Comparing fractions means determining the larger and the smaller fraction between two or more fractions. Knowledge of comparing fractions is also needed while arranging the fractions in ascending or descending order.
It’s comparatively easier to compare like fractions compared to unlike fractions which involve a specific set of rules.
Let’s learn how to compare fractions.
Comparing Fractions
The process of comparing fractions is different for different types of fractions. Comparing like fractions is quite simple as compared to the other types.
Comparing Like Fractions
Like fractions are the fractions with the same denominators. When denominators of the fractions are the same, then it is very easy to compare fractions. Fractions with numerators smaller in value are less compared to the fractions whose numerators are larger in value.
Steps to Compare Like Fractions
Consider two like fractions $\frac {a}{c}$ and $\frac {b}{c}$ with common denominators $c$.
Step 1: Compare the numerators.
Step 2: If $a \gt b$, then $\frac {a}{c}$ is greater, otherwise $\frac {b}{c}$ is greater.
Examples
Ex 1: Which of these fractions is greater – $\frac {2}{5}$ or $\frac {4}{5}$?
Observe that denominators of the fractions $\frac {2}{5}$ and $\frac {4}{5}$ are the same which is equal to $5$.
Since, $4 \gt 2$, therefore, $\frac {4}{5} \gt \frac {2}{5}$.
You can also observe the difference between these two fractions visually.
Ex 2: Which of these fractions is greater – $\frac {3}{8}$ or $\frac {7}{8}$?
Observe that denominators of the fractions $\frac {3}{8}$ and $\frac {7}{8}$ are the same which is equal to $8$.
Since, $7 \gt 3$, therefore, $\frac {7}{8} \gt \frac {3}{8}$.
You can also observe the difference between these two fractions visually.
Note: Denominator in a fraction represents the total number of parts in a whole.
Comparing Unlike Fractions
To compare unlike fractions, i.e., the fractions with different denominators, the first step is to convert them to fractions with the same denominator, i.e., to convert the unlike fractions to like fractions.
Steps to Compare Like Fractions
Consider two like fractions $\frac {a}{b}$ and $\frac {c}{d}$.
Note: The denominators of the fractions $b$ and $d$ are different.
Step 1: Find the LCM of their denominators, i.e., LCM of $b$ and $d$
Step 2: Now, multiply each denominator by a number so that they become equal to LCM
Step 3: Multiply the numerator of each fraction by a number multiplied by its denominator
Step 4: Compare the numerators
Step 5: The fraction with the larger numerator is the larger fraction
To check conversion of unlike fraction to like fraction, click here.
Examples
Ex 1: Which of the given fractions is greater – $\frac {1}{2}$ or $\frac {3}{4}$?
Denominators of the two fractions $2$ and $4$ are different.
LCM of $2$ and $4$ is $4$.
$\frac {1}{2} = \frac {1 \times 2}{2 \times 2} = \frac {2}{4}$
Now the denominator of the two fractions are equal, so compare their numerators.
Since, $3 \gt 2$, therefore, $\frac {3}{4} \gt \frac {1}{2}$
You can also observe the difference between these two fractions visually.
Ex 2: Which of the given fractions is greater – $\frac {3}{4}$ or $\frac {5}{14}$?
Denominators of the two fractions $4$ and $14$ are different.
LCM of $4$ and $14$ is $28$.
Now, we’ll find the numbers to be multiplied with each of the fractions. To do so, divide LCM by each of the denominators.
For fraction $\frac {3}{4}$, the number is $28 \div 4 = 7$
And for fraction $\frac {5}{14}$, the number is $28 \div 14 = 2$
Now, multiply the numerator and denominator of $\frac {3}{4}$ by $7$ and multiply the numerator and denominator of $\frac {5}{14}$ by $2$
$\frac {3 \times 7}{4 \times 7} = \frac {21}{28}$
$\frac {5 \times 2}{14 \times 2} = \frac {10}{28}$
Since, $21 \gt 10$, therefore, $\frac {3}{4} \gt \frac {5}{14}$
You can also observe the difference between these two fractions visually.
Comparing Fractions With Same Numerators
Comparing fractions with the same numerators is completely opposite to that comparing fractions with the same denominators. When you compare two fractions with the same numerator, then the fraction with a lesser denominator is greater than the fraction with a larger denominator.
Steps to Compare Fractions With Same Numerators
Consider two like fractions $\frac {a}{b}$ and $\frac {a}{c}$ with common numerators $a$.
Step 1: Compare the denominators
Step 2: If $b \lt c$, then $\frac {a}{b}$ is greater, otherwise $\frac {a}{c}$ is greater.
Examples
Ex 1: Which of the given fractions is greater – $\frac {1}{2}$ or $\frac {1}{3}$?
Observe that the numerators of the two fractions are the same and equal to $1$.
Since, $2 \lt 3$, therefore, $\frac {1}{2} \gt \frac {1}{3}$.
You can also observe the difference between these two fractions visually.
Ex 2: Which of the given fractions is greater – $\frac {2}{4}$ or $\frac {2}{8}$?
Observe that the numerators of the two fractions are the same and equal to $1$.
Since, $2 \lt 4$, therefore, $\frac {2}{4} \gt \frac {2}{8}$.
You can also observe the difference between these two fractions visually.
Arranging Fractions
The fractions can be arranged either in ascending order or descending order. Depending on the types of fractions, here also different steps are involved to arrange the fractions.
- Like Fractions
- Unlike Fractions
- Fractions With Same Numerators
Note:
- Ascending order means from lowest to largest.
- Descending order means from largest to smallest.
Arranging Like Fractions
As seen above in the case of like fractions, the fraction with the larger numerator is greater than the fraction with the smaller numerator. So, you can arrange the like fractions by comparing their numerators.
Steps to Arrange Like Fractions
The steps to arrange the like fractions are
Step 1: Compare the numerators.
Step 2: Arrange the fractions in desired order according to their numerators
Examples
Ex 1: Arrange the fractions in ascending order
$\frac {2}{9}$, $\frac {7}{9}$, $\frac {8}{9}$, $\frac {6}{9}$, $\frac {1}{9}$, $\frac {5}{9}$, $\frac {3}{9}$ and $\frac {4}{9}$
Arranging the numerators in ascending order: $1 \lt 2 \lt 3 \lt 4 \lt 5 \lt 6 \lt 7 \lt 8$, therefore,
$ \frac {1}{9} \lt \frac {2}{9} \lt \frac {3}{9} \lt \frac {4}{9} \lt \frac {5}{9} \lt \frac {6}{9} \lt \frac {7}{9} \lt \frac {8}{9}$.
Ex 2: Arrange the fractions in descending order
$\frac {5}{15}$, $\frac {9}{15}$, $\frac {1}{15}$, $\frac {11}{15}$, $\frac {7}{15}$
Arranging the numerators in descending order: $ 11 \gt 9 \gt 7 \gt 5 \gt 1$, therefore, $ \frac {11}{15} \gt \frac {9}{15} \gt \frac {7}{15} \gt \frac {5}{15} \gt \frac {1}{15}$
Arranging Unlike Fractions
As seen above, to compare unlike fractions first the common denominator (LCM) is computed and then numerators of fractions are changed according to the LCM as denominators.
The same process is used while arranging the, unlike fractions.
Steps to Arrange Unlike Fractions
The steps to arrange the unlike fractions are
Step 1: Find the LCM of denominators of fractions
Step 2: Now, multiply each denominator by a number so that they become equal to LCM
Step 3: Multiply the numerator of each fraction by a number multiplied by its denominator
Step 4: Arrange the numerators
Examples
Ex 1: Arrange the fractions in ascending order $\frac {5}{6}, \frac {2}{3}, \frac {1}{2}, \frac {3}{4}, \frac {2}{5}$ in ascending order.
The denominators in the fractions are $2$, $3$, $4$, $5$ and $6$.
LCM of $2$, $3$, $4$, $5$ and $6$ is $60$.
Now converting fractions to equivalent fractions, we get
$\frac {5}{6} = $\frac {5 \times 10}{6 \times 10} = $\frac {50}{60}$
$\frac {2}{3} = $\frac {2 \times 20}{3 \times 20} = $\frac {40}{60}$
$\frac {1}{2} = $\frac {1 \times 30}{2 \times 30} = $\frac {30}{60}$
$\frac {3}{4} = $\frac {3 \times 15}{4 \times 15} = $\frac {45}{60}$
$\frac {2}{5} = $\frac {2 \times 12}{5 \times 12} = $\frac {24}{60}$
Arranging the numerators in ascending order $24 \lt 30 \lt 40 \lt 45 \lt 50$, therefore, $\frac {2}{5} \lt \frac {1}{2} \lt \frac {2}{3} \lt \frac {3}{4} \lt \frac {5}{6}$.
Ex 2: Arrange the fractions in descending order $\frac {3}{5}$, $\frac {6}{8}$, $\frac {2}{3}$ and $\frac {4}{7}$ in descending order.
LCM of denominators $5$, $8$, $3$, and $7$ is $840$.
Converting the given fractions to their equivalent fractions.
$\frac {3}{5} = \frac {3 \times 168}{5 \times 168} = \frac {504}{840}$
$\frac {6}{8} = \frac {6 \times 105}{8 \times 105} = \frac {630}{840}$
$\frac {2}{3} = \frac {2 \times 280}{3 \times 280} = \frac {560}{840}$
$\frac {4}{7} = \frac {4 \times 120}{7 \times 120} = \frac {480}{840}$
Arranging 504, 630, 560, and 480 in descending order
$630 \gt 560 \gt 504 \gt 480$, therefore, $ \frac {6}{8} \gt \frac {2}{3} \gt \frac {3}{5} \gt \frac {4}{7}$
Arranging Fractions With Same Numerators
As seen above, when the numerators of the fractions are equal, then the fraction with a lesser denominator is greater than the fraction with a greater denominator, therefore, you can arrange these fractions by comparing their denominators.
Steps to Arrange Fractions With Same Numerators
The following steps are used to compare with the same numerators.
Step 1: Compare the denominators
Step 2: Arrange the fractions in descending order of their denominators to arrange them in ascending order and vice-versa
Examples
Ex 1: Arrange the fractions in ascending order $ \frac {2}{9}$, $ \frac {2}{4}$, $ \frac {2}{5}$, $ \frac {2}{6}$, $ \frac {2}{8}$ in ascending order.
Observe that the numerators of the fractions are equal.
Arranging the denominators 9, 4, 5, 6, and 8 in descending order, we get $9 \lt 8 \lt 6 \lt 5 \lt 4$.
Therefore, the fractions in ascending order will be $ \frac {2}{9} \lt \frac {2}{8} \lt \frac {2}{6} \lt \frac {2}{5} \lt \frac {2}{4}$.
Ex 2: Arrange the fractions in descending order $ \frac {4}{7}$, $ \frac {4}{19}$, $ \frac {4}{14}$, $ \frac {4}{17}$ and $ \frac {4}{8}$ in descending order.
Here also the numerators of the fractions are equal.
Arranging the denominators $7$, $19$, $14$, $17$, and $8$ in ascending order, we get $7 \lt 8 \lt 14 \lt 17 \lt 19$.
Therefore $ \frac {4}{7} \gt \frac {4}{8} \gt \frac {4}{14} \gt \frac {4}{17} \gt \frac {4}{19}$.
Conclusion
A comparison of fractions is needed while solving many types of problems. Depending on the type of fractions you are comparing, a suitable method is required to compare the fractions.
Problems
- Select the greater fraction among the given pairs of fractions
- $\frac {8}{15}$ and $\frac {8}{17}$
- $\frac {2}{9}$ and $\frac {7}{9}$
- $\frac {15}{32}$ and $\frac {21}{56}$
- $\frac {5}{21}$ and $\frac {5}{42}$
- $\frac {9}{11}$ and $\frac {5}{11}$
- $\frac {14}{29}$ and $\frac {4}{12}$
- Arrange the following fractions in ascending order
- $\frac {2}{7}$, $\frac {4}{11}$, $\frac {1}{9}$ and $\frac {5}{6}$
- $\frac {3}{5}$, $\frac {1}{4}$, $\frac {6}{7}$ and $\frac {3}{11}$
- Arrange the following fractions in descending order
- $\frac {5}{13}$, $\frac {2}{7}$, $\frac {1}{4}$, and $\frac {6}{7}$
- $\frac {4}{9}$, $\frac {7}{15}$, $\frac {2}{8}$, and $\frac {4}{11}$
