{300}$, we simplify the comparison between $\frac{3}{10}$ and $\frac{16}{30}$ as a comparison between the numerators 90 and 160:

90<160

Since the numerator of the second equivalent fraction is greater than the numerator of the first equivalent fraction, we conclude that $\frac{160}{300}$>$\frac{90}{300}$. Furthermore, this implies that the second fraction $\frac{16}{30}$ is greater than $\frac{3}{10}$:

$\frac{16}{30}$>$\frac{3}{10}$

Sample Problem 8:

Suppose we use the two given fractions from Sample Problem 7. Again, we are asked the following questions:

Solution:

  1. What is the LCD of the two fractions?
  1. What are the equivalent fractions of each fraction in terms of the least common denominator?
  1. Which fraction is smaller, $\frac{3}{10}$ or $\frac{16}{30}$?

Solution:

  1. To get the least common denominator of the given fractions, we determine the least common multiple of both denominators. Since both denominators are multiples of 10, the least common denominator of both fractions is 30.
  1. From the first fraction $\frac{3}{10}$, we multiply its denominator by 3 to arrive at the least common denominator 30. To get its equivalent fraction, we also multiply the numerator by the same number:

$\frac{3×3}{10×3}$=$\frac{9}{30}$

Therefore, the equivalent fraction of the first fraction is given by $\frac{9}{30}$.

Then, we note that the second fraction is already expressed in terms of the least common denominator. Hence, we do not need to rewrite the second fraction $\frac{16}{30}$.

  1. By comparing the equivalent like fractions $\frac{9}{30}$ and $\frac{16}{30}$, we simplify the comparison between $\frac{3}{30}$ and $\frac{16}{30}$ as a comparison between the numerators 9 and 16:

9<16

Since the numerator of the second equivalent fraction is greater than the numerator of the first equivalent fraction, we conclude that $\frac{16}{30}$>$\frac{9}{30}$. Furthermore, this implies that the second fraction $\frac{16}{30}$ is greater than $\frac{3}{10}$:

$\frac{16}{30}$>$\frac{3}{10}$

We note that the same conclusion was obtained from the previous sample problem.

Summary

  • A Fraction can be imagined as parts of a whole. Mathematically speaking, it is a ratio between the numerator and the denominator.
  • There are three types of fractions according to how their numerators and denominators are related:
  • A Proper Fraction is a fraction wherein its numerator is less than its denominator. The ratio between the numerator and denominator is always less than one.
  • An Improper Fraction, on the other hand, is a fraction wherein its numerator is greater than its denominator. The ratio between the numerator and denominator is always greater than one.
  • We can also express an improper fraction as a Mixed Number. A mixed number can be thought of as a combination of a whole number and a fraction.
  • Two fractions can be related according to the value of their denominators:
  • Like Fractions are fractions whose denominators are equal.
  • Unlike Fractions are fractions whose denominators are not equal.
  • Equivalent Fractions are unlike fractions that yield the same value.
  • Generally, we can compare fractions using three approaches:
  • An Intuitive Approach allows us to visualize each fraction, and compare the areas covered by each fraction to determine which one is larger/smaller.
  • The Decimal Approach tells us the exact decimal values of each fraction as a basis to compare them.
  • By performing Cross Multiplication, we take the cross-product of both fractions to compare which fraction is larger/smaller.
  • In the case we are comparing like fractions, we can treat the comparison between like fractions as a comparison between their numerators.
  • In the case we are comparing unlike fractions, we first rewrite their equivalent like fractions then compare their numerators and subsequently the original fractions.
  • We can find the Common Denominator of two unlike fractions and express both fractions in terms of their common denominator.Alternatively, we can find their Least Common Denominator by looking at the Least Common Multiple of their denominators and then expressing both fractions in terms of their least common denominator.

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