GSEB Class 6 Maths Solutions Chapter 7 Fractions Exercise 7.3

 

Question 1. Write the fractions. Are all these fractions equivalent?
(a)

(i)

(ii)

(iii)

(iv)
(b)

(i)

(ii)

(iii)

(iv)

(v)
Answer:
(a) Fraction represented by:
(i) \( \frac{1}{2} \)
(ii) \( \frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \)
(iii) \( \frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \)
(iv) \( \frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \)
Since, \( \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} \). Thus, all the fractions are equivalent.
(b) Fraction represented by:
(i) \( \frac{4}{12} = \frac{4 \div 4}{12 \div 4} = \frac{1}{3} \)
(ii) \( \frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \)
(iii) \( \frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \)
(iv) \( \frac{1}{3} \)
(v) \( \frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5} \)
Since, all the fractions do not represent the same fraction, i.e., \( \frac{4}{12} \ne \frac{2}{4} \ne \frac{3}{6} \ne \frac{1}{3} \ne \frac{2}{5} \). The fractions are not equivalent.
In simple words: For part (a), all the pictures show half of the shape shaded, so their fractions are all equal to \( \frac{1}{2} \). This means they are equivalent. For part (b), the pictures show different amounts shaded, like one-third, one-half, and two-fifths. Because they are not all the same fraction, they are not equivalent.

Exam Tip: To find if fractions are equivalent, simplify each fraction to its simplest form. If the simplest forms are identical, the original fractions are equivalent.

 

Question 2. Write the fractions and pair up the equivalent fractions from each row.
(a)

(b)

Answer:
(a) Fraction represented by the figure:
(a) \( \frac{1}{2} \)
(b) \( \frac{2}{4} = \frac{1}{2} \)
(c) \( \frac{3}{3} = \frac{1}{1} \)
(d) \( \frac{4}{8} = \frac{1}{2} \)
(e) \( \frac{3}{4} \)

(b) Fraction represented by the figure:
(i) \( \frac{1}{2} \)
(ii) \( \frac{2}{3} \)
(iii) \( \frac{3}{4} \)
(iv) \( \frac{2}{8} = \frac{1}{4} \)
(v) \( \frac{1}{4} \)

Now, the equivalent fractions are:
(a) \( \rightarrow \) (ii) \( \left[ \frac{1}{2} = \frac{2}{4} \right] \)
(b) \( \rightarrow \) (iv) \( \left[ \frac{2}{3} = \frac{8}{12} \right] \)
(c) \( \rightarrow \) (i) \( \left[ \frac{3}{3} = \frac{9}{3} \right] \)
(d) \( \rightarrow \) (v) \( \left[ \frac{1}{4} = \frac{4}{16} \right] \)
(e) \( \rightarrow \) (iii) \( \left[ \frac{3}{4} = \frac{12}{16} \right] \)
In simple words: First, you write down the fraction that each picture shows. Then, you simplify these fractions to their smallest form if possible. After that, you look for fractions in the first row that are equal to fractions in the second row to find the matching pairs. For each pair, you also write two more fractions that are equivalent by multiplying the numerator and denominator by the same number.

Exam Tip: When matching equivalent fractions, it is often easiest to simplify all fractions to their lowest terms first. This makes direct comparison straightforward.

 

Question 3. Replace \( \boxed{\phantom{X}} \) in each of the following by the correct number:
(a) \( \frac{2}{7} = \frac{8}{\boxed{\phantom{X}}} \)
(b) \( \frac{5}{8} = \frac{10}{\boxed{\phantom{X}}} \)
(c) \( \frac{3}{5} = \frac{\boxed{\phantom{X}}}{20} \)
(d) \( \frac{45}{60} = \frac{15}{\boxed{\phantom{X}}} \)
(e) \( \frac{18}{24} = \frac{\boxed{\phantom{X}}}{4} \)
Answer:
(a) We have \( \frac{2}{7} = \frac{8}{\boxed{\phantom{X}}} \)
\( 2 \times \boxed{\phantom{X}} = 7 \times 8 \)
\( \boxed{\phantom{X}} = \frac{7 \times 8}{2} = 28 \)
\( \implies \frac{2}{7} = \frac{8}{28} \)
(b) We have \( \frac{5}{8} = \frac{10}{\boxed{\phantom{X}}} \)
\( 5 \times \boxed{\phantom{X}} = 10 \times 8 \)
\( \boxed{\phantom{X}} = \frac{10 \times 8}{5} = \frac{80}{5} = 16 \)
\( \implies \frac{5}{8} = \frac{10}{16} \)
(c) We have \( \frac{3}{5} = \frac{\boxed{\phantom{X}}}{20} \)
\( 3 \times 20 = \boxed{\phantom{X}} \times 5 \)
\( \boxed{\phantom{X}} = \frac{3 \times 20}{5} = \frac{60}{5} = 12 \)
\( \implies \frac{3}{5} = \frac{12}{20} \)
(d) We have \( \frac{45}{60} = \frac{15}{\boxed{\phantom{X}}} \)
\( 45 \times \boxed{\phantom{X}} = 15 \times 60 \)
\( \boxed{\phantom{X}} = \frac{15 \times 60}{45} = \frac{900}{45} = 20 \)
\( \implies \frac{45}{60} = \frac{15}{20} \)
(e) We have \( \frac{18}{24} = \frac{\boxed{\phantom{X}}}{4} \)
\( 18 \times 4 = 24 \times \boxed{\phantom{X}} \)
\( \boxed{\phantom{X}} = \frac{18 \times 4}{24} = \frac{72}{24} = 3 \)
\( \implies \frac{18}{24} = \frac{3}{4} \)
In simple words: To find the missing number, you can cross-multiply. Multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first by the numerator of the second. Then, solve the resulting equation to get the unknown value.

Exam Tip: Remember that when two fractions are equivalent, their cross-products are always equal. This is a very useful property for finding missing numbers.

 

Question 4. Find the equivalent fraction of \( \frac{3}{5} \) having
(a) denominator 20
(b) numerator 9
(c) denominator 30
(d) numerator 27
Answer:
Let 'N' stand for the numerator and 'D' stand for the denominator.
(a) Here, it is required that the denominator be 20.
\( \frac{N}{20} = \frac{3}{5} \)
\( N \times 5 = 3 \times 20 \)
\( N = \frac{3 \times 20}{5} = \frac{60}{5} = 12 \)
The required fraction is \( \frac{12}{20} \).
(b) Here, it is required that the numerator be 9.
\( \frac{9}{D} = \frac{3}{5} \)
\( 5 \times 9 = 3 \times D \)
\( D = \frac{5 \times 9}{3} = \frac{45}{3} = 15 \)
The required fraction is \( \frac{9}{15} \).
(c) Here, it is required that the denominator be 30.
\( \frac{N}{30} = \frac{3}{5} \)
\( N \times 5 = 3 \times 30 \)
\( N = \frac{3 \times 30}{5} = \frac{90}{5} = 18 \)
The required fraction is \( \frac{18}{30} \).
(d) Here, it is required that the numerator be 27.
\( \frac{27}{D} = \frac{3}{5} \)
\( 27 \times 5 = 3 \times D \)
\( D = \frac{27 \times 5}{3} = \frac{135}{3} = 45 \)
The required fraction is \( \frac{27}{45} \).
In simple words: To find an equivalent fraction with a specific numerator or denominator, set up a proportion (two fractions equal to each other). Use cross-multiplication to solve for the unknown value. This method helps you find the matching number while keeping the fraction's value the same.

Exam Tip: When dealing with equivalent fractions, remember you multiply or divide both the numerator and denominator by the exact same non-zero number.

 

Question 5. Find the equivalent fraction of \( \frac{36}{48} \) with
(a) numerator 9
(b) denominator 4
Answer:
(a) Here, it is required that the numerator be 9.
\( \frac{9}{D} = \frac{36}{48} \)
\( 9 \times 48 = D \times 36 \)
\( D = \frac{9 \times 48}{36} = \frac{432}{36} = 12 \)
The required fraction is \( \frac{9}{12} \).
(b) Here, it is required that the denominator be 4.
\( \frac{N}{4} = \frac{36}{48} \)
\( N \times 48 = 4 \times 36 \)
\( N = \frac{4 \times 36}{48} = \frac{144}{48} = 3 \)
The required fraction is \( \frac{3}{4} \).
In simple words: To get an equivalent fraction, you either multiply or divide the top and bottom numbers by the same value. If you need a smaller numerator, you divide. If you need a smaller denominator, you also divide. Just make sure to use the same number for both!

Exam Tip: Always make sure to perform the same operation (multiplication or division) on both the numerator and the denominator to maintain the equivalence of the fraction.

 

Question 6. Check whether the given fractions are equivalent:
(a) \( \frac{5}{9}, \frac{30}{54} \)
(b) \( \frac{3}{10}, \frac{12}{50} \)
(c) \( \frac{7}{13}, \frac{5}{11} \)
Answer:
(a) For \( \frac{5}{9} \) and \( \frac{30}{54} \)
We have \( 5 \times 54 = 270 \) and \( 30 \times 9 = 270 \).
Since the cross-products are equal, \( 5 \times 54 = 30 \times 9 \). Thus, \( \frac{5}{9} \) and \( \frac{30}{54} \) are equivalent fractions.
(b) For \( \frac{3}{10} \) and \( \frac{12}{50} \)
We have \( 3 \times 50 = 150 \) and \( 12 \times 10 = 120 \).
Since the cross-products are not equal, \( 3 \times 50 \ne 12 \times 10 \). Thus, \( \frac{3}{10} \) and \( \frac{12}{50} \) are not equivalent fractions.
(c) For \( \frac{7}{13} \) and \( \frac{5}{11} \)
We have \( 7 \times 11 = 77 \) and \( 5 \times 13 = 65 \).
Since the cross-products are not equal, \( 7 \times 11 \ne 5 \times 13 \). Thus, \( \frac{7}{13} \) and \( \frac{5}{11} \) are not equivalent fractions.
In simple words: To check if two fractions are equivalent, multiply the top number of the first fraction by the bottom number of the second, and the bottom number of the first by the top number of the second. If both results are the same, the fractions are equivalent. If they are different, the fractions are not equivalent.

Exam Tip: Cross-multiplication is a quick and reliable method to determine if two fractions are equivalent without simplifying them.

 

Question 7. Reduce the following fractions to simplest form.
(a) \( \frac{48}{60} \)
(b) \( \frac{105}{60} \)
(c) \( \frac{84}{98} \)
(d) \( \frac{12}{52} \)
(e) \( \frac{7}{28} \)
Answer:
(a) For \( \frac{48}{60} \)
Factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Common factors are: 1, 2, 3, 4, 6 and 12.
The HCF (Highest Common Factor) of 48 and 60 is 12.
Now, \( \frac{48 \div 12}{60 \div 12} = \frac{4}{5} \).
Thus, the simplest form of \( \frac{48}{60} \) is \( \frac{4}{5} \).
(b) For \( \frac{150}{60} \)
To find the HCF of 150 and 60, we can use the division method:
Divide 150 by 60: \( 150 = 2 \times 60 + 30 \)
Divide 60 by 30: \( 60 = 2 \times 30 + 0 \)
The HCF of 150 and 60 is 30.
Now, \( \frac{150 \div 30}{60 \div 30} = \frac{5}{2} \).
Thus, the simplest form of \( \frac{150}{60} \) is \( \frac{5}{2} \).
(c) For \( \frac{84}{98} \)
To find the HCF of 84 and 98:
Divide 98 by 84: \( 98 = 1 \times 84 + 14 \)
Divide 84 by 14: \( 84 = 6 \times 14 + 0 \)
The HCF of 84 and 98 is 14.
Now, \( \frac{84 \div 14}{98 \div 14} = \frac{6}{7} \).
Thus, the simplest form of \( \frac{84}{98} \) is \( \frac{6}{7} \).
(d) For \( \frac{12}{52} \)
Factors of 12 are: 1, 2, 3, 4, 6 and 12.
Factors of 52 are: 1, 2, 4, 13, 26 and 52.
The HCF of 12 and 52 is 4.
Now, \( \frac{12 \div 4}{52 \div 4} = \frac{3}{13} \).
Thus, the simplest form of \( \frac{12}{52} \) is \( \frac{3}{13} \).
(e) For \( \frac{7}{28} \)
Factors of 7 are: 1 and 7.
Factors of 28 are: 1, 2, 4, 7, 14 and 28.
Common factors are: 1 and 7.
The HCF of 7 and 28 is 7.
Now, \( \frac{7 \div 7}{28 \div 7} = \frac{1}{4} \).
Thus, the simplest form of \( \frac{7}{28} \) is \( \frac{1}{4} \).
In simple words: To simplify a fraction, you need to divide both the top number (numerator) and the bottom number (denominator) by their biggest common factor (HCF). Keep dividing until you can't divide them any further by the same number, except for 1. The resulting fraction is the simplest form.

Exam Tip: To ensure a fraction is in its simplest form, verify that the only common factor between its numerator and denominator is 1.

 

Question 8. Ramesh had 20 pencils, Sheelu had 50 pencils and Jamaal had 80 pencils. Ramesh used up 10 pencils, Sheelu used up 25 pencils and Jamaal used up 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of her/his pencils?
Answer:
Fraction of pencils used by Ramesh \( = \frac{10}{20} \)
Fraction of pencils used by Sheelu \( = \frac{25}{50} \)
Fraction of pencils used by Jamaal \( = \frac{40}{80} \)
Now, let's simplify each fraction:
For Ramesh: \( \frac{10}{20} = \frac{10 \div 10}{20 \div 10} = \frac{1}{2} \)
For Sheelu: \( \frac{25}{50} = \frac{25 \div 25}{50 \div 25} = \frac{1}{2} \)
For Jamaal: \( \frac{40}{80} = \frac{40 \div 40}{80 \div 40} = \frac{1}{2} \)
Therefore, \( \frac{10}{20} = \frac{25}{50} = \frac{40}{80} \) because each equals \( \frac{1}{2} \).
Each person used up an equal fraction \( \left( \frac{1}{2} \right) \) of their pencils.
In simple words: First, write down how many pencils each person used as a fraction of their total pencils. Then, simplify each of these fractions. If all the simplified fractions are the same, it means they all used up an equal share of their pencils. In this case, everyone used exactly half of their pencils.

Exam Tip: When comparing fractions, always simplify them to their lowest terms. This makes it much easier to see if they are equal or not.

 

Question 9. Match the equivalent fractions and write two more for each.
(i) \( \frac{250}{400} \)
(ii) \( \frac{180}{200} \)
(iii) \( \frac{660}{990} \)
(iv) \( \frac{180}{360} \)
(v) \( \frac{220}{550} \)
(a) \( \frac{2}{3} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{5}{8} \)
(e) \( \frac{9}{10} \)
Answer:
The fractions (a), (b), (c), (d) and (e) are already in their simplest form.
We will reduce (i), (ii), (iii), (iv) and (v) into their simplest form.
(i) For \( \frac{250}{400} \)
Factors of 250 = \( 2 \times 5 \times 5 \times 5 \)
Factors of 400 = \( 2 \times 2 \times 2 \times 2 \times 5 \times 5 \)
The HCF of 250 and 400 = \( 2 \times 5 \times 5 = 50 \).
Now, \( \frac{250}{400} = \frac{250 \div 50}{400 \div 50} = \frac{5}{8} \).
So, \( \frac{250}{400} \rightarrow \frac{5}{8} \).
Match: (i) \( \rightarrow \) (d)
Two more equivalent fractions:
\( \frac{5}{8} \times \frac{2}{2} = \frac{10}{16} \)
\( \frac{5}{8} \times \frac{3}{3} = \frac{15}{24} \)

(ii) For \( \frac{180}{200} \)
Factors of 180 = \( 2 \times 2 \times 3 \times 3 \times 5 \)
Factors of 200 = \( 2 \times 2 \times 2 \times 5 \times 5 \)
The HCF of 180 and 200 = \( 2 \times 2 \times 5 = 20 \).
Now, \( \frac{180}{200} = \frac{180 \div 20}{200 \div 20} = \frac{9}{10} \).
So, \( \frac{180}{200} \rightarrow \frac{9}{10} \).
Match: (ii) \( \rightarrow \) (e)
Two more equivalent fractions:
\( \frac{9}{10} \times \frac{2}{2} = \frac{18}{20} \)
\( \frac{9}{10} \times \frac{3}{3} = \frac{27}{30} \)

(iii) For \( \frac{660}{990} \)
Factors of 660 = \( 2 \times 2 \times 3 \times 5 \times 11 \)
Factors of 990 = \( 2 \times 3 \times 3 \times 5 \times 11 \)
The HCF of 660 and 990 = \( 2 \times 3 \times 5 \times 11 = 330 \).
Now, \( \frac{660}{990} = \frac{660 \div 330}{990 \div 330} = \frac{2}{3} \).
So, \( \frac{660}{990} \rightarrow \frac{2}{3} \).
Match: (iii) \( \rightarrow \) (a)
Two more equivalent fractions:
\( \frac{2}{3} \times \frac{2}{2} = \frac{4}{6} \)
\( \frac{2}{3} \times \frac{3}{3} = \frac{6}{9} \)

(iv) For \( \frac{180}{360} \)
Factors of 180 = \( 2 \times 2 \times 3 \times 3 \times 5 \)
Factors of 360 = \( 2 \times 2 \times 2 \times 3 \times 3 \times 5 \)
The HCF of 180 and 360 = \( 2 \times 2 \times 3 \times 3 \times 5 = 180 \).
Now, \( \frac{180}{360} = \frac{180 \div 180}{360 \div 180} = \frac{1}{2} \).
So, \( \frac{180}{360} \rightarrow \frac{1}{2} \).
Match: (iv) \( \rightarrow \) (c)
Two more equivalent fractions:
\( \frac{1}{2} \times \frac{2}{2} = \frac{2}{4} \)
\( \frac{1}{2} \times \frac{3}{3} = \frac{3}{6} \)

(v) For \( \frac{220}{550} \)
Factors of 220 = \( 2 \times 2 \times 5 \times 11 \)
Factors of 550 = \( 2 \times 5 \times 5 \times 11 \)
The HCF of 220 and 550 = \( 2 \times 5 \times 11 = 110 \).
Now, \( \frac{220}{550} = \frac{220 \div 110}{550 \div 110} = \frac{2}{5} \).
So, \( \frac{220}{550} \rightarrow \frac{2}{5} \).
Match: (v) \( \rightarrow \) (b)
Two more equivalent fractions:
\( \frac{2}{5} \times \frac{2}{2} = \frac{4}{10} \)
\( \frac{2}{5} \times \frac{3}{3} = \frac{6}{15} \)
In simple words: To match equivalent fractions, first simplify each given fraction to its lowest form by finding the highest common factor (HCF) and dividing both the numerator and denominator by it. Then, pair the simplified fractions from the first list with the matching simplified fractions from the second list. To write two more equivalent fractions, simply multiply the numerator and denominator of the simplified fraction by the same number (like 2/2, 3/3, etc.).

Exam Tip: Always calculate the HCF correctly to ensure the fraction is reduced to its true simplest form. This is crucial for accurate matching and generating other equivalent fractions.