• Home
• /
• Blog
• /
• Reducing Fractions – Lowest Form of A Fraction

# Reducing Fractions – Lowest Form of A Fraction

July 30, 2022 While performing calculations on fractions, you are required to find the lowest form of a fraction because it makes the calculations simple and easy. There are three commonly used methods of reducing fractions to the lowest terms. Let’s start finding out what is the lowest form of fraction and how to arrive at it.

## Lowest Form of Fraction

The lowest form also called the simplest form of a fraction is the fraction, whose numerator and denominator are relatively prime. It means the numerator and the denominator of the fraction do not have any common factor between them, apart from 1.

For example lowest or simplest form of the fraction $\frac{42}{108}$ is $\frac {2}{3}$. Here the numbers $7$ and $18$ are called relatively prime numbers because there are no common factors between $7$ and $18$ other than $1$.

Note: Two numbers are called relatively prime (or coprime) if there is no number greater than $1$ that divides them both (that is, their greatest common divisor is $1$).

## Why Reducing Fractions To Lowest Form Required?

When you solve problems based on fractions, you perform operations like addition and subtraction, or multiplication and division. Performing these operations will be easy and less time-consuming if the fractions are in their lowest or simplest form.

Though we reduce the fractions to simplify, their value remains unchanged. The reduced fraction is equivalent to the original fraction. In fact, the original fraction and the reduced fractions form a pair of equivalent fractions

For example, $\frac {1}{4}$ is the reduced form of $\frac {3}{12}$, and there is no difference in the value represented by these two fractions. Both represent the same value. Maths can be really interesting for kids

## Reducing Fractions To Lowest Terms

Reducing fractions means converting a fraction to a form where the numerator and denominator do not have any factor in common between them other than $1$.

When the numerator and the denominator of a fraction have only $1$ as a factor, then the fraction is in its lowest or simplest form. A fraction in the lowest form cannot be reduced further.

For example $\frac {2}{3}$, $\frac {4}{11}$, $\frac{8}{21}$ are in their lowest or simplest form.

Note: $1$ is a factor of every number.

## Methods of Reducing Fractions To Lowest Terms

Reducing a fraction means making a fraction as simple as possible. In order to find the reduced form of fractions, you have to get such values of the numerator and the denominator that are relatively prime (co-prime) to each other.

The following are the methods of reducing fractions to the lowest terms.

• Equivalent Fractions Method
• Prime Factorization Method
• Highest Common Factor Method

### Equivalent Fractions Method

Equivalent fractions are the fractions that have the same value irrespective of their numerators and denominators. In this method, we try to find the common factors and cancel out those from the numerator and denominator thus reducing the fraction to its lowest form.

Is your child struggling with Maths? We can help!
Country
• Afghanistan 93
• Albania 355
• Algeria 213
• American Samoa 1-684
• Andorra 376
• Angola 244
• Anguilla 1-264
• Antarctica 672
• Antigua & Barbuda 1-268
• Argentina 54
• Armenia 374
• Aruba 297
• Australia 61
• Austria 43
• Azerbaijan 994
• Bahamas 1-242
• Bahrain 973
• Belarus 375
• Belgium 32
• Belize 501
• Benin 229
• Bermuda 1-441
• Bhutan 975
• Bolivia 591
• Bosnia 387
• Botswana 267
• Bouvet Island 47
• Brazil 55
• British Indian Ocean Territory 246
• British Virgin Islands 1-284
• Brunei 673
• Bulgaria 359
• Burkina Faso 226
• Burundi 257
• Cambodia 855
• Cameroon 237
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• Chile 56
• China 86
• Christmas Island 61
• Cocos (Keeling) Islands 61
• Colombia 57
• Comoros 269
• Congo - Brazzaville 242
• Congo - Kinshasa 243
• Cook Islands 682
• Costa Rica 506
• Croatia 385
• Cuba 53
• Cyprus 357
• Czech Republic 420
• Denmark 45
• Djibouti 253
• Dominica 1-767
• Egypt 20
• Equatorial Guinea 240
• Eritrea 291
• Estonia 372
• Ethiopia 251
• Falkland Islands 500
• Faroe Islands 298
• Fiji 679
• Finland 358
• France 33
• French Guiana 594
• French Polynesia 689
• French Southern Territories 262
• Gabon 241
• Gambia 220
• Georgia 995
• Germany 49
• Ghana 233
• Gibraltar 350
• Greece 30
• Greenland 299
• Guam 1-671
• Guatemala 502
• Guernsey 44
• Guinea 224
• Guinea-Bissau 245
• Guyana 592
• Haiti 509
• Heard & McDonald Islands 672
• Honduras 504
• Hong Kong 852
• Hungary 36
• Iceland 354
• India 91
• Indonesia 62
• Iran 98
• Iraq 964
• Ireland 353
• Isle of Man 44
• Israel 972
• Italy 39
• Jamaica 1-876
• Japan 81
• Jersey 44
• Jordan 962
• Kazakhstan 7
• Kenya 254
• Kiribati 686
• Kuwait 965
• Kyrgyzstan 996
• Laos 856
• Latvia 371
• Lebanon 961
• Lesotho 266
• Liberia 231
• Libya 218
• Liechtenstein 423
• Lithuania 370
• Luxembourg 352
• Macau 853
• Macedonia 389
• Malawi 265
• Malaysia 60
• Maldives 960
• Mali 223
• Malta 356
• Marshall Islands 692
• Martinique 596
• Mauritania 222
• Mauritius 230
• Mayotte 262
• Mexico 52
• Micronesia 691
• Moldova 373
• Monaco 377
• Mongolia 976
• Montenegro 382
• Montserrat 1-664
• Morocco 212
• Mozambique 258
• Myanmar 95
• Namibia 264
• Nauru 674
• Nepal 977
• Netherlands 31
• New Caledonia 687
• New Zealand 64
• Nicaragua 505
• Niger 227
• Nigeria 234
• Niue 683
• Norfolk Island 672
• North Korea 850
• Northern Mariana Islands 1-670
• Norway 47
• Oman 968
• Pakistan 92
• Palau 680
• Palestine 970
• Panama 507
• Papua New Guinea 675
• Paraguay 595
• Peru 51
• Philippines 63
• Pitcairn Islands 870
• Poland 48
• Portugal 351
• Puerto Rico 1
• Qatar 974
• Romania 40
• Russia 7
• Rwanda 250
• Samoa 685
• San Marino 378
• Saudi Arabia 966
• Senegal 221
• Serbia 381 p
• Seychelles 248
• Sierra Leone 232
• Singapore 65
• Slovakia 421
• Slovenia 386
• Solomon Islands 677
• Somalia 252
• South Africa 27
• South Georgia & South Sandwich Islands 500
• South Korea 82
• South Sudan 211
• Spain 34
• Sri Lanka 94
• Sudan 249
• Suriname 597
• Svalbard & Jan Mayen 47
• Swaziland 268
• Sweden 46
• Switzerland 41
• Syria 963
• Sao Tome and Principe 239
• Taiwan 886
• Tajikistan 992
• Tanzania 255
• Thailand 66
• Timor-Leste 670
• Togo 228
• Tokelau 690
• Tonga 676
• Tunisia 216
• Turkey 90
• Turkmenistan 993
• Turks & Caicos Islands 1-649
• Tuvalu 688
• U.S. Outlying Islands
• U.S. Virgin Islands 1-340
• UK 44
• US 1
• Uganda 256
• Ukraine 380
• United Arab Emirates 971
• Uruguay 598
• Uzbekistan 998
• Vanuatu 678
• Vatican City 39-06
• Venezuela 58
• Vietnam 84
• Wallis & Futuna 681
• Western Sahara 212
• Yemen 967
• Zambia 260
• Zimbabwe 263
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

### Steps For Equivalent Fractions Method

These are the steps to reduce fractions by the equivalent fractions method.

Step 1: Find any common factor of the numerator and the denominator

Step 2: Divide the numerator and denominator by the common factor

Step 3: Repeat steps 2 and 3 until there are no more common factors other than 1

### Examples

Ex 1: $\frac {84}{126}$

In the given fraction both $84$ and $126$ are even numbers, so $2$ is a common factor.

$\frac {84}{126}$ = $\frac {42 \times 2}{63 \times 2}$

Canceling out $2$ from both numerator and denominator, we get $\frac {42}{63}$

Still, the fraction is not in the lowest form, as both the numerator and denominator are divisible by $3$.

$\frac {42}{63}$ = $\frac {14 \times 3}{21 \times 3}$

Canceling out $3$ from both numerator and denominator, we get $\frac {14}{21}$

Now also, the numerator and the denominator have a common factor $7$.

$\frac {14}{21}$ = $\frac {2 \times 7}{3 \times 7}$

Again removing $7$ from the numerator and the denominator, we get $\frac {2}{3}$

The numbers $2$ and $3$ are coprime and hence cannot be reduced further. Therefore, the lowest form of $\frac {84}{126}$ is $\frac {2}{3}$.

Ex 2: $\frac {315}{735}$

$\frac {315}{735} = \frac {63 \times 5}{147 \times 5}$

Canceling $5$, we get $\frac {63}{147}$

Again $\frac {63}{147} = {21 \times 3}{49 \times 3}$

Canceling $3$, the fraction further reduces to $\frac {21}{49}$

$\frac {21}{49} = \frac {3 \times 7}{7 \times 7}$

Finally canceling $7$, the fraction reduces to $\frac {3}{7}$

So, $\frac {3}{7}$ is the lowest form of the fraction $\frac {315}{735}$.

### Prime Factorization Method

Prime factorization is a way of expressing any composite number as a product of its prime factors. In this method, the numerator and the denominator are converted to their prime factorization respectively.

### Steps For Prime Factorization Method

These are the steps to reduce fractions by the prime factorization method.

Step 1: Find the prime factorization of the numerator and the denominator

Step 2: Cancel out the common prime factors of the numerator and the denominator

Step 3: The remaining numbers in the numerator and the denominator is the reduced fraction

### Examples

Ex 1: $\frac {72}{108}$

Prime factorization of $72$ and $108$ in $\frac {72}{108}$ is $\frac {2 \times 2 \times 2 \times 3 \times 3}{2 \times 2 \times 3 \times 3 \times 3}$.

Canceling two $2$s and two $3$s from the numerator and the denominator, the fraction $\frac {72}{108}$ reduces to $\frac {2}{3}$.

Thus the lowest form of $\frac {72}{108}$ is $\frac {2}{3}$.

Ex 2: $\frac {180}{252}$

Prime factorization of $180$ and $252$ in $\frac {180}{252}$ is $\frac {2 \times 2 \times 3 \times 3 \times 5}{2 \times 2 \times 3 \times 3 \times 7}$.

Canceling two $2$s and two $3$s from the numerator and the denominator, the fraction $\frac {180}{252}$ reduces to $\frac {5}{7}$.

Thus the lowest form of $\frac {180}{252}$ is $\frac {5}{7}$.

### Highest Common Factor Method

The highest common factor or HCF of two or more numbers is the greatest number that divides exactly the given numbers. Using HCF, you can reduce a fraction to its lowest or simplest form.

### Steps For Highest Common Factor Method

These are the steps to reduce fractions by the HCF method.

Step 1: Find the highest common factor (HCF) of the numerator and the denominator

Step 2: Divide the numerator and denominator by the HCF. The fraction so obtained is the reduced fraction

### Examples

Ex 1: $\frac {84}{174}$

HCF of $84$ and $174$ is $6$.

Dividing both the numerator and the denominator by $6$.

$\frac {84 \div 6}{174 \div 6}$ = $\frac {14}{29}$

The lowest form of $\frac {84}{174}$ is $\frac {14}{29}$.

Ex 2: $\frac {48}{152}$

HCF of $48$ and $152$ is $8$.

Dividing both the numerator and the denominator by $8$.

$\frac {48 \div 8}{152 \div 8}$ = $\frac {6}{19}$

The lowest form of $\frac {48}{152}$ is $\frac {6}{19}$.

## Conclusion

Reducing a fraction to its lowest form generally forms the first step in calculations involving fractions. You can use any of the three methods, viz. equivalent fractions method, prime factorization method, or HCF method to reduce any fraction.

## Practice Problems

1. Find the lowest form of the fractions using the equivalent fractions method
• $\frac {21}{56}$
• $\frac {48}{64}$
• $\frac {44}{72}$
2. Find the lowest form of the fractions using the prime factorization method
• $\frac {120}{360}$
• $\frac {28}{140}$
• $\frac {48}{168}$
3. Find the lowest form of the fractions using the HCF method
• $\frac {7}{28}$
• $\frac {9}{72}$