Informative line

Multiplication Of Fractions

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.

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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.

Illustration Questions

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).

Illustration Questions

What is the product of the given shaded squares?

A

B

C

D

×

Given: Two shaded squares

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.

image

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.

image

Hence, option (C) is correct.

What is the product of the given shaded squares?

image
A image
B image
C image
D image

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 \(\dfrac{6}{4}\).

Illustration Questions

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.

image

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

\(3 \times \dfrac{4}{5} = \dfrac{12}{5}\)

image

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

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

image

Hence, option (A) is correct. 

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

A image
B image
C image
D image

Option A is Correct

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