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# Multiplication of Fractions with Whole Numbers

• When we have a fraction that is being added multiple times, then it can be easily solved as a product of the fraction and the number of times it is being added.
• To multiply a fraction by a whole number, we should follow the given steps:
• Let us consider an example:

$$\underbrace 9_\text{a whole number}×\underbrace{\dfrac{12}{7}}_\text{a fraction}$$

Step 1: Convert the whole number into a fraction.

• Remember that every whole number can be put over $$1$$ and its value remains the same.
• Convert $$9$$ into a fraction by putting it over $$1$$.

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

Step 2: Rewrite the problem,

$$\dfrac{9}{1}×\dfrac{12}{7}$$

Step 3: Multiply numerator by numerator and denominator by denominator,

$$\dfrac{9}{1}×\dfrac{12}{7}\; \begin {matrix} \longleftarrow\text{Numerators}\\ \longleftarrow\text{Denominators}\end {matrix}$$

$$=\dfrac{9×12}{1×7}$$

$$=\dfrac{108}{7}$$

Step 4: Simplify the result, $$\dfrac{108}{7}$$

$$108$$ and $$7$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{108}{7}$$ is in its simplest form.

Step 5: If the result is an improper fraction then convert it into a mixed number, if the problem demands.

$$108>7$$

$$\therefore\;\dfrac{108}{7}$$ is an improper fraction.

$$\dfrac{108}{7}=15\dfrac{3}{7}$$

• Thus, it is our answer.

Note: Sometimes, we will see the word "of" in a problem. The word "of" means multiply.

For example: $$\dfrac{1}{3}$$ of $$4$$

• Convert "of" to multiplication sign,

$$\dfrac{1}{3}×4$$

$$=\dfrac{1}{3}×\dfrac{4}{1}$$

$$=\dfrac{1×4}{3×1}=\dfrac{4}{3}$$

$$4$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{4}{3}$$ is in its simplest form.

Thus, $$\dfrac{4}{3}$$ is our answer.

#### Which one of the following options represents the solution of the expression, $$3×\dfrac{5}{2}$$?

A $$\dfrac{10}{3}$$

B $$\dfrac{6}{5}$$

C $$\dfrac{15}{2}$$

D $$\dfrac{5}{6}$$

×

Given: $$3×\dfrac{5}{2}$$

Converting $$3$$ into a fraction,

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

Rewriting the problem and multiplying,

$$\dfrac{3}{1}×\dfrac{5}{2}$$

$$\dfrac{\text{numerator×numerator}}{\text{denominator×denominator}}$$

$$=\dfrac{3×5}{1×2}$$

$$=\dfrac{15}{2}$$

Result $$=\dfrac{15}{2}$$

$$15$$ and $$2$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{15}{2}$$ is in its simplest form.

Thus, $$\dfrac{15}{2}$$ is our answer.

Hence, option (C) is correct.

### Which one of the following options represents the solution of the expression, $$3×\dfrac{5}{2}$$?

A

$$\dfrac{10}{3}$$

.

B

$$\dfrac{6}{5}$$

C

$$\dfrac{15}{2}$$

D

$$\dfrac{5}{6}$$

Option C is Correct

# Multiplication of a Fraction with Another Fraction

• To multiply a fraction with another fraction, we should follow the given steps:
• Let us consider an example: $$\dfrac{3}{2}×\dfrac{4}{3}$$

Step 1: Multiply numerator by numerator and denominator by denominator,

$$3×4=12\;\;\leftarrow \text{Numerator}$$

$$2×3=6\;\;\leftarrow \text{Denominator}$$

Fraction $$=\dfrac{12}{6}$$

Step 2: Simplify the fraction obtained.

• Greatest common factor of $$12$$ and $$6=6$$
• Divide $$12$$ and $$6$$ by the Greatest Common Factor,

$$\dfrac{12\div6}{6\div6}$$ $$=\dfrac{2}{1}=2$$

Step 3: If we get an improper fraction on multiplication, convert it into a mixed number.

$$2>1$$

$$\therefore$$ $$\dfrac{2}{1}$$ is an improper fraction.

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

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

#### Which one of the following options represents the solution of the expression, $$\dfrac{1}{2}×\dfrac{2}{5}$$?

A $$\dfrac{1}{5}$$

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

C $$\dfrac{1}{10}$$

D $$\dfrac{3}{10}$$

×

Given: $$\dfrac{1}{2}×\dfrac{2}{5}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{1×2}{2×5}=\dfrac{2}{10}$$

Result $$=\dfrac{2}{10}$$

Simplifying the result [Greatest Common Factor of $$2$$ and $$10$$ is $$2$$]

$$\dfrac{2\div 2}{10\div2}=\dfrac{1}{5}$$

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

$$\therefore\;\dfrac{1}{5}$$ is in its simplest form.

Hence, option (A) is correct.

### Which one of the following options represents the solution of the expression, $$\dfrac{1}{2}×\dfrac{2}{5}$$?

A

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

.

B

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

C

$$\dfrac{1}{10}$$

D

$$\dfrac{3}{10}$$

Option A is Correct

# Multiplication of a Mixed Number with a Fraction

• To multiply a mixed number with a fraction, we should follow the given steps:
• Let us consider an example: $$\dfrac{2}{5}×1\dfrac{1}{3}$$

Step 1: Convert the mixed fraction into an improper fraction.

$$1\dfrac{1}{3}=\dfrac{(3×1)+1}{3}$$$$=\dfrac{3+1}{3}=\dfrac{4}{3}$$

Step 2: Rewrite the problem.

$$\dfrac{2}{5}×\dfrac{4}{3}$$

Step 3: Multiply numerator by numerator and denominator by denominator.

$$\dfrac{2×4}{5×3}$$$$=\dfrac{8}{15}$$

Step 4: Simplify the fraction.

$$8$$ and $$15$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{8}{15}$$ is in its simplest form.

Step 5: If we get an improper fraction, then convert it into a mixed number.

$$8<15$$

$$\therefore\;\dfrac{8}{15}$$ is a proper fraction.

• Thus, $$\dfrac{8}{15}$$ is our final answer.

#### Which one of the following options represents the solution of the expression, $$\dfrac{1}{4}×1\dfrac{1}{4}$$?

A $$\dfrac{5}{16}$$

B $$\dfrac{3}{16}$$

C $$\dfrac{1}{16}$$

D $$\dfrac{7}{16}$$

×

Given: $$\dfrac{1}{4}×1\dfrac{1}{4}$$

Converting $$1\dfrac{1}{4}$$ into an improper fraction,

$$1\dfrac{1}{4}=\dfrac{(4×1)+1}{4}$$$$=\dfrac{4+1}{4}=\dfrac{5}{4}$$

Rewriting the problem,

$$\dfrac{1}{4}×\dfrac{5}{4}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{1×5}{4×4}=\dfrac{5}{16}$$

Simplifying the fraction obtained, $$\dfrac{5}{16}$$

$$5$$ and $$16$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{5}{16}$$ is in its simplest form.

Thus, $$\dfrac{5}{16}$$ is our answer.

Hence, option (A) is correct.

### Which one of the following options represents the solution of the expression, $$\dfrac{1}{4}×1\dfrac{1}{4}$$?

A

$$\dfrac{5}{16}$$

.

B

$$\dfrac{3}{16}$$

C

$$\dfrac{1}{16}$$

D

$$\dfrac{7}{16}$$

Option A is Correct

# Multiplication of Mixed Numbers

• To multiply a mixed number with another mixed number, we should follow the given steps:
• Let us consider an example: $$2\dfrac{1}{3}×3\dfrac{1}{2}$$

Step 1: Convert the mixed fractions into improper fractions.

$$2\dfrac{1}{3}=\dfrac{(3×2)+1}{3}=\dfrac{6+1}{3}=\dfrac{7}{3}$$

$$3\dfrac{1}{2}=\dfrac{(2×3)+1}{2}=\dfrac{6+1}{2}=\dfrac{7}{2}$$

Step 2: Rewrite the problem.

$$\dfrac{7}{3}×\dfrac{7}{2}$$

Step 3: Multiply numerator by numerator and denominator by denominator.

$$\dfrac{7×7}{3×2}=\dfrac{49}{6}$$

Step 4: Simplify the fraction obtained, $$\dfrac{49}{6}$$

$$49$$ and $$6$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{49}{6}$$ is in its simplest form.

Step 5: If we get an improper fraction, convert it into a mixed fraction.

$$49>6$$

$$\therefore\;\dfrac{49}{6}$$ is an improper fraction.

So,  $$\dfrac{49}{6}=8\dfrac{1}{6}$$

• Thus, $$8\dfrac{1}{6}$$ is our answer.

#### Which one of the following options represents the solution of the expression, $$3\dfrac{1}{3}×2\dfrac{1}{2}$$?

A $$7\dfrac{1}{3}$$

B $$8\dfrac{1}{3}$$

C $$8\dfrac{1}{2}$$

D $$7\dfrac{1}{2}$$

×

Given: $$3\dfrac{1}{3}×2\dfrac{1}{2}$$

Converting mixed fractions into improper fractions,

$$3\dfrac{1}{3}=\dfrac{(3×3)+1}{3}=\dfrac{9+1}{3}=\dfrac{10}{3}$$

$$2\dfrac{1}{2}=\dfrac{(2×2)+1}{2}=\dfrac{4+1}{2}=\dfrac{5}{2}$$

Rewriting the problem,

$$\dfrac{10}{3}×\dfrac{5}{2}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{10×5}{3×2}=\dfrac{50}{6}$$

Simplifying the fraction obtained,  $$\dfrac{50}{6}$$

The greatest common factor (G.C.F) of $$50$$ and $$6=2$$

Dividing $$50$$ and $$6$$ by G.C.F,

$$\dfrac{50\div2}{6\div2}$$$$=\dfrac{25}{3}$$

$$25$$ and $$3$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{25}{3}$$ is in its simplest form.

Changing the improper fraction into a mixed fraction,

$$25>3$$

$$\therefore\;\dfrac{25}{3}$$ is an improper fraction.

So, $$\dfrac{25}{3}=8\dfrac{1}{3}$$

Thus, $$8\dfrac{1}{3}$$ is our answer.

Hence, option (B) is correct.

### Which one of the following options represents the solution of the expression, $$3\dfrac{1}{3}×2\dfrac{1}{2}$$?

A

$$7\dfrac{1}{3}$$

.

B

$$8\dfrac{1}{3}$$

C

$$8\dfrac{1}{2}$$

D

$$7\dfrac{1}{2}$$

Option B is Correct

# Multiplication of a Mixed Number with a Whole Number

• To multiply a mixed number with a whole number, we should follow the given steps:
• Let us consider an example: $$5×2\dfrac{1}{2}$$

Step 1: Convert the mixed number into an improper fraction.

Convert $$2\dfrac{1}{2}=\dfrac{(2×2)+1}{2}$$$$=\dfrac{4+1}{2}=\dfrac{5}{2}$$

$$5>2$$

$$\therefore\;\dfrac{5}{2}$$ is an improper fraction.

Step 2: Change the whole number to a fraction by putting it over $$1$$

$$5=\dfrac{5}{1}$$

Step 3: Rewrite the problem,

$$\dfrac{5}{1}×\dfrac{5}{2}$$

Step 4: Multiply numerator by numerator and denominator by denominator.

$$=\dfrac{5×5}{1×2}\;\;\dfrac{\leftarrow\text{Numerator}}{\leftarrow\text{Denominator}}$$

$$=\dfrac{25}{2}$$

Step 5: Simplify the fraction, if needed.

Result $$=\dfrac{25}{2}$$

$$25$$ and $$2$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{25}{2}$$ is in its simplest form.

Step 6: If we get an improper fraction, then convert it into a mixed fraction (if problem demands).

$$25>2$$

$$\therefore\;\dfrac{25}{2}$$ is an improper fraction.

$$\dfrac{25}{2}=12\dfrac{1}{2}$$

• Thus, $$12\dfrac{1}{2}$$ is our answer.

#### Which one of the following options represents the solution of the problem, $$3×1\dfrac{1}{2}$$?

A $$1\dfrac{2}{3}$$

B $$4\dfrac{1}{2}$$

C $$2\dfrac{1}{3}$$

D $$1\dfrac{1}{6}$$

×

Given: $$3×1\dfrac{1}{2}$$

Converting $$1\dfrac{1}{2}$$ into an improper fraction,

$$1\dfrac{1}{2}=\dfrac{(1×2)+1}{2}$$$$=\dfrac{2+1}{2}$$$$=\dfrac{3}{2}$$

Changing $$3$$ into a fraction,

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

Rewriting the problem,

$$\dfrac{3}{1}×\dfrac{3}{2}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{3×3}{1×2}=\dfrac{9}{2}$$

Simplifying the fraction obtained, $$\dfrac{9}{2}$$

$$9$$ and $$2$$ do not have any common factor other than $$1$$.

$$\therefore\;\dfrac{9}{2}$$ is in its simplest form.

Changing the improper fraction into a mixed fraction,

$$9>2$$

$$\therefore\;\dfrac{9}{2}$$ is an improper fraction.

So,  $$\dfrac{9}{2}=4\dfrac{1}{2}$$

Thus, $$4\dfrac{1}{2}$$ is our answer.

Hence, option (B) is correct.

### Which one of the following options represents the solution of the problem, $$3×1\dfrac{1}{2}$$?

A

$$1\dfrac{2}{3}$$

.

B

$$4\dfrac{1}{2}$$

C

$$2\dfrac{1}{3}$$

D

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

Option B is Correct

# Multiplication of Fractions through Models

• When we multiply two fractions, whether like or unlike ones, it means we are capturing a part of another part.
• For example: If the problem is $$\dfrac{2}{3}×\dfrac{4}{5}$$,  we can also call it as $$\dfrac{2}{3}$$ of $$\dfrac{4}{5}$$.
• This means, we are taking two-third part of four-fifth part of a fraction.

• Let's understand with an example:

Consider the two squares, M and N, having some shaded parts as shown.

• Square M has $$\dfrac{3}{4}$$ part shaded as $$3$$ columns out of $$4$$ are shaded and square N has $$\dfrac{2}{3}$$ part shaded as $$2$$ rows out of $$3$$ are shaded.

Now to get the product, we have to superimpose the two squares.

• The figure shown has $$12$$ parts, in which $$\dfrac{6}{12}$$ part is the common shaded area.
• The common shaded area is the product of the squares, M and N.

Note: To represent the multiplication in models, we have to draw one fraction horizontally (means in rows) and another fraction vertically (means in columns).

#### What is the product of the given shaded squares?

A

B

C

D

×

One square has $$\dfrac{1}{3}$$ part shaded and another has $$\dfrac{2}{3}$$ part shaded. Square A has $$3$$ rows and $$1$$ column and square B has $$3$$ columns and $$1$$ row.

To multiply them, we will superimpose the two squares. The common shaded area will be the answer (product).

The common shaded area is the product of squares, A and B.

Hence, option (C) is correct.

### What is the product of the given shaded squares?

A
B
C
D

Option C is Correct

# Multiplication of Fractions on a Number Line

Multiplication of fractions means repetitive addition of fractions.

To understand it properly, let's consider an example:

Represent $$2 \times \dfrac{3}{4}$$ on a number line.

Mark $$\dfrac{3}{4}$$ on the number line. For this, we first divide each interval into 4 equal segments as the denominator is 4.

Now, we have to move $$3$$  segments at a time, each of length  $$\dfrac{1}{4}$$ on a number line, twice.

$$2 \times \dfrac{3}{4} = \dfrac{6}{4}$$

We reach at $$\dfrac{6}{4}$$.

Thus, the result of the product is $$\dfrac{6}{4}$$.

#### Which number line represents the solution of $$3 \times \dfrac{4}{5}$$ as point $$P$$?

A

B

C

D

×

Given: $$3 \times \dfrac{4}{5}$$

If we want to represent $$C \times \dfrac{a}{b}$$ on a number line, we have to move $$a$$  units at a time, each of length $$\dfrac{1}{b}$$  on the number line, $$C$$ times.

Mark $$\dfrac{4}{5}$$ on the number line. For this, we first divide each interval into 5 equal segments as the denominator is 5.

Now, we have to move $$4$$  segments at a time, each of length  $$\dfrac{1}{5}$$ on a number line, thrice.

$$3 \times \dfrac{4}{5} = \dfrac{12}{5}$$

Thus, the product is $$\dfrac{12}{5}$$.

Here, point $$P$$ represents $$\dfrac{12}{5}$$.

Hence, option (A) is correct.

### Which number line represents the solution of $$3 \times \dfrac{4}{5}$$ as point $$P$$?

A
B
C
D

Option A is Correct