Informative line

Mixed Fractions And Equivalent Fractions

Mixed Number

  • A mixed number is a number which has a whole number and a fraction together.
  • For example: \(1\dfrac {1}{3}\)

    In \(1\dfrac {1}{3}\), whole number = 1, fraction = \(\dfrac {1}{3}\)

  • \(\therefore\;1\dfrac {1}{3}\) is a mixed number.

For example: \(1\dfrac {1}{3}\)

\(\because\) In \(1\dfrac {1}{3}\), whole number = 1

Fraction = \(\dfrac {1}{3}\)

\(\therefore\;1\dfrac {1}{3}\) is a mixed number.

Illustration Questions

Which one of the following options represents a mixed number?

A \(\dfrac {1}{2}\)

B \(3\)

C \(\dfrac {1}{5}\)

D \(2\dfrac {1}{3}\)

×

A mixed number has a whole number and a fraction together.

In option (D), i.e., \(2\dfrac {1}{3}\)

Whole number = 2

Fraction = \(\dfrac {1}{3}\)

\(\therefore\; 2\dfrac {1}{3}\) is a mixed number.

Hence, option (D) is correct.

Which one of the following options represents a mixed number?

A

\(\dfrac {1}{2}\)

.

B

\(3\)

C

\(\dfrac {1}{5}\)

D

\(2\dfrac {1}{3}\)

Option D is Correct

Mixed Numbers Using Models

  • A mixed number is a number which has a whole number and a fraction together.

 

  • To understand it easily, let us consider an example:
  • Ana buys three caramel chocolates.
  • She eats one-half of a chocolate.
  • Then her two cousins arrive.
  • They also want to have chocolates.
  • Ana gives them the remaining chocolates.

  • Numerically, 

Total number of chocolates = \(3\)

Ana ate = \(\dfrac {1}{2}\)

Chocolate remaining = \(2\) and a half part 

\(2+\dfrac {1}{2}\)

\(=2\dfrac {1}{2}\)

\(2\dfrac {1}{2}\) has both, a whole number and a fraction part.

\(\therefore\) It is a mixed number.

Illustration Questions

Robin had three pizzas. Each pizza is cut into 4 equal slices. Robin eats one slice of a pizza. How much pizza is left?

A \(\dfrac {3}{4}\)

B \(1\dfrac {3}{4}\)

C \(2\dfrac {1}{4}\)

D \(2\dfrac {3}{4}\)

×

Total number of pizzas = \(3\)

image

Number of slices in one pizza = \(4\)

Number of slices Robin eats = \(1\)

Number of slices left = \(3\)

Fraction of slices left = \(\dfrac {3}{4}\)

image

Pizza left = \(2\) whole pizzas and \(\dfrac {3}{4}\) slices of another pizza

\(2+\dfrac {3}{4}\)

\(=2\dfrac {3}{4}\)

image

Hence, option (D) is correct.

image

Robin had three pizzas. Each pizza is cut into 4 equal slices. Robin eats one slice of a pizza. How much pizza is left?

image
A

\(\dfrac {3}{4}\)

.

B

\(1\dfrac {3}{4}\)

C

\(2\dfrac {1}{4}\)

D

\(2\dfrac {3}{4}\)

Option D is Correct

Equivalent Fractions

  • Equivalent fractions are fractions which have the same value, even though they look different.

For example:

\(\dfrac {1}{2}=\dfrac {2}{4}=\dfrac {4}{8}\) are equivalent fractions.

  • The fundamental fact about equivalent fractions is that a fraction does not change when its numerator and denominator are multiplied or divided by a same non-zero whole number.
  • To understand it easily, let us consider the following example for the fraction \(\dfrac {1}{3}\):

Multiply both numerator and denominator by the same non-zero whole number, like 2 in this case.

\(\Rightarrow\dfrac {1×2}{3×2}=\dfrac {2}{6}\)

The Greatest Common Factor of 2 and 6 is 2, so we divide both the numerator and the denominator by 2.

We get \(\dfrac{1}{3}\) which is same as the original fraction.

The new fraction \(\left(\dfrac{2}{6}\right)\)  is the equivalent fraction of the original fraction.

Equivalent fraction of \(\dfrac {1}{3}=\dfrac {2}{6}\)

Though, \(\dfrac {1}{3}\) and \(\dfrac {2}{6}\) look different but they have the same value.

Illustration Questions

Which one of the following is an equivalent fraction of \(\dfrac {3}{2}\)?

A \(\dfrac {9}{6}\)

B \(\dfrac {6}{9}\)

C \(\dfrac {12}{4}\)

D \(\dfrac {15}{8}\)

×

Given fraction: \(\dfrac {3}{2}\)

Option (A): \(\dfrac {9}{6}\)

To get 9 from 3, we need to multiply the numerator (3) by 3.

\(\therefore\) The denominator (2) will also get multiplied by 3.

Thus, the equivalent fraction of \(\dfrac {3}{2}=\dfrac {3×3}{2×3}=\dfrac {9}{6}\)

Hence, option (A) is correct.

Option (B): \(\dfrac {6}{9}\)

To get 6 from 3, we need to multiply the numerator (3) by 2.

\(\therefore\) The denominator (2) will also get multiplied by 2.

Thus, the equivalent fraction of \(\dfrac{3}{2}=\dfrac {3×2}{2×2}=\dfrac {6}{4}\)

Hence, option (B) is incorrect.

Option (C): \(\dfrac {12}{4}\)

To get 12 from 3, we need to multiply the numerator (3) by 4.

\(\therefore\) The denominator (2) will also get multiplied by 4.

Thus, the equivalent fraction of \(\dfrac {3}{2}\) =  \(\dfrac {3×4}{2×4}=\dfrac {12}{8}\)

Hence, option (C) is incorrect.

Option (D): \(\dfrac {15}{8}\)

To get 15 from 3, we need to multiply the numerator (3) by 5.

\(\therefore\) The denominator (2) will also get multiplied by 5.

Thus, the equivalent fraction of \(\dfrac {3}{2}\) =  \(\dfrac {3×5}{2×5}=\dfrac {15}{10}\)

Hence, option (D) is incorrect.

Which one of the following is an equivalent fraction of \(\dfrac {3}{2}\)?

A

\(\dfrac {9}{6}\)

.

B

\(\dfrac {6}{9}\)

C

\(\dfrac {12}{4}\)

D

\(\dfrac {15}{8}\)

Option A is Correct

Equivalent Fractions Using Models

  • Equivalent fractions are fractions which have the same values, even though they look different.

For example: \(\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{4}{8}\) are equivalent fractions.

Pictorial representation is a better tool to understand the equivalent fractions.

Let us consider an example:

  • Carl and Kevin go to a restaurant.
  • Carl orders a veggie pizza and Kevin orders a cheese pizza.
  • The veggie pizza is cut into 8 slices while the cheese pizza is cut into 4 slices.
  • Carl eats 4 slices of the veggie pizza.
  • Kevin eats 2 slices of the cheese pizza.

They are saying that they both have eaten the same amount of pizza.

Is it true?

To solve this problem, consider the following procedure:

  • The colored part shows the part of pizza eaten by Carl and Kevin.

By observing the figure, it is clear that both of them have eaten the same amount of pizza.

\(\therefore\) Fraction of a pizza eaten by Carl = Fraction of a pizza eaten by Kevin

Fraction of pizza eaten by Carl = 4 out of 8 \(=\dfrac{4}{8}\)

Fraction of pizza eaten by Kevin = 2 out of 4 \(=\dfrac{2}{4}\)

\(\Rightarrow\;\dfrac{4}{8}=\dfrac{2}{4}\)

So, \(\dfrac{4}{8}\) and \(\dfrac{1}{2}\) are equivalent fractions.

Thus, they are correct.

?

Illustration Questions

In each of the options, a pair of shaded bars is given. Which one of the options represents the pair of equivalent fractions? [Assume the size of both the bars in each option is same.]

A

B

C

D

×

Option (A):

In figure -I:  \(\dfrac {2}{6}\) part is shaded

In figure -II: \(\dfrac {2}{5}\) part is shaded

Since size of both the bars is same, so we compare the shaded portion.

We can see that shaded parts of both the figures are not equal.

image

Thus, \(\dfrac {2}{6}\) and \(\dfrac {2}{5}\) are not equivalent.

Hence, option (A) is incorrect.

Option (B):

In figure -I: \(\dfrac {3}{5}\) part is shaded

In figure -II: \(\dfrac {4}{12}\) part is shaded

Since size of both the bars is same, so we compare the shaded portion.

We can see that shaded parts of both the figures are not equal.

image

Thus, \(\dfrac {3}{5}\) and \(\dfrac {4}{12}\) are not equivalent.

Hence, option (B) is incorrect.

Option (C):

In figure -I: \(\dfrac {3}{7}\) part is shaded

In figure -II: \(\dfrac {6}{8}\) part is shaded

Since size of both the bars is same, so we compare the shaded portion.

We can see that shaded parts of both the figures are not equal.

image

Thus, \(\dfrac {3}{7}\) and \(\dfrac {6}{8}\) are not equivalent.

Hence, option (C) is incorrect.

Option (D):

In figure -I: \(\dfrac {4}{6}\) part is shaded

In figure -II: \(\dfrac {12}{18}\) part is shaded

Since size of both the bars is same, so we compare the shaded portion.

We can see that shaded parts of both the figures are equal.

Figure-II is obtained by partitioning each part of figure-I into 3 equal segments.

Size of each segment is \(\dfrac {1}{18}\).

image

By combining 12 segments \((\dfrac {1}{18})\), we get \(\dfrac {12}{18}\).

Number of shaded parts in fraction form:

For figure-I = \(\dfrac {4}{6}\)

For figure-II = \(\dfrac {12}{18}\)

\(\therefore\,\dfrac {4}{6}=\dfrac {12}{18}\)

Thus, these figure represents equivalent fractions.

Hence, option (D) is correct.

In each of the options, a pair of shaded bars is given. Which one of the options represents the pair of equivalent fractions? [Assume the size of both the bars in each option is same.]

A image
B image
C image
D image

Option D is Correct

Number Line Representation of Equivalent Fractions

Two fractions are equivalent or equal if they represent the same point on the number line.

For example: Consider \(\dfrac {1}{2}\) and \(\dfrac {3}{6}\).

Step 1: Define the interval from \(0\) to \(1\) and divide it into \(2\) equal segments.

Step 2: The size of each segment is \(\dfrac {1}{2}\).

Step 3: Now, if we divide each segment of length \(\dfrac {1}{2}\) into 3 equal parts, we will get:

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

Step 4: We can see that the fractions \(\dfrac {1}{2}\) and \(\dfrac {3}{6}\) represent the same point on the number line.

Thus, they are equivalent fractions.

Illustration Questions

Point \(K\) represents the equivalent fraction of \(\dfrac {3}{4}\). Which one of the following options represents the value of \(K\)?

A \(\dfrac {5}{8}\)

B \(\dfrac {9}{12}\)

C \(\dfrac {18}{20}\)

D \(\dfrac {6}{12}\)

×

Given number line:

image

Two fractions are equivalent or equal if they represent the same point on the number line.

To find the value of \(K\), we need to calculate an equivalent fraction for \(\dfrac {3}{4}\).

Thus, we will check each option one by one.

Option (A):

To get \(\dfrac {5}{8}\) from \(\dfrac {3}{4}\), we need to divide each segment into two equal segments \((4×2=8)\).

 By counting  \(5\) segments on the number line, we plot  \(\dfrac {5}{8}\) . We observe that \(\dfrac {5}{8}\) and \(\dfrac {3}{4}\) do not represent the same point on the number line.

\(\therefore\) \(\dfrac {5}{8}\) is not equivalent to \(\dfrac {3}{4}\).

Hence, option (A) is incorrect.

image

Option (B):

To get \(\dfrac {9}{12}\) from \(\dfrac {3}{4}\), we need to divide each segment into \(3\) equal segments \((4×3=12)\).

By counting \(9\) segments on the number line, we plot \(\dfrac {9}{12}\).  We observe that \(\dfrac {9}{12}\) and \(\dfrac {3}{4}\) represent the same point on the number line.

\(\therefore\) \(\dfrac {9}{12}\) is equivalent to \(\dfrac {3}{4}\).

Hence, option (B) is correct.

image

Option (C):

To get \(\dfrac {18}{20}\) from \(\dfrac {3}{4}\), we need to divide each segment into \(5\) equal segments \((4×5=20)\).

By counting  \(18\) segments on the number line, we plot  \(\dfrac {18}{20}\).  We observe that \(\dfrac {18}{20}\) and \(\dfrac {3}{4}\) do not represent the same point on the number line.

\(\therefore\) \(\dfrac {18}{20}\) is not equivalent to \(\dfrac {3}{4}\).

Hence, option (C) is incorrect.

image

Option (D):

To get \(\dfrac {6}{12}\) from \(\dfrac {3}{4}\), we need to divide each segment into \(3\) equal segments \((4×3=12)\).

 By counting \(6\) segments on the number line, we plot \(\dfrac {6}{12}\).  We observe that \(\dfrac {6}{12}\) and \(\dfrac {3}{4}\) do not represent the same point on the number line.

\(\therefore\) \(\dfrac {6}{12}\) is not equivalent to\(\dfrac {3}{4}\).

Hence, option (D) is incorrect.

image

Point \(K\) represents the equivalent fraction of \(\dfrac {3}{4}\). Which one of the following options represents the value of \(K\)?

image
A

\(\dfrac {5}{8}\)

.

B

\(\dfrac {9}{12}\)

C

\(\dfrac {18}{20}\)

D

\(\dfrac {6}{12}\)

Option B is Correct

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