Informative line

# Proper Fraction

• A proper fraction is a fraction in which the numerator is smaller than the denominator.

For example:

$$\dfrac{2}{3}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}$$

Since $$2$$ is smaller than $$3$$, therefore $$\dfrac {2}{3}$$ is a proper fraction.

#### Which one of the following fractions represents a proper fraction?

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

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

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

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

×

In a proper fraction, the numerator is smaller than the denominator.

Only in option (C), i.e., $$\dfrac {1}{2}$$

Numerator (1) is smaller than the denominator (2).

$$\therefore\,\dfrac {1}{2}$$ is a proper fraction.

Hence, option (C) is correct.

### Which one of the following fractions represents a proper fraction?

A

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

.

B

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

C

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

D

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

Option C is Correct

# Unit Fraction

• A unit fraction is a fraction in which the numerator is 1.

For example:

$$\dfrac {1}{2},\,\dfrac {1}{3},\,\dfrac {1}{8},\,\dfrac {1}{11}$$ are unit fractions.

#### Which one of the following options represents the unit fraction?

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

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

C $$\dfrac {4}{5}$$

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

×

A unit fraction has the numerator 1.

Only in option (B), i.e., $$\dfrac {1}{3}$$

Numerator = 1

$$\therefore\,\dfrac {1}{3}$$ is a unit fraction.

Hence, option (B) is correct.

### Which one of the following options represents the unit fraction?

A

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

.

B

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

C

$$\dfrac {4}{5}$$

D

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

Option B is Correct

# Proper Fractions Using Models

• A proper fraction is a fraction in which the numerator is smaller than the denominator.

For example:

$$\dfrac{2}{3}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}$$

Since $$2$$ is smaller than $$3$$, therefore $$\dfrac {2}{3}$$ is a proper fraction.

• To understand the concept of proper fractions using fraction for shaded or unshaded parts of a figure, consider this example:
• Ronnie and Jacob together buy 3 pizzas, as shown in figure b1.

• Each pizza has 4 equal slices.
• If they eat 9 slices in total and leave the remaining pizza for Ronnie's sister, then

'how many pizza slices are left for Ronnie's sister?'

Consider figure b2 for the remaining pizza slices.

• We can observe that each slice represents $$\dfrac{1}{4}$$ or one-quarter and only 3 slices are left.

$$\therefore$$ In fraction, we can represent it as-

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

Now, in $$\dfrac{3}{4}$$, the numerator 3 is less than the denominator 4.

Thus, $$\dfrac{3}{4}$$ is a proper fraction.

#### Consider the given figures. Some parts of the 2 big squares are shaded. Find the fraction for the shaded parts of the given squares altogether.

A $$\dfrac{1}{4}$$

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

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

D $$\dfrac{3}{8}$$

×

Given:

By observing the given squares, we can say that each big square has 4 equal parts.

$$\therefore$$ One part of a big square represents one-fourth or $$\dfrac{1}{4}$$.

Calculating fraction for the shaded parts 'altogether'

Each part represents one-fourth $$\left(\dfrac{1}{4}\right)$$ and here, total 3 parts are shaded.

So, we have three-fourths $$=\dfrac{3}{4}$$

Hence, option (C) is correct.

### Consider the given figures. Some parts of the 2 big squares are shaded. Find the fraction for the shaded parts of the given squares altogether.

A

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

.

B

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

C

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

D

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

Option C is Correct

# Number Line Representation of Unit Fractions

A unit fraction is a fraction in which the numerator is $$1$$.

For example: $$\dfrac {1}{2},\dfrac {1}{3},\dfrac {1}{8},\dfrac {1}{11}$$ are unit fractions.

Let us consider $$\dfrac {1}{b}$$ as a unit fraction.

To represent $$\dfrac {1}{b}$$ on a number line:

• Define interval from $$0$$ to $$1$$.
• Divide it into 'b' equal parts.
• The size of each part is $$\dfrac {1}{b}$$.
• The length from point $$0$$ to $$\dfrac {1}{b}$$ represents the unit fraction.

For example: Represent $$\dfrac {1}{5}$$ on a number line.

Step 1 : Define the interval from $$0$$ to $$1$$.

Step 2 : Divide it into 5 equal parts.

Step 3 : Size of each part is $$\dfrac {1}{5}$$.

Step 4 : By observing the number line, we can say that the distance from $$0$$ to point $$\dfrac {1}{5}$$ represents the unit fraction.

#### Which one of the following number lines represents the correct position of the fraction $$\dfrac {1}{6}$$?

A

B

C

D

×

Given fraction: $$\dfrac{1}{6}$$

Define the interval from $$0$$ to $$1$$ .

Divide it into 6 equal parts.

Size of each part is $$\dfrac{1}{6}$$.

$$\dfrac{1}{6}$$ represents the length of the first segment.

Hence, option (D) is correct.

### Which one of the following number lines represents the correct position of the fraction $$\dfrac {1}{6}$$?

A
B
C
D

Option D is Correct

# Improper Fractions

• An improper fraction is a fraction in which the numerator is larger than the denominator.

For example:

$$\dfrac{5}{2}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}$$

• Since $$5$$ is larger than $$2$$, therefore $$\dfrac{5}{2}$$ is an improper fraction.

• Since $$5$$ is larger than $$2$$

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

#### Which one of the following fractions represents an improper fraction?

A $$\dfrac{12}{5}$$

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

C $$\dfrac{3}{5}$$

D $$\dfrac{13}{15}$$

×

In an improper fraction, the numerator is larger than the denominator.

Only in option (A), i.e., $$\dfrac{12}{5}$$

The numerator $$(12)$$ is larger than the denominator $$(5)$$ .

$$\therefore\; \dfrac{12}{5}$$ is an improper fraction.

Hence, option (A) is correct.

### Which one of the following fractions represents an improper fraction?

A

$$\dfrac{12}{5}$$

.

B

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

C

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

D

$$\dfrac{13}{15}$$

Option A is Correct

# Improper Fractions using Models

• An improper fraction is a fraction in which the numerator is larger than the denominator.

For example: $$\dfrac{5}{2}$$

$$5$$ is larger than $$2.$$

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

• To understand improper fractions by using fraction for shaded or unshaded parts of a figure, consider this example:
• Mr. James buys two pizzas, as shown in figure $$a_1$$.
• Each pizza has $$4$$ equal slices.

• If he eats $$3$$ slices out of $$4$$ of one pizza then "how many pizza slices are left altogether?"
• Consider figure $$a_2$$ for the remaining pizza slices.
• We can observe that each slice represents $$\dfrac{1}{4}$$ or one-quarter.
• $$\therefore$$ We have five quarters, i.e., $$\dfrac{5}{4}$$

• Now, in $$\dfrac{5}{4}$$, the denominator $$5$$ is greater than the numerator $$4.$$
• Thus, it is an improper fraction.

• We can also write $$\dfrac{5}{4}$$ as:
• $$\dfrac{5}{4}=\dfrac{4}{4}+\dfrac{1}{4}$$$$=1+\dfrac{1}{4}$$
• Here, $$1$$ represents one whole pizza and $$\dfrac{1}{4}$$ represents one-fourth part(one slice of second pizza) that is left.

• We can observe that each slice represents $$\dfrac{1}{4}$$ or one-quarter.

$$\therefore$$ We have five quarters, i.e., $$\dfrac{5}{4}$$

• Now, in $$\dfrac{5}{4}$$, the denominator $$5$$ is greater than the numerator $$4.$$

Thus, it is an improper fraction.

• We can also write $$\dfrac{5}{4}$$ as:

$$\dfrac{5}{4}=\dfrac{4}{4}+\dfrac{1}{4}$$

$$1+\dfrac{1}{4}$$

• Here, $$1$$ represents one whole pizza and $$\dfrac{1}{4}$$ represents one-fourth part that is left.

#### Consider the given figures, some parts of these circles are shaded. Which one of the following options represents fraction for the shaded parts of the given figures altogether?

A $$\dfrac{16}{25}$$

B $$\dfrac{15}{8}$$

C $$\dfrac{4}{64}$$

D $$\dfrac{64}{15}$$

×

By observing the given figures, we can say that each circle has $$8$$ equal parts.

$$\therefore$$ One part of a circle represents one-eighth or $$\dfrac{1}{8}$$.

Calculating fraction for shaded parts 'altogether':

Each part represents one-eighth $$\left(\dfrac{1}{8}\right)$$.

Here, total 15 parts are shaded.

$$\therefore$$ We have fifteen-eighths  $$=\dfrac{15}{8}$$.

Hence, option (B) is correct.

### Consider the given figures, some parts of these circles are shaded. Which one of the following options represents fraction for the shaded parts of the given figures altogether?

A

$$\dfrac{16}{25}$$

.

B

$$\dfrac{15}{8}$$

C

$$\dfrac{4}{64}$$

D

$$\dfrac{64}{15}$$

Option B is Correct

# Number Line Representation of a Fraction

To represent the fraction $$\dfrac {a}{b}$$ on a number line:

• Define the interval starting from zero.
• Divide each interval into 'b' equal parts.
• The size of each part is $$\dfrac {1}{b}$$.
• The fraction $$\dfrac {a}{b}$$ represents the combined length of 'a' parts of size $$\dfrac {1}{b}$$.

For example: Represent $$\dfrac {3}{4}$$  on a number line.

Step 1: Define the interval from 0 to 1.

Step 2: Divide the interval into 4 equal parts.

Step 3 : The size of each part is $$\dfrac {1}{4}$$.

Step 4: Thus, $$\dfrac {3}{4}$$ represents the combined length of $$3$$ parts.

Here,

$$\dfrac {0}{4}+\dfrac {1}{4} =\dfrac {1}{4}$$

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

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

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

Step 5: The resulting number line representation is:

#### Which one of the following number lines represents the correct position of the fraction $$\dfrac {5}{2}$$?

A

B

C

D

×

Given fraction: $$\dfrac {5}{2}$$

Define the interval from $$0$$ to $$3$$.

Divide each interval into $$2$$ equal parts.

The size of each part is $$\dfrac {1}{2 }$$ .

Thus, $$\dfrac {5}{2}$$ represents the combined length of  $$5$$  parts.

Here,

$$0+\dfrac {1}{2}=\dfrac {1}{2}$$

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

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

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

$$\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2} =\dfrac {5}{2}$$

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

The resulting number line representation is:

Hence, option (A) is correct.

### Which one of the following number lines represents the correct position of the fraction $$\dfrac {5}{2}$$?

A
B
C
D

Option A is Correct