Definition

A complete or whole circle is taken as 1 and parts of the circles are represented as fractions. For example, if a circle is divided into 8 equal parts, each of the parts represents the fraction 1/8. Three parts of such a circle would represent 3/8 and on. Here we are dealing with a type of problems, where fractions representing certain parts in a circle are given and we are required to find the fraction representing the remaining unknown part of the circle. To solve such problems, we add up the fractions representing the fractional parts and then subtract the sum from 1, the whole circle. The result gives the fraction representing the unknown fractional part of the circle.

Problem 1

How much of the circle is unshaded? Write your answer as a fraction in simplest form.

Solution

Step 1:

First we find what total part of figure is shaded.

14 + 47 = 728 + 1628 = (7+16)28 = 2328

Step 2:

To find the fraction of the figure that is unshaded we subtract the result we got (2328) from 1.

1 2328 = 2828 2328 = (28−2328 = 528

So, the fraction of the figure that is unshaded is 528.

Problem 2

How much of the circle is shaded? Write your answer as a fraction in simplest form.

Solution

Step 1:

First we figure out how much of the figure is unshaded.

15 + 13 = 315 + 515 = (3+5)15 = 815

Step 2:

To find the fraction of the figure that is unshaded we subtract the result we got (815) from 1.

1 815 = 1515 815 = (15−8)15 = 715

So, the fraction of the figure that is shaded is 715.

Problem 1

Jamie bought a box of fruit weighing 325 kilograms. If she bought a second box that weighed 713 kilograms, what is the combined weight of both boxes?

Solution

Step 1:

Weight of the first box of fruit = 325 kilograms

Weight of the second box of fruit = 713 kilograms

The combined of the two boxes of fruit = 325 + 713 = 175 + 223

Step 2:

The denominators are different. So the LCD of the fractions or LCM of denominators 3 and 5 is 15.

Rewriting to get equivalent fractions with LCD as denominator

17×35×3 + 22×53×5 = 5115 + 11015 = (51+110)15 = 16115 = 101115

Problem 2

During the weekend, Nancy spent a total 513 hours studying. If she spent 314 hours studying on Saturday, how long did she study on Sunday?

Solution

Step 1:

Time spent studying on the weekend = 513 hours

Time spent studying on Saturday = 314 hours

Time spent studying on Sunday =

Time spent studying on the weekend − Time spent studying on Saturday

= 513 − 314 = 163 − 134

Step 2:

LCD of the fractions or the LCM of the denominators 3 and 4 is 12

Rewriting to get equivalent fractions with LCD as denominator

16×43×4 − 13×34×3 = 6412 − 3912 = 64−3912 = 2512 = 2112 hours

So, the time spent studying on Sunday = 2112 hours

Problem 3

Marcos bought apples that weighed 623 kilograms. If he gave away 315 kilograms of apples to his friends, how many kilograms of apples does he have left?

Solution

Step 1:

Weight of the apples bought = 623 kilograms

Weight of the apples given to friends = 315 kilograms

Weight of the apples left =

Weight of the apples bought Weight of the apples given to friends

= 623 − 315 = 203 − 165

Step 2:

LCD of the fractions or LCM of the denominators 3 and 5 is 15

Rewriting to get equivalent fractions with LCD as denominator

20×53×5 − 16×35×3 = 10015 − 4815 = 100−4815 = 5215 = 3715 kilograms

So, the weight of the apples left = 3715 kilograms

Definition

When the denominators of any fractions are unequal or are different those fractions are called unlike fractions.

Operations like addition and subtraction cannot be done directly on unlike fractions.

These unlike fractions are first converted into like fractions by finding the least common denominator of these fractions and rewriting the fractions into equivalent fractions with same denominators (LCD)Adding unlike fractions − Formula

When fractions with different or unlike fractions are to be added, first the least common denominator of the fractions is found. The equivalent fractions of given fractions are found with LCD as the common denominator. The numerators are now added and the result is put over the LCD to get the sum of fractions.

• We find the least common denominator of all the fractions.
• We rewrite the fractions to have the denominators equal to the LCD obtained in first step .
• We add the numerators of all the fractions keeping the denominator value equal to the LCD obtained in first step.
• We then express the fraction in lowest terms.

Subtracting unlike fractions − Formula

When fractions with different or unlike fractions are to be subtracted, first the least common denominator of the fractions is found. The equivalent fractions of given fractions are found with LCD as the common denominator. The numerators are now subtracted and the result is put over the LCD to get the difference of the given fractions.

• We find the least common denominator of all the fractions.
• We rewrite the fractions to have the denominators equal to the LCD obtained in step 1.
• We subtract the numerators of all the fractions keeping the denominator value equal to the LCD obtained in step 1.
• We express the fraction in lowest terms.

Problem 1

Add 15 + 27

Solution

Step 1:

Add 15 + 27

Here the denominators are different. As 5 and 7 are prime the LCD is their product 35.

Step 2:

Rewriting

15 + 27 = (1×7)(5×7) + (2×5)(7×5) = 735 + 1035

Step 3:

As the denominators have become equal

735 + 1035 = (7+10)35 = 1735

Step 4:

So, 15 + 27 = 1735

Problem 2

Subtract 215 − 110

Solution

Step 1:

Subtract 215 − 110

Here the denominators are different. The LCM of 10 and 15 is 30.

Step 2:

Rewriting

215 − 110 = (2×2)(15×2) − (1×3)(10×3) = 430 − 330

Step 3:

As the denominators have become equal

430 − 330 = (4−3)30 = 130

Step 4:

So, 215 − 110 = 130

Definition

A unit fraction is a fraction where the numerator is always one and the denominator is a positive integer. Addition or subtraction of unit fractions can be of two types; one, where the denominators are same; two, where the denominators are different.

Rules for Addition of unit fractions

• When the unit fractions have like denominators, we add the numerators and put the result over the common denominator to get the answer.
• When the unit fractions have unlike or different denominators, we first find the LCD of the fractions. Then we rewrite all unit fractions to equivalent fractions using the LCD as the denominator. Now that all denominators are alike, we add the numerators and put the result over the common denominator to get the answer.

Rules for Subtraction of unit fractions

• When the unit fractions have like denominators, we subtract the numerators and put the result over the common denominator to get the answer.
• When the unit fractions have unlike or different denominators, we first find the LCD of the fractions. Then we rewrite all unit fractions to equivalent fractions using the LCD as the denominator. Now that all denominators are alike, we subtract the numerators and put the result over the common denominator to get the answer.

Problem 1

Add 13 + 19

Solution

Step 1:

Add 13 + 19

Here the denominators are different. As 9 is a multiple of 3, the LCD is 9 itself.

Step 2:

Rewriting

13 + 19 = (1×3)(3×3) + 19 = 39 + 19

Step 3:

As the denominators have become equal

39 + 19 = (3+1)9 = 49

Step 4:

So, 13 + 19 = 49

Problem 2

Subtract 19 − 112

Solution

Step 1:

Subtract 19 − 112

Here the denominators are different. The LCD of the fractions is 36.

Step 2:

Rewriting

19 − 112 = (1×4)(9×4) − (1×3)(12×3) = 436 − 336

Step 3:

As the denominators have become equal

436 − 336 = (4−3)36 = 136

Step 4:

So, 19 − 112 = 136

Definition

When we add or subtract fractions, their denominators need to be same or common. If they are different, we need to find the LCD (least common denominator) of the fractions before we add or subtract.

To find the LCD of the fractions, we find the least common multiple (LCM) of their denominators. LCD can be found by two methods. In the first method, LCD of two or more fractions is found as the smallest of all the possible common denominators.In second method, we find the prime factors of the denominators. Then we look for the most occurrence of each of those prime factors and then take their product. This gives the LCD of the fractions.

Formula 1

Here is how to find out LCD of any two fractions; for example 1/3 and 1/6:

Their denominators are 3 and 6 and the multiples of 3 and 6 are

List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, …

List the multiples of 6: 6, 12, 18, 24, …

The common multiples are 6, 12, 18…The least among these common multiples is 6. So, 6 is the Least Common Denominator of 1/3 and 1/6.

Formula 2

Here is how to find out LCD of any two fractions; for example 1/8 and 7/12:

The denominators of the fractions are 8 and 12

Their prime factorizations are

8 = 2 2 2

12 = 2 2 3

The most occurrences of the primes 2 and 3 are 2 2 2 (in 8) and 3 (in 12).

Their product is 2 2 2 3 = 24

So, 24 is the LCD of these two fractions.

Problem 1

Find the LCD of 38, 512

Solution

Step 1:

Since the denominators of the fractions are different, we need to find the LCD of the fractions.

The denominators of the fractions are 8 and 12.

Step 2:

To find their LCD, we find their multiples

8: 8, 16, 24, 32, 40, 48…

12: 12, 24, 36, 48,….

Step 3:

The common multiples of 8 and 12 are 24, 48….

Step 4:

The least of the common multiples is 24. So, 24 is the LCD of these two fractions.

Problem 2

Find the LCD of 34, 79

Solution

Step 1:

Since the denominators of the fractions are different, we need to find the LCD of the fractions.

The denominators of the fractions are 4 and 9.

Step 2:

To find their LCD, we find their prime factorization.

4 = 2 2

9 = 3 3

Step 3:

The most occurrences of the primes 2 and 3 are 2 2 (in 4) and 3 3 (in 9). Their product is 2 2 3 3 = 36

Step 4:

So 36 is the LCD of these two fractions.

Adding like fractions and simplification − Formula

If fractions with same denominators are to be added, we add the numerators only and keep the same denominator. If necessary, we simplify the resulting fraction to lowest terms.

• Sum of the fractions = ac + bc = \frac{(a + b)}{c}, where a, b and c are any three real numbers.

Subtracting like fractions and simplification − Formula

If fractions with same denominators are to be subtracted, we subtract the numerators only and keep the same denominator. If necessary, we simplify the resulting fraction to lowest terms.

• Difference of the fractions = ac − bc = (a−b)c, where a, b and c are any three real numbers.

Problem 1

Add 38 + 18

Solution

Step 1:

Add 38 + 18

Here, the denominators are the same 8. Since this is an addition operation,

We add the numerators 3 + 1 = 4 and put the result 4 over the common denominator to get the answer.

So 38 + 18 = (3+1)8 = 48

Step 2:

Reducing the fraction to lowest terms

48 = 12

So, 38 + 18 = 12

Problem 2

Subtract 56 16

Solution

Step 1:

Subtract 56 16

Here, the denominators are same 6. Since this is a subtraction operation, we subtract the numerators, 5 1 = 4 and put the result 4 over the common denominator 6.

So 56 16 = (5−1)6 = 46

Step 2:

Simplifying to the lowest terms,

46 = 23

So, 56 16 = 23

Definition

Fractions that have exact same denominators are called like fractions.

Fractions such as 15 and 45 are like fractions because they have a common denominator 5.

In other words, fractions with like denominators are categorized as like fractions. Performing any mathematical operations on like fractions is comparatively easier as we can make use of the common denominator for fraction operations like addition and subtraction.

Adding like fractions − Formula

If fractions with same denominators are to be added, we need to add the numerators only and keep the same denominator.

• We add the numerators.
• We keep the common denominator.
• Then the Sum of the fractions = (Sum−of−the−Numerators)(Common−Denominator)
• Sum of the fractions = ac + bc = \frac{(a + b)}{c}, where a, b and c are any three real numbers.

Subtracting like fractions − Formula

If fractions with same denominators are to be subtracted, we need to subtract the numerators only and keep the same denominator.

• We subtract the numerators.
• We keep the common denominator.
• Then the Difference of the fractions = (Difference−of−the−Numerators)(Common−Denominator)
• Difference of the fractions = ac − bc = (a−b)c, where a, b and c are any three real numbers

Problem 1

Add 37 + 27

Solution

Step 1:

Here, the denominators are the same 7. We add the numerators 3 + 2 = 5 and put the result 5 over the common denominator 7 to get the answer.

37 + 27 = (3+2)7 = 57

Step 2:

So, 37 + 27 = 57

Problem 2

Subtract 56 46

Solution

Step 1:

Here, the denominators are the same 6. We subtract the numerators; 5 4 = 1 and put the result 1 over the common denominator to get the answer.

56 46 = (5−4)6 = 16

Step 2:

So, 56 46 = 16

This tutorial provides comprehensive coverage of adding and subtracting of fractions based on Common Core (CCSS) and State Standards and its prerequisites. Students can navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. This simple tutorial uses appropriate examples to help you understand adding and subtracting of fractions in a general and quick way.

Audience

This tutorial has been prepared for beginners to help them understand the basics of adding and subtracting of fractions. After completing this tutorial, you will find yourself at a moderate level of expertise in adding and subtracting of fractions, from where you can advance further.

Prerequisites

Before proceeding with this tutorial, you need a basic knowledge of elementary math concepts such as number sense, addition, subtraction, multiplication, division, whole numbers, fractions, equivalent fractions, least common denominator and so on.