GSEB Class 6 Maths Solutions Chapter 7 Fractions Exercise 7.4

Gujarat Board Textbook Solutions Class 6 Maths Chapter 7 Fractions Ex 7.4

Question 1. Write shaded portion as fraction. Arrange them in ascending and descending order using correct sign '<', '=' '>' between the fractions:
(a)
(b)
(c) (i) show \( \frac { 2 }{ 6 } \), \( \frac { 4 }{ 6 } \), \( \frac { 8 }{ 6 } \) and \( \frac { 6 }{ 6 } \) on the number line.
(ii) Put appropriate signs between the fractions given.
\( \frac { 5 }{ 6 } \), \( \frac { 2 }{ 6 } \)
\( \frac { 3 }{ 6 } \), \( 0 \)
\( \frac { 1 }{ 6 } \), \( \frac { 8 }{ 6 } \)
\( \frac { 5 }{ 6 } \), \( \frac { 6 }{ 6 } \)
Answer:
(a) Figure (i) shows the fraction \( \frac { 3 }{ 8 } \).
Figure (ii) shows the fraction \( \frac { 6 }{ 8 } \).
Figure (iii) shows the fraction \( \frac { 4 }{ 8 } \).
Figure (iv) shows the fraction \( \frac { 1 }{ 8 } \).
These fractions are similar, so we arrange them based on their top numbers (numerators).
Ascending order: \( \frac { 1 }{ 8 } < \frac { 3 }{ 8 } < \frac { 4 }{ 8 } < \frac { 6 }{ 8 } \)
Descending order: \( \frac { 6 }{ 8 } > \frac { 4 }{ 8 } > \frac { 3 }{ 8 } > \frac { 1 }{ 8 } \)
(b) Figure (i) shows the fraction \( \frac { 8 }{ 9 } \).
Figure (ii) shows the fraction \( \frac { 4 }{ 9 } \).
Figure (iii) shows the fraction \( \frac { 3 }{ 9 } \).
Figure (iv) shows the fraction \( \frac { 6 }{ 9 } \).
These fractions are similar, but with different top numbers (numerators).
Ascending order: \( \frac { 3 }{ 9 } < \frac { 4 }{ 9 } < \frac { 6 }{ 9 } < \frac { 8 }{ 9 } \)
Descending order: \( \frac { 8 }{ 9 } > \frac { 6 }{ 9 } > \frac { 4 }{ 9 } > \frac { 3 }{ 9 } \)
(c) (i) Given fractions are: \( \frac { 2 }{ 6 } \), \( \frac { 4 }{ 6 } \), \( \frac { 8 }{ 6 } \) and \( \frac { 6 }{ 6 } \).
We can show them on a number line as follows:

(ii) We have:
\( \frac { 5 }{ 6 } > \frac { 2 }{ 6 } \)
\( \frac { 3 }{ 6 } > 0 \)
\( \frac { 6 }{ 6 } > \frac { 1 }{ 6 } \)
\( \frac { 8 }{ 6 } > \frac { 5 }{ 6 } \)
In simple words: For fractions with the same bottom number (denominator), compare their top numbers (numerators) to see which is bigger or smaller. For fractions like \( \frac{6}{6} \) which is 1, and 0, compare based on their actual values.

Exam Tip: When comparing like fractions, always focus on the numerators. The larger numerator means the larger fraction. For unlike fractions, either make denominators common or use cross-multiplication.

Question 2. Compare the fractions and put an appropriate sign.
(a) \( \frac { 3 }{ 6 } \square \frac { 5 }{ 6 } \)
(b) \( \frac { 1 }{ 7 } \square \frac { 1 }{ 4 } \)
(c) \( \frac { 4 }{ 5 } \square \frac { 5 }{ 5 } \)
(d) \( \frac { 3 }{ 5 } \square \frac { 3 }{ 7 } \)
Answer:
(a) Here, these are 'like fractions', so we compare them only by their numerators.
Therefore, \( \frac { 3 }{ 6 } < \frac { 5 }{ 6 } \)
(b) Here, these are 'unlike fractions' with the same numerators, so we compare them only by their denominators.
Therefore, \( \frac { 1 }{ 7 } < \frac { 1 }{ 4 } \)
(c) These are like fractions, so we compare by their numerators only.
Therefore, \( \frac { 4 }{ 5 } < \frac { 5 }{ 5 } \)
(d) These are 'unlike fractions' with the same numerators, so we compare them only by their denominators.
Therefore, \( \frac { 3 }{ 5 } > \frac { 3 }{ 7 } \)
In simple words: If fractions have the same bottom number, the one with the bigger top number is larger. If they have the same top number, the one with the smaller bottom number is larger.

Exam Tip: Remember the rule for fractions with the same numerator: the fraction with the smaller denominator is actually the larger value because the whole is divided into fewer, larger pieces.

Question 3. Make five more such pairs and put appropriate signs.
(i) \( \frac { 4 }{ 9 } \square \frac { 6 }{ 9 } \)
(ii) \( \frac { 5 }{ 7 } \square \frac { 0 }{ 7 } \)
(iii) \( \frac { 7 }{ 13 } \square \frac { 9 }{ 13 } \)
(iv) \( \frac { 11 }{ 17 } \square \frac { 13 }{ 17 } \)
(v) \( \frac { 9 }{ 10 } \square \frac { 11 }{ 10 } \)
Answer:
After writing the appropriate signs, we have:
(i) \( \frac { 4 }{ 9 } < \frac { 6 }{ 9 } \)
(ii) \( \frac { 5 }{ 7 } > \frac { 0 }{ 7 } \)
(iii) \( \frac { 7 }{ 13 } < \frac { 9 }{ 13 } \)
(iv) \( \frac { 11 }{ 17 } < \frac { 13 }{ 17 } \)
(v) \( \frac { 9 }{ 10 } < \frac { 11 }{ 10 } \)
In simple words: When comparing fractions with the same bottom number, simply look at the top numbers. The fraction with the bigger top number is the larger one.

Exam Tip: Practicing with diverse fraction pairs helps build intuition for comparison. Pay attention to fractions where one is zero or where numerators are larger than denominators.

Question 4. Look at the figures and write '<' '>'or '=' between the given pairs of fractions.
(a) \( \frac { 1 }{ 6 } \square \frac { 1 }{ 3 } \)
(b) \( \frac { 3 }{ 4 } \square \frac { 2 }{ 6 } \)
(c) \( \frac { 2 }{ 3 } \square \frac { 2 }{ 4 } \)
(d) \( \frac { 6 }{ 6 } \square \frac { 3 }{ 3 } \)
(e) \( \frac { 5 }{ 6 } \square \frac { 5 }{ 5 } \)
Make five more such problems and solve them with your friends.
Answer:
(a) In the figure, \( \frac { 1 }{ 6 } \) is to the left of \( \frac { 1 }{ 3 } \).
Therefore, \( \frac { 1 }{ 6 } < \frac { 1 }{ 3 } \)
(b) In the figure, \( \frac { 3 }{ 4 } \) is to the right of \( \frac { 2 }{ 6 } \).
Therefore, \( \frac { 3 }{ 4 } > \frac { 2 }{ 6 } \)
(c) In the figure, \( \frac { 2 }{ 3 } \) is to the right of \( \frac { 2 }{ 4 } \).
Therefore, \( \frac { 2 }{ 3 } > \frac { 2 }{ 4 } \)
(d) In the figure, \( \frac { 6 }{ 6 } \) is at the same point as \( \frac { 3 }{ 3 } \).
Therefore, \( \frac { 6 }{ 6 } = \frac { 3 }{ 3 } \)
(e) In the figure, \( \frac { 5 }{ 6 } \) is at the same point as \( \frac { 5 }{ 5 } \).
Therefore, \( \frac { 5 }{ 6 } < \frac { 5 }{ 5 } \)
Five more examples can be given as below, looking at the previous figure and writing '<', '>' or '=' between the following pairs of fractions:
(i) \( \frac { 1 }{ 2 } > \frac { 1 }{ 5 } \)
(ii) \( \frac { 2 }{ 6 } < \frac { 3 }{ 5 } \)
(iii) \( \frac { 2 }{ 5 } < \frac { 2 }{ 4 } \)
(iv) \( \frac { 3 }{ 3 } = \frac { 5 }{ 5 } \)
(v) \( \frac { 0 }{ 5 } = \frac { 0 }{ 2 } \)
In simple words: By observing the positions on the number line or visual representation, you can easily tell if one fraction is larger, smaller, or equal to another. For example, a fraction further to the right on a number line is bigger.

Exam Tip: Visual models like number lines or shaded diagrams are useful for understanding fraction comparisons, especially when developing initial intuition. Always double-check with mathematical methods.

Question 5. How quickly can you do this? Fill appropriate sign. ('<', '=', '>')
(a) \( \frac { 1 }{ 2 } \square \frac { 1 }{ 5 } \)
(b) \( \frac { 2 }{ 4 } \square \frac { 3 }{ 6 } \)
(c) \( \frac { 3 }{ 5 } \square \frac { 2 }{ 3 } \)
(d) \( \frac { 3 }{ 4 } \square \frac { 2 }{ 8 } \)
(e) \( \frac { 3 }{ 5 } \square \frac { 6 }{ 5 } \)
(f) \( \frac { 7 }{ 9 } \square \frac { 3 }{ 9 } \)
(g) \( \frac { 1 }{ 4 } \square \frac { 2 }{ 8 } \)
(h) \( \frac { 6 }{ 10 } \square \frac { 4 }{ 5 } \)
(i) \( \frac { 3 }{ 4 } \square \frac { 7 }{ 8 } \)
(j) \( \frac { 6 }{ 10 } \square \frac { 4 }{ 5 } \)
(k) \( \frac { 5 }{ 7 } \square \frac { 15 }{ 21 } \)
Answer:
We use the cross-product method for faster calculations.
(a) \( 1 \times 5 = 5 \); \( 1 \times 2 = 2 \). Since \( 5 > 2 \),
Therefore, \( \frac { 1 }{ 2 } > \frac { 1 }{ 5 } \)
(b) \( 2 \times 6 = 12 \); \( 4 \times 3 = 12 \). Since \( 12 = 12 \),
Therefore, \( \frac { 2 }{ 4 } = \frac { 3 }{ 6 } \)
(c) \( 3 \times 3 = 9 \); \( 2 \times 5 = 10 \). Since \( 9 < 10 \),
Therefore, \( \frac { 3 }{ 5 } < \frac { 2 }{ 3 } \)
(d) \( 3 \times 8 = 24 \); \( 4 \times 2 = 8 \). Since \( 24 > 8 \),
Therefore, \( \frac { 3 }{ 4 } > \frac { 2 }{ 8 } \)
(e) \( 3 \times 5 = 15 \); \( 5 \times 6 = 30 \). Since \( 15 < 30 \),
Therefore, \( \frac { 3 }{ 5 } < \frac { 6 }{ 5 } \)
(f) \( 7 \times 9 = 63 \); \( 9 \times 3 = 27 \). Since \( 63 > 27 \),
Therefore, \( \frac { 7 }{ 9 } > \frac { 3 }{ 9 } \)
(g) \( 1 \times 8 = 8 \); \( 4 \times 2 = 8 \). Since \( 8 = 8 \),
Therefore, \( \frac { 1 }{ 4 } = \frac { 2 }{ 8 } \)
(h) \( 6 \times 5 = 30 \); \( 4 \times 10 = 40 \). Since \( 30 < 40 \),
Therefore, \( \frac { 6 }{ 10 } < \frac { 4 }{ 5 } \)
(i) \( 3 \times 8 = 24 \); \( 4 \times 7 = 28 \). Since \( 24 < 28 \),
Therefore, \( \frac { 3 }{ 4 } < \frac { 7 }{ 8 } \)
(j) \( 6 \times 5 = 30 \); \( 10 \times 4 = 40 \). Since \( 30 < 40 \),
Therefore, \( \frac { 6 }{ 10 } < \frac { 4 }{ 5 } \)
(k) \( 5 \times 21 = 105 \); \( 7 \times 15 = 105 \). Since \( 105 = 105 \),
Therefore, \( \frac { 5 }{ 7 } = \frac { 15 }{ 21 } \)
In simple words: The cross-product method involves multiplying the numerator of the first fraction by the denominator of the second, and vice-versa. Then, you compare these two products to determine the relationship between the fractions.

Exam Tip: The cross-product method is a reliable and quick way to compare any two fractions, especially useful when denominators are different and finding a common denominator would take more time.

Question 6. The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form.
(a) \( \frac { 2 }{ 12 } \)
(b) \( \frac { 3 }{ 15 } \)
(c) \( \frac { 8 }{ 50 } \)
(d) \( \frac { 16 }{ 100 } \)
(e) \( \frac { 10 }{ 60 } \)
(f) \( \frac { 15 }{ 75 } \)
(g) \( \frac { 12 }{ 60 } \)
(h) \( \frac { 16 }{ 96 } \)
(i) \( \frac { 12 }{ 75 } \)
(j) \( \frac { 12 }{ 72 } \)
(k) \( \frac { 3 }{ 18 } \)
(l) \( \frac { 4 }{ 25 } \)
Answer:
We simplify each fraction:
(a) \( \frac { 2 }{ 12 } = \frac { 2 \div 2 }{ 12 \div 2 } = \frac { 1 }{ 6 } \) [HCF of 2 and 12 = 2]
(b) \( \frac { 3 }{ 15 } = \frac { 3 \div 3 }{ 15 \div 3 } = \frac { 1 }{ 5 } \) [HCF of 3 and 15 = 3]
(c) \( \frac { 8 }{ 50 } = \frac { 8 \div 2 }{ 50 \div 2 } = \frac { 4 }{ 25 } \) [HCF of 8 and 50 = 2]
(d) \( \frac { 16 }{ 100 } = \frac { 16 \div 4 }{ 100 \div 4 } = \frac { 4 }{ 25 } \) [HCF of 16 and 100 = 4]
(e) \( \frac { 10 }{ 60 } = \frac { 10 \div 10 }{ 60 \div 10 } = \frac { 1 }{ 6 } \) [HCF of 10 and 60 = 10]
(f) \( \frac { 15 }{ 75 } = \frac { 15 \div 15 }{ 75 \div 15 } = \frac { 1 }{ 5 } \) [HCF of 15 and 75 = 15]
(g) \( \frac { 12 }{ 60 } = \frac { 12 \div 12 }{ 60 \div 12 } = \frac { 1 }{ 5 } \) [HCF of 12 and 60 = 12]
(h) \( \frac { 16 }{ 96 } = \frac { 16 \div 16 }{ 96 \div 16 } = \frac { 1 }{ 6 } \) [HCF of 16 and 96 = 16]
(i) \( \frac { 12 }{ 75 } = \frac { 12 \div 3 }{ 75 \div 3 } = \frac { 4 }{ 25 } \) [HCF of 12 and 75 = 3]
(j) \( \frac { 12 }{ 72 } = \frac { 12 \div 12 }{ 72 \div 12 } = \frac { 1 }{ 6 } \) [HCF of 12 and 72 = 12]
(k) \( \frac { 3 }{ 18 } = \frac { 3 \div 3 }{ 18 \div 3 } = \frac { 1 }{ 6 } \) [HCF of 3 and 18 = 3]
(l) \( \frac { 4 }{ 25 } = \frac { 4 \div 1 }{ 25 \div 1 } = \frac { 4 }{ 25 } \) [HCF of 4 and 25 = 1]
Grouping the simplest forms, we have:
(I) \( \frac { 2 }{ 12 } = \frac { 10 }{ 60 } = \frac { 16 }{ 96 } = \frac { 12 }{ 72 } = \frac { 3 }{ 18 } \) [each \( \frac { 1 }{ 6 } \)]
(II) \( \frac { 3 }{ 15 } = \frac { 15 }{ 75 } = \frac { 12 }{ 60 } \) [each \( \frac { 1 }{ 5 } \)]
(III) \( \frac { 8 }{ 50 } = \frac { 16 }{ 100 } = \frac { 12 }{ 75 } = \frac { 4 }{ 25 } \) [each \( \frac { 4 }{ 25 } \)]
In simple words: To group equivalent fractions, first reduce each fraction to its simplest form by dividing both the top and bottom numbers by their largest common factor. Then, collect all fractions that simplify to the same value into one group.

Exam Tip: Always look for the Highest Common Factor (HCF) to simplify fractions in one step. This method is efficient and helps avoid errors from multiple division steps.

Question 7. Find answers to the following. Write and indicate how you solved them,
(a) Is \( \frac { 5 }{ 9 } \) equal to \( \frac { 4 }{ 5 } \) ?
(b) Is \( \frac { 9 }{ 16 } \) equal to \( \frac { 5 }{ 9 } \) ?
(c) Is \( \frac { 4 }{ 5 } \) equal to \( \frac { 16 }{ 20 } \) ?
(d) Is \( \frac { 1 }{ 15 } \) equal to \( \frac { 4 }{ 30 } \) ?
Answer:
(a) By cross product, we have
\( 5 \times 5 = 25 \) and \( 9 \times 4 = 36 \)
Since \( 25 \neq 36 \), this means \( 5 \times 5 \neq 9 \times 4 \).
So, \( \frac { 5 }{ 9 } \) is not equal to \( \frac { 4 }{ 5 } \).
(b) By cross product, we have
\( 9 \times 9 = 81 \) and \( 16 \times 5 = 80 \)
Since \( 81 \neq 80 \), this means \( 9 \times 9 \neq 16 \times 5 \).
So, \( \frac { 9 }{ 16 } \) is not equal to \( \frac { 5 }{ 9 } \).
(c) By cross product, we have
\( 4 \times 20 = 80 \) and \( 5 \times 16 = 80 \)
Since \( 80 = 80 \), this means \( 4 \times 20 = 5 \times 16 \).
So, \( \frac { 4 }{ 5 } \) and \( \frac { 16 }{ 20 } \) are equal.
(d) By cross product, we have
\( 1 \times 30 = 30 \) and \( 4 \times 15 = 60 \)
Since \( 30 \neq 60 \), this means \( 1 \times 30 \neq 4 \times 15 \).
So, \( \frac { 1 }{ 15 } \) is not equal to \( \frac { 4 }{ 30 } \).
In simple words: To check if two fractions are equal, you can multiply diagonally (cross-multiply). If the products are the same, the fractions are equal. If the products are different, the fractions are not equal.

Exam Tip: The cross-multiplication method is a straightforward way to compare or check for equality between fractions, especially useful for fractions that cannot be easily simplified or made to have common denominators.

Question 8. Ila read 25 pages of a book containing 100 pages. Lauta read \( \frac { 2 }{ 5 } \) of the same book. Who read less?
Answer:
Fraction of book read by Ila = \( \frac { 25 }{ 100 } \)
To simplify this, \( \frac { 25 \div 25 }{ 100 \div 25 } = \frac { 1 }{ 4 } \).
Fraction of book read by Lauta = \( \frac { 2 }{ 5 } \).
To compare \( \frac { 1 }{ 4 } \) and \( \frac { 2 }{ 5 } \), we cross-multiply:
\( 1 \times 5 = 5 \)
\( 4 \times 2 = 8 \)
Since \( 5 < 8 \), this means \( \frac { 1 }{ 4 } < \frac { 2 }{ 5 } \).
Thus, Ila read fewer pages.
In simple words: First, turn Ila's pages read into a fraction and simplify it. Then, compare her fraction with Lauta's fraction using cross-multiplication. The smaller cross-product means a smaller fraction.

Exam Tip: Always simplify fractions to their lowest terms before comparing, if possible. This makes the numbers smaller and comparisons easier. Cross-multiplication is an effective way to compare fractions with different denominators.

Question 9. Rafiq exercised for \( \frac { 3 }{ 6 } \) of an hour, while Rohit exercised for \( \frac { 3 }{ 4 } \) of an hour Who exercised for a longer time?
Answer:
Fraction of an hour for which Rafiq exercised = \( \frac { 3 }{ 6 } \).
Fraction of an hour for which Rohit exercised = \( \frac { 3 }{ 4 } \).
To compare \( \frac { 3 }{ 6 } \) and \( \frac { 3 }{ 4 } \), we observe that these are 'unlike fractions' with the same numerator.
For 'unlike fractions' with the same numerator, the greater fraction has the lesser denominator.
Since \( 4 < 6 \),
Therefore, \( \frac { 3 }{ 4 } > \frac { 3 }{ 6 } \).
Rohit exercised for a longer time.
In simple words: When two fractions have the same top number, the one with the smaller bottom number is actually the bigger fraction. So, compare the bottom numbers to see who exercised more.

Exam Tip: When comparing fractions with the same numerator, the fraction with the smaller denominator represents a larger portion of the whole. This is a crucial rule for quick comparisons.

Question 10. In a class A of 25 students, 20 passed in first class; in another class B of 30 students, 24 passed in first class. In which class was a greater fraction of students getting first class?
Answer:
In class A, the fraction of students who got 1st division is:
\( \frac { 20 }{ 25 } = \frac { 20 \div 5 }{ 25 \div 5 } = \frac { 4 }{ 5 } \) ..........(i)
In class B, the fraction of students who got 1st division is:
\( \frac { 24 }{ 30 } = \frac { 24 \div 6 }{ 30 \div 6 } = \frac { 4 }{ 5 } \) ..........(ii)
From (i) and (ii), the same fraction \( \left(\frac{4}{5}\right) \) of students got 1st class in both classes.
In simple words: First, find the fraction of students who passed in first class for each class and simplify these fractions. If the simplified fractions are the same, then both classes had an equal proportion of students passing in the first class.

Exam Tip: Always simplify fractions to their simplest form before comparing them. This makes it easy to see if they are equivalent or which one is larger/smaller. In this case, simplifying quickly showed they were equal.