Informative line

Concept of Reciprocal

• A reciprocal is the inverse or opposite form of a fraction.

For example: Reciprocal of $$\dfrac{3}{2}$$ is $$\dfrac{2}{3}$$.

• To find the reciprocal, we should consider the following cases:

Case I: Reciprocal of a whole number:

• Let us consider an example.

Find the reciprocal of $$4$$.

Step 1: Convert the whole number into a fraction by putting it over $$1$$.

$$4=\dfrac{4}{1}$$

Step 2: Flip the numerator and the denominator of the fraction.

Fraction $$=\dfrac{4}{1}$$

Flipping $$4$$ and $$1,$$ i.e. $$\downarrow\dfrac{4}{1}\uparrow$$

$$=\dfrac{1}{4}$$

Step 3: The resulting fraction is the reciprocal of the given fraction.

$$\therefore$$ Reciprocal of $$\dfrac{4}{1}=\dfrac{1}{4}$$

Case II: Reciprocal of a fraction:

• Let us consider an example.

Find the reciprocal of $$\dfrac{2}{5}$$.

Step 1: Flip the numerator and the denominator of the fraction.

Fraction $$=\dfrac{2}{5}$$

Flipping $$2$$ and $$5,$$ i.e. $$\uparrow\dfrac{2}{5}\downarrow$$

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

Step 2: The resulting fraction is the reciprocal of the given fraction.

$$\therefore$$ Reciprocal of $$\dfrac{2}{5}=\dfrac{5}{2}$$

Case III: Reciprocal of a mixed number:

• Let us consider an example.

Find the reciprocal of $$2\dfrac{1}{3}$$.

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

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

Step 2: Flip the numerator and the denominator of the fraction.

Fraction $$=\dfrac{7}{3}$$

Flipping $$7$$ and $$3,$$ i.e. $$\uparrow\dfrac{7}{3}\downarrow$$

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

Step 3: The resulting fraction is the reciprocal of the given fraction.

$$\therefore$$ Reciprocal of $$\dfrac{7}{3}=\dfrac{3}{7}$$

Note: If we multiply a fraction with its reciprocal, their product is always $$1.$$

For example: Multiply $$\dfrac{3}{2}$$ and $$\dfrac{2}{3}$$.

$$\dfrac{3}{2}×\dfrac{2}{3}=\dfrac{3×2}{2×3}=\dfrac{6}{6}=1$$

Which one of the following represents the reciprocal of $$\dfrac{6}{7}$$?

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

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

C $$\dfrac{7}{6}$$

D $$\dfrac{7}{3}$$

×

Given: $$\dfrac{6}{7}$$

Flipping the numerator and the denominator of $$\dfrac{6}{7}$$ i.e. $$\uparrow\dfrac{6}{7}\downarrow$$

$$=\dfrac{7}{6}$$

$$\therefore$$ Reciprocal of $$\dfrac{6}{7}=\dfrac{7}{6}$$

Hence, option (C) is correct.

Which one of the following represents the reciprocal of $$\dfrac{6}{7}$$?

A

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

.

B

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

C

$$\dfrac{7}{6}$$

D

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

Option C is Correct

Division of a Fraction by a Whole Number

Dividing a fraction by a whole number means splitting a part into more parts.

• To divide a fraction by a whole number, we should follow the given steps:
• Let us consider an example.

Divide $$\dfrac{4}{5}$$ by $$2,$$ i.e. $$\dfrac{4}{5}\div2$$

Step 1: Change the whole number into a fraction by putting it over $$1$$.

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

Step 2: Rewrite the problem.

$$\dfrac{4}{5}\div\dfrac{2}{1}$$

Step 3: Find the reciprocal of the second fraction.

Reciprocal of $$\dfrac{2}{1}=\dfrac{1}{2}$$     $$\Bigg($$Flip $$2$$ and $$1,$$ i.e. $$\uparrow\dfrac{2}{1}\downarrow\;\Bigg)$$

Step 4: Change the division sign to multiplication.

$$\dfrac{4}{5}\div\dfrac{2}{1}\\\;\;\;\downarrow\\\dfrac{4}{5}×$$

Step 5: Replace the second fraction with its reciprocal.

$$\dfrac{4}{5}×\dfrac{1}{2}$$

Step 6: Multiply numerator by numerator and denominator by denominator.

$$\dfrac{4}{5}×\dfrac{1}{2}=\dfrac{4×1}{5×2}=\dfrac{4}{10}$$

Step 7: Simplify the fraction obtained.

• The greatest common factor of $$4$$ and $$10=2$$
• Divide $$4$$ and $$10$$ by $$2,$$ i.e.

$$\dfrac{4\div2}{10\div2}=\dfrac{2}{5}$$

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

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

Step 8: If we get an improper fraction, change it to a mixed number (if needed).

$$2<5$$

$$\therefore\;\dfrac{2}{5}$$ is a proper fraction.

Thus, $$\dfrac{2}{5}$$ is the answer.

Which one of the following options represents the solution of the expression $$\dfrac{5}{2}\div5$$?

A $$\dfrac{3}{4}$$

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

C $$\dfrac{15}{4}$$

D $$\dfrac{1}{8}$$

×

Given: $$\dfrac{5}{2}\div5$$

Changing $$5$$ into a fraction,

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

Rewriting the problem,

$$\dfrac{5}{2}\div\dfrac{5}{1}$$

Finding reciprocal of the second fraction, $$\dfrac{5}{1}$$

Flipping $$5$$ and $$1,$$ i.e. $$\uparrow\dfrac{5}{1}\downarrow$$

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

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

Changing the division sign to multiplication,

$$\dfrac{5}{2}\div\dfrac{5}{1}\\\;\;\;\downarrow\\\dfrac{5}{2}×$$

Replacing the second fraction with its reciprocal.

$$\dfrac{5}{2}×\dfrac{1}{5}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{5×1}{2×5}=\dfrac{5}{10}$$

Simplifying $$\dfrac{5}{10}$$,

The greatest common factor of $$5$$ and $$10=5$$

Dividing  $$5$$ and $$10$$ by $$5,$$ $$\dfrac{5\div5}{10\div5}=\dfrac{1}{2}$$

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

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

Hence, option (B) is correct.

Which one of the following options represents the solution of the expression $$\dfrac{5}{2}\div5$$?

A

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

.

B

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

C

$$\dfrac{15}{4}$$

D

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

Option B is Correct

Division of a Fraction by Another Fraction

• To divide a fraction by another fraction, we should follow the given steps:
• Let us consider an example.

Divide $$\dfrac{9}{4}$$ by $$\dfrac{3}{2}$$  i.e.

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

Step 1: Change the division sign to multiplication.

$$\dfrac{9}{4}\div\dfrac{3}{2}\\\;\;\;\downarrow\\\dfrac{9}{4}×$$

Step 2: Find the reciprocal of the second fraction.

Reciprocal of $$\dfrac{3}{2}=\dfrac{2}{3}$$     $$\Bigg($$Flip $$3$$ and $$2,$$ i.e. $$\uparrow\dfrac{3}{2}\downarrow\;\Bigg)$$

Step 3: Replace the second fraction with its reciprocal.

$$\dfrac{9}{4}×\dfrac{2}{3}$$

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

$$\dfrac{9×2}{4×3}=\dfrac{18}{12}$$

Step 5: Simplify the fraction obtained.

Fraction $$=\dfrac{18}{12}$$

• The greatest common factor of $$18$$ and $$12=6$$
• Divide $$18$$ and $$12$$ by $$6,$$

$$\dfrac{18\div6}{12\div6}=\dfrac{3}{2}$$

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

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

Step 6: If we get an improper fraction, convert it to a mixed fraction (if needed).

$$3>2$$

$$\therefore\;\dfrac{3}{2}$$ is an improper fraction.

So, $$\dfrac{3}{2}=1\dfrac{1}{2}$$

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

Which one of the following options represents the solution of the expression $$\dfrac{3}{8}\div\dfrac{9}{2}$$?

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

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

C $$\dfrac{27}{16}$$

D $$\dfrac{9}{16}$$

×

Given: $$\dfrac{3}{8}\div\dfrac{9}{2}$$

Changing division sign to multiplication,

$$\dfrac{3}{8}\div\dfrac{9}{2}\\\;\;\;\downarrow\\\dfrac{3}{8}×$$

Finding reciprocal of the second fraction,

Reciprocal of $$\dfrac{9}{2}=\dfrac{2}{9}$$    $$\Bigg($$Flipping  $$2$$ and $$9,$$ i.e. $$\uparrow\dfrac{9}{2}\downarrow\;\Bigg)$$

Replacing the second fraction with its reciprocal,

$$\dfrac{3}{8}×\dfrac{2}{9}$$

Multiplying numerator by numerator and denominator by denominator,

$$\dfrac{3×2}{8×9}=\dfrac{6}{72}$$

Simplifying $$\dfrac{6}{72}$$,

The greatest common factor of $$6$$ and $$72=6$$

Dividing $$6$$ and $$72$$ by $$6,$$ $$\dfrac{6\div6}{72\div6}=\dfrac{1}{12}$$

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

$$\therefore\;\dfrac{1}{12}$$ is in its simplest form.

Thus, $$\dfrac{1}{12}$$ is the answer.

Hence, option (A) is correct.

Which one of the following options represents the solution of the expression $$\dfrac{3}{8}\div\dfrac{9}{2}$$?

A

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

.

B

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

C

$$\dfrac{27}{16}$$

D

$$\dfrac{9}{16}$$

Option A is Correct

Division of a Whole Number by a Fraction (Through Figures)

• Dividing a whole number by a fraction means we are taking one complete figure as the whole and dividing it into new wholes.
• To understand it easily, consider an example where $$2$$ bars are taken, A and B.

• In the given division problem, we want to find out, "how many $$\dfrac{1}{5}$$ 's are there in $$1$$?"
• Firstly, to solve this problem, we have to subdivide bar (A) into same equal parts as of bar (B) as shown in figure (C).

• Now we divide bar (A) by bar (B) to get the answer.

• From figure (D), we can observe that the shaded part (blue) of bar (B) fits into the shaded part (red) of bar (A), $$5$$ times. Thus, the result of the problem is $$5.$$
• Thus, there are five $$\dfrac{1}{5}$$ 's in $$1.$$

The figures shown represent $$3$$ cakes. How many $$\dfrac{4}{8}$$ parts can these $$3$$ cakes contain?

A $$3$$

B $$5$$

C $$4$$

D $$6$$

×

Given: $$3$$ similar cakes. These cakes are already divided into $$\dfrac{1}{4}$$ parts.

Here, each cake is divided into $$4$$ equal parts means, one slice is equal to $$\dfrac{1}{4}$$ part.

We want to have $$\dfrac{4}{8}$$ part, so we have to subdivide them so that each slice becomes $$\dfrac{1}{8}$$ part.

Now, we have to calculate the total number of $$\dfrac{4}{8}$$ parts ($$4$$ parts of $$\dfrac{1}{8}$$ as $$1$$ part) in these $$3$$ cakes.

Thus, we can observe that there are $$6$$ groups of $$\dfrac{4}{8}$$ in these $$3$$ cakes, so, the result is $$6.$$

Hence, option (D) is correct.

The figures shown represent $$3$$ cakes. How many $$\dfrac{4}{8}$$ parts can these $$3$$ cakes contain?

A

$$3$$

.

B

$$5$$

C

$$4$$

D

$$6$$

Option D is Correct

Dividing a Fraction by a Fraction using Fraction Bars

• Division of a fraction by another fraction means the number of parts of the second fraction that the first fraction can contain.
• Let's understand with an example.
• Example 1: There are $$2$$ different bars, blue and green.

• Blue bar has $$\dfrac{2}{3}$$ part shaded and green bar has $$\dfrac{1}{6}$$ part shaded.
• Now, to divide $$\dfrac{2}{3}$$ by $$\dfrac{1}{6}\;\left(\dfrac{2}{3}\div\dfrac{1}{6}\right)$$, we have to place the fraction bars under each other to compare their shaded region.

• On observing the fraction bars, we get to know how many times the shaded region of $$\dfrac{2}{3}$$ is bigger than the shaded region of $$\dfrac{1}{6}$$.
• Here, the shaded region of $$\dfrac{2}{3}$$ is $$4$$ times bigger than the shaded region of $$\dfrac{1}{6}$$.

$$\therefore\;\dfrac{2}{3}\div\dfrac{1}{6}=4$$

• We can also say that the shaded part of the green fraction bar fits into the shaded part of the blue fraction bar, $$4$$ times.

• Example 2: Consider figures (I) and (II) having $$\dfrac{1}{2}$$ and $$\dfrac{1}{4}$$ parts shaded, respectively.

• Before performing the division, we have to subdivide figures (I) and (II) so as to make the total number of parts equal in both, as shown in figures (III) and (IV).

• For division, we see how many $$\dfrac{1}{4}$$'s fit into $$\dfrac{1}{2}$$.
• Thus, we can observe from figure (C) that two times the shaded part of figure (B) fits into the shaded part of figure (A). Thus, the result is $$2.$$

• Thus we can observe from figure (C) that two times the shaded part of figure (B) fits into the shaded part of figure (A). Thus the result is $$2.$$

What is the result of the following problem?

A $$1$$

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

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

D $$3$$

×

Given two shaded circles: P and Q

$$\dfrac{3}{4}$$ and $$\dfrac{1}{4}$$ parts are shaded of P and Q respectively.

Both have the equal number of parts.

Therefore, there is no need to subdivide.

Now, in order to divide P by Q, we find out how many $$\dfrac{1}{4}$$'s of Q fit into $$\dfrac{3}{4}$$ of P.

Thus, three $$\dfrac{1}{4}$$'s of Q fit into $$\dfrac{3}{4}$$ of P.

Hence, option (D) is correct.

What is the result of the following problem?

A

$$1$$

.

B

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

C

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

D

$$3$$

Option D is Correct

Division of a Fraction by a Whole Number

• Division of a fraction by a whole number means to consider a part of something and then splitting it up into more parts.

Example: Tyler has two-third length of a rope for a project, from which he requires $$4$$ equal sized pieces. What will be the size of each piece?

• In the figure, the two third shaded part represents the length of the rope.

• The rope is required to be divided into $$4$$ equal pieces, i.e., $$\dfrac{2}{3}\div4$$
• According to the problem, two-third part is to be divided into $$4$$ equal parts which means each shaded part is to be divided into $$2$$ segments.

• We can not divide only the  $$2$$ parts. So, we divide all the parts.

• From the figure shown, we get to know that each part is $$\dfrac{1}{6}$$ part of the rope.

• So, the size of each piece of the rope is obtained as:

$$\dfrac{2}{3}\div4=\dfrac{1}{6}$$

The given rectangle is divided into $$5$$ equal sections, of which $$3$$ are shaded. Choose the option representing the solution of $$\dfrac{3}{5}\div9$$.

A

B

C

D

×

In the figure, $$\dfrac{3}{5}$$ part is shaded.

We are asked to calculate $$\dfrac{3}{5}\div9$$.

According to the problem, $$\dfrac{3}{5}$$ part is to be divided into $$9$$ equal sections, which means that the shaded section is to be divided into $$9$$ sections [each part into three].

We cannot divide only the $$3$$ parts.

So, we divide all the parts.

From the figure shown, we get to know that each part is $$\dfrac{1}{15}$$ part of the rectangle.

Each part of the rectangle represents $$\dfrac{1}{15}$$ part.

$$\therefore\;\dfrac{3}{5}\div9=\dfrac{1}{15}$$

Hence, option (B) is correct.

The given rectangle is divided into $$5$$ equal sections, of which $$3$$ are shaded. Choose the option representing the solution of $$\dfrac{3}{5}\div9$$.

A
B
C
D

Option B is Correct

Division of Fractions on a Number Line

• Division of fractions means we are dividing a part of a whole into more parts.
• To understand it properly, let's consider an example:
• Represent $$\dfrac{5}{4}\div\dfrac{2}{8}$$ on a number line.
• It means we are asked, "How many $$\dfrac{2}{8}$$ parts are there in $$\dfrac{5}{4}$$?"
• To get the answer, we represent $$\dfrac{5}{4}$$ on a number line.

• Divide each part into two and get $$\dfrac{2}{8}$$ $$(4\times2=8)$$ and make move of $$\dfrac{2}{8}$$ segments till we reach $$\dfrac{5}{4}$$.

• It is clear from the figure that five segments of $$\dfrac{2}{8}$$ fit in $$\dfrac{5}{4}$$.

There are five $$\dfrac{2}{8}'s$$ in $$\dfrac{5}{4}$$.

Which number line represents the solution of $$\dfrac{5}{2}\div\dfrac{1}{4}$$ as point R?

A

B

C

D

×

Given: $$\dfrac{5}{2}\div\dfrac{1}{4}$$

Represent $$\dfrac{5}{2}$$ on a number line.

Split each part into 2 parts and get $$\dfrac{1}{4}$$ $$(2\times2=4)$$.

Make the moves of $$\dfrac{1}{4}$$ till we reach $$\dfrac{5}{2}$$.

We can observe that there are ten $$\dfrac{1}{4}'s$$ in $$\dfrac{5}{2}$$.

The diagram represents $$\dfrac{5}{2}\div\dfrac{1}{4}=10$$

The resulting number line is:

Hence, option (B) is correct.

Which number line represents the solution of $$\dfrac{5}{2}\div\dfrac{1}{4}$$ as point R?

A
B
C
D

Option B is Correct