In mathematics, ordering fractions refers to the two ways we might organiz

In mathematics, ordering fractions refers to the two ways we might organize the fractions, smallest to greatest and largest to smallest. We must compare the fractions for this reason.

When comparing fractions, the denominator is checked to see if it is the same by using the LCM (least common multiple) of both denominators. After equating the denominators, we can rapidly establish the order of the fractions. The fraction with the greater numerator is presumed to be larger than the other.

In this article, the process of ordering fractions, ways to determine the order of fractions, examples, and applications in everyday life will be discussed.

Comparing two or more fractions to discover whether one is bigger or smaller than the others is known as ordering fractions.

Usually, this process is done to find the following orders of fractions

- Descending order (To arrange fractions from greater to smaller)
- Ascending order (To arrange fractions from smaller to greater)

To order fractions from least to greatest (or vice versa), follow these steps:

**Find a Common Denominator:**Determine the least common multiple (LCM) of all the denominators of the fractions you're comparing. This will be your common denominator.**Convert to Equivalent Fractions:**Using the common denominator, rewrite each fraction as an equivalent fraction with the common denominator.**Compare Numerators:**Once all fractions have the same denominator, order them by comparing their numerators. The fraction with the smallest numerator will be the least, and so on.**Use a Calculator:**If you'd like to speed up the process, consider using a least to greatest calculator. These online tools can quickly sort fractions for you. However, it's still essential to understand the process manually, especially if you're learning or teaching the concept.**Simplify if Necessary:**After ordering, if you need the fractions in their simplest form, reduce each fraction to its lowest terms.

There are two common methods used for ordering fractions:

This method involves finding a common denominator for all the fractions and then converting each fraction so they all have this common denominator. Once they all have the same denominator, you can compare the numerators to determine the order.

*Example***:** To order 1/4 and 3/5, find a common denominator, which in this case is 20. Convert the fractions to 5/20 and 12/20, respectively. Now, it's clear that 1/4 is smaller.

This method involves converting each fraction into its decimal equivalent. Once in decimal form, the numbers can be easily compared.

*Example***:** 1/4 converts to 0.25 and 3/5 converts to 0.6. Comparing the decimals, 0.25 is smaller than 0.6, so 1/4 is smaller than 3/5.

(By the method of equating denominators)

**Example 1:**Arrangethe following in descending order 8/15, 5/7, 1/7, 9/5, 3/10

**Solution:**

**Step 1:**Taking the L.C.M of all denominators of the above fractions, we have

2 |
15,7,7,5,10 |

3 |
15,7,7,5,5 |

5 |
5,7,7,5,5 |

7 |
1,7,7,1,1 |

1,1,1,1,1 |

**L.C.M = **2×3×5×7 =210

**Step 2:** To equate all the denominators i.e. 210

(8/15) × 210/210 = 112/210

(5/7) × 210/210 = 150/210

(1/7) × 210/210 = 30/210

(9/5) × 210/210 = 378/210

(3/10) × 210/210 = 63/210

**Step 3:** Arrange all fractions according to numerators like biggest to smallest, we have

378/210, 150 /210, 112/210, 63/210, 30/210

**Hence the ascending order is **9/5, 5/7, 8/15, 3/10, 1/7

**Example 2:**Arrangethe following in ascending order 1/3, 4/7, 2/9, 4/5, 7/10

**Solution:**

**Step 1:**Taking the L.C.M of all denominators of the above fractions, we have

3 |
3,7,9,5,10 |

3 |
1,7,3,5,10 |

5 |
1,7,1,5,10 |

2 |
1,7,1,1,2 |

7 |
1,7,1,1,1 |

1,1,1,1,1 |

**L.C.M = **3×3×5×2×7 = 630

**Step 2:** To equate all the denominators i.e. 630

(1/3) × 630/630 = 210/630

(4/7) × 630/630 = 360/630

(2/9) × 630/630 = 140/630

(4/5) × 630/630 = 504/630

(7/10) × 630/630 = 441/630

**Step 3:** Arrange all fractions according to numerators like lowest to greatest, we have

140 /630, 210/630, 360/630, 441/630, 504/630

**Hence the descending order is **2/9, 1/3, 4/7, 7/10, 4/5

**Example 3: **(By the method of converting the fractions into decimals)

Arrange 2/3, 1/5, and 5/6 in ascending order.

**Solution:**

**Step 1:**

Convert the fractions to decimals

Divide the numerator by the denominator using long division or a calculator to change fractions to decimals. Once all of the fractions have been converted to decimals, you may compare them to discover which one is the smallest or largest.

**For example**, let's say we want to order the fractions 2/3, 1/5, and 5/6 in decimal form:

2/3 = 0.666666...

1/5 = 0.20

5/6 = 0.833333...

Now that we have to change the fractions to decimals, we can see that the order from least to greatest is:

1/5 = 0.20

< 2/3 = 0.666666...

< 5/6 = 0.833333...

Now,

**1/5 < 2/3< 5/6**

**Example 4: **Check the order in the following fractions by the method of converting decimals.

1/5, 3/10, 2/3, 5/7, 8/11

**Solution:**

Converting the given fractions into decimals, we have

1/5 = 0.20

3/10 = 0.30

2/3 = 0.66

5/7 = 0.71

8/11 = 0.82

Clearly, we can say that the given order of fractions is **ascending** as it starts from the lowest one and ends at the greatest one.

**1/5 < 3/10 < 2/3 < 5/7 < 8/11**

In daily life, fractional thinking is incredibly helpful. Fractions usually assist us in finding solutions to our problems. The ordering-of-fractions strategy is widely used in the following situations:

**In Education:**The teaching of fractions is a key idea in mathematics. They aid in laying the groundwork for comprehending more complex mathematical ideas like decimals and percentages.**In Shopping:**Every time we go outdoors to buy, we frequently encounter sales on various items from various companies. For instance, we must utilize the notion of fractions to determine the price we must pay for a pair of shoes that is 40% off.**In Cooking:**We must balance the elements needed for a particular dish while we are cooking. Our recipe won't be tasty if the order or sequence is broken. We may serve a delectable dish in front of visitors including you after ordering the ingredients needed for the recipe. Pizza serves as the ideal illustration for how to adjust the idea of fractions as a big pizza typically comprises 8 pieces. If there are four of you, everyone will receive an equal number of 8/2 slices.**In Sports:**We are required to abide by the regulations established by the game authorities at all times. We cannot be eligible for it if we don't maintain balance in them. Fractions are utilized for this. For instance, a single bowler may only bowl 20/4 over in a T20 Cricket match. All other bowlers are also limited to a maximum of 20/4 over.**In the Medical field:**For medical purposes, the dosage of suitable medicine prescribed by a doctor varies from age to age. For example for an adult person, a specific medicine is prescribed 4/5 tablespoon as per dosage by a doctor and only 2/5 is prescribed for a child of age less than 10.

We learned about the concept of ordering fractions as well as several methods and examples in the discussion above. To find an exact answer anywhere, any method can be applied. The usage of fractions in everyday life recognizes both the significance of fractions and their broad application in terms of ordering.