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# Subtraction of Like Fractions

Like denominators

Two or more fractions are said to have like denominators if they have same denominators.

For example: $$\dfrac{3}{5},\;\dfrac{2}{5}$$

Here, like denominator $$=5$$

• Let us take an example: $$\dfrac{7}{10}-\dfrac{3}{10}$$

Here, both the fractions have the same denominator = $$10$$

Step 1: Find the difference of the numerators.

$$7-3=4$$

Step 2: Difference of numerators becomes the new numerator.

Numerator $$=4$$

Step 3: Denominator remains the same.

Denominator $$=10$$

Step 4: Write the resulting fraction.

Fraction $$=\dfrac{4}{10}$$

Step 5: Simplify the result.

• The G.C.F. (Greatest Common Factor) of $$4$$ and $$10=2$$
• Divide by $$2$$,  $$\dfrac{4\div2}{10\div2}=\dfrac{2}{5}$$

$$2$$ and $$5$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{2}{5}$$ is in its simplest form.

Thus, $$\dfrac{2}{5}$$ is our final answer.

#### Which one of the following options represents the solution of the expression $$\dfrac{23}{9}-\dfrac{20}{9}$$?

A $$\dfrac{6}{9}$$

B $$-\dfrac{1}{3}$$

C $$\dfrac{1}{3}$$

D $$-\dfrac{2}{9}$$

×

Given: $$\dfrac{23}{9}-\dfrac{20}{9}$$

Finding the difference of numerators,

$$23-20=3$$

New numerator $$=3$$

Denominator remains the same.

Denominator $$=9$$

Fraction $$=\dfrac{3}{9}$$

Simplifying the result.

The G.C.F. of $$3$$ and $$9=3$$

Divide by $$3$$,  $$\dfrac{3\div3}{9\div3}=\dfrac{1}{3}$$

$$1$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{1}{3}$$ is in its simplest form.

Hence, option (C) is correct.

### Which one of the following options represents the solution of the expression $$\dfrac{23}{9}-\dfrac{20}{9}$$?

A

$$\dfrac{6}{9}$$

.

B

$$-\dfrac{1}{3}$$

C

$$\dfrac{1}{3}$$

D

$$-\dfrac{2}{9}$$

Option C is Correct

# Subtraction of Mixed Numbers

• Mixed Number = A whole number + A proper fraction
• To understand the subtraction of mixed fractions easily, let us consider the following example: $$5\dfrac{4}{5}-3\dfrac{1}{4}$$

Step 1: Subtract the fraction part first.

$$\dfrac{4}{5}-\dfrac{1}{4}$$

The least common multiple of $$5$$ and $$4$$:

Multiples of $$5=5,\;10,\;15,\;20\,...$$

Multiples of $$4=4,\,8,\;12,\;16,\;20\,...$$

Least common multiple (L.C.M) $$=20$$

L.C.M becomes the lowest common denominator (L.C.D).

L.C.D $$=20$$

Convert each fraction into equivalent fraction with L.C.D as the denominator.

Equivalent fraction of $$\dfrac{4}{5}=\dfrac{4×4}{5×4}=\dfrac{16}{20}$$

Equivalent fraction of $$\dfrac{1}{4}=\dfrac{1×5}{4×5}=\dfrac{5}{20}$$

Subtract the equivalent fractions of $$\dfrac{4}{5}$$ and $$\dfrac{1}{4}$$.

$$\dfrac{16}{20}-\dfrac{5}{20}=\dfrac{16-5}{20}=\dfrac{11}{20}$$

Simplify the resulting fraction.

$$11$$ and $$20$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{11}{20}$$ is in its simplest form.

Step 2: Subtract the whole numbers.

$$5-3=2$$

Step 3: Write the difference of whole numbers and fractions together as a mixed number, i.e.

$$2\dfrac{11}{20}$$

Thus, $$2\dfrac{11}{20}$$ is our final answer.

#### Which one the following options represents the solution of the expression $$21\dfrac{8}{9}-18\dfrac{5}{9}$$?

A $$3\dfrac{1}{3}$$

B $$4\dfrac{2}{9}$$

C $$5\dfrac{1}{9}$$

D $$2\dfrac{1}{9}$$

×

Given: $$21\dfrac{8}{9}-18\dfrac{5}{9}$$

Subtracting the fraction part,

$$\dfrac{8}{9}-\dfrac{5}{9}=\dfrac{3}{9}$$

Simplifying the fraction obtained, $$\dfrac{3}{9}$$

The greatest common factor of $$3$$ and $$9=3$$

Divide by $$3$$$$\dfrac{3\div3}{9\div3}=\dfrac{1}{3}$$

$$1$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{1}{3}$$ is in its simplest form.

Subtracting the whole numbers,

$$21-18=3$$

Putting the difference of whole numbers and fractions together as a mixed number,

$$3\dfrac{1}{3}$$

Thus, $$3\dfrac{1}{3}$$ is our final answer.

Hence, option (A) is correct.

### Which one the following options represents the solution of the expression $$21\dfrac{8}{9}-18\dfrac{5}{9}$$?

A

$$3\dfrac{1}{3}$$

.

B

$$4\dfrac{2}{9}$$

C

$$5\dfrac{1}{9}$$

D

$$2\dfrac{1}{9}$$

Option A is Correct

# Subtraction of Like Fractions through Figures

• Subtraction of fractions having like denominators means subtraction of fractions that have the same number of total parts.
• Let us consider an example:
• Emma makes two squares and divides each of them into $$3$$ rows and $$2$$ columns as shown in figure (A) and figure (B).

• She shades $$5$$ parts of figure (A), i.e. $$\dfrac{5}{6}$$ part as shown in figure (C) and $$2$$ parts of figure (B), i.e. $$\dfrac{2}{6}$$ part as shown in figure (D).

• Now, she wants to subtract the shaded part of figure (D) from the shaded part of figure (C).

So, when she subtracts these two squares, she gets:

Thus, figure (E) shows the subtraction of the shaded part of figure (D) from the shaded part of figure (C) and represents the fraction equals $$\dfrac{3}{6}$$.

#### What is the difference of the shaded parts of the given circles?

A

B

C

D

×

In both the given figures, there are same number of equal parts, i.e. $$8$$.

Thus, we get to know that these are like fractions.

Each part of the figures represents $$\dfrac{1}{8}$$ part.

In Figure (I), there are $$6$$ shaded parts =  $$\dfrac{6}{8}$$ part

In Figure (II), there are $$2$$ shaded parts =  $$\dfrac{2}{8}$$ part

On subtracting the shaded parts of both the circles, we get:

Hence, option (A) is correct.

### What is the difference of the shaded parts of the given circles?

A
B
C
D

Option A is Correct

# Subtraction of Unlike Fractions using Models

• Subtraction of fractions having unlike denominators means subtraction of fractions that do not have the same number of total parts.
• Let us consider an example:
• John draws two different squares. He divides the first square into $$3$$ rows as shown in figure (A) and the second square into $$2$$ columns as shown in figure (B).

• He shades the first two rows of figure (A) as shown in figure (C) and the first column of figure (B) as shown in figure (D).

• Now, he wants to subtract the shaded part of figure (D) from the shaded part of figure (C).
• First, he subdivides the rows and columns of figure (C) and figure (D) in such a way that the number of parts becomes equal.

• On subtracting the subdivided shaded parts of both the squares, he gets:

Thus, figure (G) shows the subtraction of the shaded part of figure (F) from the shaded part of figure (E) and the fraction represented equals $$\dfrac{1}{6}$$.

#### Three friends are sitting in a pizza parlor and order one large pizza with $$8$$ slices. If two of them eat $$2$$ slices each, how many slices are left for the third one?

A

B

C

D

×

The total number of pizza slices equals $$8$$ which means one slice represents $$\dfrac{1}{8}$$ part.

We know that two of the friends eat $$2$$ slices each, which means total $$4$$ slices.

Now, let's calculate the number of slices left for the third friend. For this, we subtract the shaded part of figure (G) from the shaded part of figure (F).

So, the number of slices left for the third friend is $$\dfrac{4}{8}$$ part as shown in figure (I).

Hence, option (C) is correct.

### Three friends are sitting in a pizza parlor and order one large pizza with $$8$$ slices. If two of them eat $$2$$ slices each, how many slices are left for the third one?

A
B
C
D

Option C is Correct

# Subtraction of Unlike Fractions

• In order to perform subtraction of fractions having different denominators, we first need to convert them into fractions with like denominators.

• To understand it easily, let us consider the following example: $$\dfrac{3}{2}-\dfrac{7}{8}$$

Step 1: Find the least common multiple of denominators.

The least common multiple (L.C.M) of $$2$$ and $$8$$

Multiples of $$2=2,\;4,\;6,\;8\;...$$

Multiples of $$8=8,\;16\,...$$

L.C.M of $$2$$  and $$8=8$$

Step 2: L.C.M becomes the lowest common denominator (L.C.D).

L.C.D $$=8$$

Step 3: Convert the fractions into equivalent fractions with L.C.D as the denominator.

• Equivalent fraction of $$\dfrac{3}{2}=\dfrac{3×4}{2×4}=\dfrac{12}{8}$$

$$\dfrac{7}{8}$$ already has $$8$$ as its denominator.

$$\therefore$$ It does not need to be converted.

Step 4: Rewrite the expression with fractions having same denominators.

$$\dfrac{12}{8}-\dfrac{7}{8}$$

Step 5: Find the difference of numerators and it becomes the new numerator.

$$12-7=5$$

Numerator $$=5$$

Step 6: Denominator remains the same.

Denominator $$=8$$

Step 7: Write the resulting fraction.

Resulting fraction $$=\dfrac{5}{8}$$

Step 8: Simplify the result.

$$5$$ and $$8$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{5}{8}$$ is in its simplest form.

Thus, $$\dfrac{5}{8}$$ is our final answer.

#### Which one of the following options represents the solution of the expression $$\dfrac{4}{3}-\dfrac{3}{4}$$?

A $$\dfrac{1}{12}$$

B $$\dfrac{2}{6}$$

C $$\dfrac{7}{12}$$

D $$\dfrac{3}{6}$$

×

Given: $$\dfrac{4}{3}-\dfrac{3}{4}$$

The least common multiple of denominators:

Multiples of $$3=3,\;6,\;9,\;12\,...$$

Multiples of $$4=4,\;8,\;12...$$

L.C.M of $$3$$ and $$4=12$$

L.C.M becomes the lowest common denominator (L.C.D).

L.C.D $$=12$$

Converting the fractions into equivalent fractions with L.C.D as the denominator,

Equivalent fraction of $$\dfrac{4}{3}=\dfrac{4×4}{3×4}=\dfrac{16}{12}$$

Equivalent fraction of $$\dfrac{3}{4}=\dfrac{3×3}{4×3}=\dfrac{9}{12}$$

Rewriting the expression with fractions having same denominators,

$$\dfrac{16}{12}-\dfrac{9}{12}$$

Finding the difference of numerators and it becomes the new numerator,

$$16-9=7$$

Numerator $$=7$$

Denominator remains the same.

Denominator $$=12$$

Writing the resulting fraction,

Fraction $$=\dfrac{7}{12}$$

Simplifying the result,

$$7$$ and $$12$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{7}{12}$$ does not need to be changed.

Hence, option (C) is correct.

### Which one of the following options represents the solution of the expression $$\dfrac{4}{3}-\dfrac{3}{4}$$?

A

$$\dfrac{1}{12}$$

.

B

$$\dfrac{2}{6}$$

C

$$\dfrac{7}{12}$$

D

$$\dfrac{3}{6}$$

Option C is Correct

# Subtraction of Mixed Numbers by Renaming

• To rename the mixed numbers, we find the equivalent fractions of the given fractions or the whole numbers.

For example: $$\dfrac{3}{3}$$ or $$\dfrac{5}{5}$$ or $$\dfrac{2}{2}$$ are equivalent to $$1.$$

• Let us consider the following two cases:

Case I: If we subtract a mixed number from a whole number:

For example: $$5-3\dfrac{1}{2}$$

• First, rename the whole number to a mixed number.
• To do this, follow the given steps:

Step 1: Borrow $$1$$ from $$5$$  to make it four, i.e.

$$5=4+1$$

Step 2: Take $$1$$ and change it into a fraction having $$2$$ as the denominator.

$$1=\dfrac{2}{2}$$

[In $$3\dfrac{1}{2}$$, the denominator is $$2]$$

$$[\,\therefore$$ We take $$1=\dfrac{2}{2}]$$

Step 3: Rewrite the problem.

$$5=4+1$$

$$=4+\dfrac{2}{2}$$

$$=4\dfrac{2}{2}$$

Step 4: Subtract the mixed numbers having same denominators.

$$\underline{\;\;\;\;4\dfrac{2}{2}\\-3\dfrac{1}{2}}$$

Subtract the fraction part, i.e.

$$\dfrac{2}{2}-\dfrac{1}{2}=\dfrac{1}{2}$$

Simplify the resulting fraction.

$$1$$ and $$2$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{1}{2}$$ is in its simplest form.

Subtract the whole numbers, i.e.

$$4-3=1$$

Put the difference of whole numbers and fractions together as a mixed number, i.e.

$$1\dfrac{1}{2}$$

Thus, $$1\dfrac{1}{2}$$ is our final answer.

Case II: If fraction part of the subtrahend is greater than the fraction part of minuend:

For example: $$\underline{\;\;\;\;7\dfrac{1}{8}\;\;(\text{minuend})\\-4\dfrac{5}{8}\;\;\text{(subtrahend)}}$$

• Since $$\dfrac{5}{8}>\dfrac{1}{8}$$, so we cannot subtract $$\dfrac{5}{8}$$ from $$\dfrac{1}{8}$$.
• To do this subtraction, follow the given steps:

Step 1: First rename the top mixed number, i.e.

$$7\dfrac{1}{8}=7+\dfrac{1}{8}$$

Borrow $$1$$ from $$7$$ to make it $$6,$$ i.e.

$$7=6+1$$

Take $$1$$ and change it into a fraction having $$8$$ as the denominator, i.e.

$$1=\dfrac{8}{8}$$

[ In $$7\dfrac{1}{8}$$ and $$4\dfrac{5}{8}$$, the denominators are $$8]$$

$$[\,\therefore$$ We take $$1=\dfrac{8}{8}]$$

Thus, $$7\dfrac{1}{8}=6+\dfrac{8}{8}+\dfrac{1}{8}$$

$$=6+\dfrac{8+1}{8}$$

$$=6+\dfrac{9}{8}$$

$$=6\dfrac{9}{8}$$

Step 2: Rewrite the problem and subtract the mixed numbers with same denominators.

$$\underline{\;\;\;\;6\dfrac{9}{8}\\-4\dfrac{5}{8}}$$

Subtract the fraction part, i.e.

$$\dfrac{9}{8}-\dfrac{5}{8}=\dfrac{9-5}{8}=\dfrac{4}{8}$$

Simplify the resulting fraction, i.e. $$\dfrac{4}{8}$$

The greatest common factor of $$4$$ and $$8=4$$

Divide by $$4$$$$\dfrac{4\div4}{8\div4}=\dfrac{1}{2}$$

$$1$$ and $$2$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{1}{2}$$ is in its simplest form.

Subtract the whole numbers, i.e.

$$6-4=2$$

Put the difference of whole numbers and fractions together as a mixed number, i.e.

$$2\dfrac{1}{2}$$

Thus, $$2\dfrac{1}{2}$$ is our final answer.

#### Which one of the following options represents the solution of the expression $$10\dfrac{1}{3}-4\dfrac{2}{3}$$?

A $$6\dfrac{1}{3}$$

B $$5\dfrac{2}{3}$$

C $$4\dfrac{2}{3}$$

D $$4\dfrac{1}{3}$$

×

Given: $$10\dfrac{1}{3}-4\dfrac{2}{3}$$

Subtracting the fraction part.

Since $$\dfrac{2}{3}>\dfrac{1}{3}$$

So, we cannot subtract $$\dfrac{2}{3}$$ from $$\dfrac{1}{3}$$.

Renaming the first mixed number,

$$10\dfrac{1}{3}=10+\dfrac{1}{3}$$

Borrowing $$1$$ from $$10$$ to make it $$9,$$ i.e.

$$10=9+1$$

Taking $$1$$ and changing it into a fraction having $$3$$ as the denominator, i.e.

$$1=\dfrac{3}{3}$$

[In $$10\dfrac{1}{3}$$ and $$4\dfrac{2}{3}$$, the denominators are $$3]$$

$$[\,\therefore$$ We take $$1=\dfrac{3}{3}]$$

Thus, $$10\dfrac{1}{3}=9+\dfrac{3}{3}+\dfrac{1}{3}$$

$$=9+\dfrac{3+1}{3}$$

$$=9+\dfrac{4}{3}$$

$$=9\dfrac{4}{3}$$

Rewriting the problem:

$$9\dfrac{4}{3}-4\dfrac{2}{3}$$

Subtracting the fractions:

$$\dfrac{4}{3}-\dfrac{2}{3}=\dfrac{4-2}{3}=\dfrac{2}{3}$$

$$2$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore$$ $$\dfrac{2}{3}$$ is in its simplest form.

Subtracting the whole numbers:

$$9-4=5$$

Putting the difference of whole numbers and fractions together as a mixed number, i.e.

$$5\dfrac{2}{3}$$

Thus, $$5\dfrac{2}{3}$$ is our final answer.

Hence, option (B) is correct.

### Which one of the following options represents the solution of the expression $$10\dfrac{1}{3}-4\dfrac{2}{3}$$?

A

$$6\dfrac{1}{3}$$

.

B

$$5\dfrac{2}{3}$$

C

$$4\dfrac{2}{3}$$

D

$$4\dfrac{1}{3}$$

Option B is Correct

# Subtraction of Fractions using a Number Line

• Consider a fraction as a segment on a number line.
• Subtraction of fractions on a number line is represented by taking away the part which we want to subtract.

• Before performing subtraction on the number line, we have to make sure that each fraction has the same denominator.

For example:  Represent $$\dfrac {7}{5}-\dfrac {3}{5}$$ on a number line.

Step 1:  Represent $$\dfrac {7}{5}$$ on the number line.

Step 2 : Represent $$\dfrac {3}{5}$$ on the number line.

Step 3 : Subtraction means moving 3 units in backward direction, each of length $$\dfrac {1}{5}$$, from $$\dfrac {7}{5}$$.

$$\dfrac {7}{5}-\dfrac {3}{5}=\dfrac {4}{5}$$

Hence, the result is $$\dfrac {4}{5}$$.

#### Which number line represents the solution of $$\dfrac {5}{3}-\dfrac {2}{3}$$ as point $$M$$?

A

B

C

D

×

Represent $$\dfrac {5}{3}$$ on the number line.

Represent $$\dfrac {2}{3}$$ on the number line.

Now, move 2 units in backward direction, each of length  $$\dfrac {1}{3}$$, from $$\dfrac {5}{3}$$ to subtract.

$$\dfrac {5}{3}-\dfrac {2}{3}=\dfrac {3}{3}=1$$

The resulting number line is:

Hence, option (B) is correct.

### Which number line represents the solution of $$\dfrac {5}{3}-\dfrac {2}{3}$$ as point $$M$$?

A
B
C
D

Option B is Correct