RBSE Solutions Class 6 Maths Chapter 5 Fractions Exercise 5.3

Question 1. Write fraction for each of the following diagram and then arrange them in ascending and descending order.
(i)
(ii)
Answer:
(i) For the first diagram, the fractions are \( \frac{5}{8}, \frac{3}{8}, \frac{7}{8}, \frac{2}{8}, \frac{1}{8} \).
Ascending order: \( \frac{1}{8} < \frac{2}{8} < \frac{3}{8} < \frac{5}{8} < \frac{7}{8} \)
Descending order: \( \frac{7}{8} > \frac{5}{8} > \frac{3}{8} > \frac{2}{8} > \frac{1}{8} \)
(ii) For the second diagram, the fractions are \( \frac{3}{6}, \frac{4}{6}, \frac{2}{6}, \frac{1}{6}, \frac{5}{6} \).
Ascending order: \( \frac{1}{6} < \frac{2}{6} < \frac{3}{6} < \frac{4}{6} < \frac{5}{6} \)
Descending order: \( \frac{5}{6} > \frac{4}{6} > \frac{3}{6} > \frac{2}{6} > \frac{1}{6} \)
Fractions with the same denominator are easy to compare based on their numerators.
In simple words: First, look at each picture and write down the fraction it shows. Then, put them in order from smallest to biggest (ascending) and from biggest to smallest (descending). Remember, for fractions with the same bottom number, you just need to compare the top numbers.

🎯 Exam Tip: When arranging fractions, if they have the same denominator, directly compare their numerators. If the denominators are different, find a common denominator first.

Question 2. Compare the two fractions and put a sign (<, >, =).
(i) \( \frac{5}{8}, \frac{9}{11} \)
(ii) \( \frac{3}{4}, \frac{1}{5} \)
(iii) \( \frac{3}{5}, \frac{3}{7} \)
Answer:
(i) To compare \( \frac{5}{8} \) and \( \frac{9}{11} \), find the L.C.M. of 8 and 11, which is 88.
\( \frac{5}{8} = \frac{5 \times 11}{8 \times 11} = \frac{55}{88} \)
\( \frac{9}{11} = \frac{9 \times 8}{11 \times 8} = \frac{72}{88} \)
Since \( \frac{55}{88} < \frac{72}{88} \), then \( \frac{5}{8} < \frac{9}{11} \).

(ii) To compare \( \frac{3}{4} \) and \( \frac{1}{5} \), find the L.C.M. of 4 and 5, which is 20.
\( \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \)
\( \frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \)
Since \( \frac{15}{20} > \frac{4}{20} \), then \( \frac{3}{4} > \frac{1}{5} \).

(iii) To compare \( \frac{3}{5} \) and \( \frac{3}{7} \), find the L.C.M. of 5 and 7, which is 35.
\( \frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35} \)
\( \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \)
Since \( \frac{21}{35} > \frac{15}{35} \), then \( \frac{3}{5} > \frac{3}{7} \).
Finding the least common multiple helps to compare fractions easily.
In simple words: To compare two fractions, make their bottom numbers (denominators) the same by finding the smallest common multiple. Then, look at the top numbers (numerators) to see which fraction is bigger, smaller, or if they are equal.

🎯 Exam Tip: When comparing fractions, convert them to equivalent fractions with a common denominator. The fraction with the larger numerator will be the larger fraction.

Question 3. All the following fractions represent three different numbers. Convert these in simple form and write these in group of those three.
(i) \( \frac{2}{12} \)
(ii) \( \frac{3}{15} \)
(iii) \( \frac{8}{50} \)
(iv) \( \frac{16}{100} \)
(v) \( \frac{10}{60} \)
(vi) \( \frac{15}{75} \)
(vii) \( \frac{18}{90} \)
(viii) \( \frac{16}{96} \)
(ix) \( \frac{12}{75} \)
(x) \( \frac{12}{72} \)
(xi) \( \frac{10}{50} \)
(xii) \( \frac{4}{25} \)
Answer:
First, we simplify each fraction:
(i) \( \frac{2}{12} = \frac{1}{6} \)
(ii) \( \frac{3}{15} = \frac{1}{5} \)
(iii) \( \frac{8}{50} = \frac{4}{25} \)
(iv) \( \frac{16}{100} = \frac{4}{25} \)
(v) \( \frac{10}{60} = \frac{1}{6} \)
(vi) \( \frac{15}{75} = \frac{1}{5} \)
(vii) \( \frac{18}{90} = \frac{1}{5} \)
(viii) \( \frac{16}{96} = \frac{1}{6} \)
(ix) \( \frac{12}{75} = \frac{4}{25} \)
(x) \( \frac{12}{72} = \frac{1}{6} \)
(xi) \( \frac{10}{50} = \frac{1}{5} \)
(xii) \( \frac{4}{25} = \frac{4}{25} \)

Now, we group the equivalent fractions:
Group-I: Fractions equivalent to \( \frac{1}{6} \)
\( \frac{2}{12}, \frac{10}{60}, \frac{16}{96}, \frac{12}{72} \)
These all are equivalent to the fraction \( \frac{1}{6} \).

Group-II: Fractions equivalent to \( \frac{1}{5} \)
\( \frac{3}{15}, \frac{15}{75}, \frac{18}{90}, \frac{10}{50} \)
These all are equivalent to the fraction \( \frac{1}{5} \).

Group-III: Fractions equivalent to \( \frac{4}{25} \)
\( \frac{8}{50}, \frac{16}{100}, \frac{12}{75}, \frac{4}{25} \)
These all are equivalent to the fraction \( \frac{4}{25} \).
Simplifying fractions helps us see which ones are actually the same value.
In simple words: First, make each fraction as simple as possible by dividing the top and bottom numbers by their biggest common factor. Then, collect all the fractions that simplify to the same value into groups. You should find three groups in total.

🎯 Exam Tip: Always reduce fractions to their simplest form to easily compare and group them. This makes calculations less complex and clarifies relationships between fractions.

Question 4. Answer the following and show how did you solve these?
(i) Are \( \frac{12}{15} \) and \( \frac{15}{30} \) equal?
(ii) Are \( \frac{4}{5} \) and \( \frac{5}{6} \) equal?
(iii) Are \( \frac{3}{5} \) and \( \frac{9}{15} \) equal?
(iv) Are \( \frac{9}{16} \) and \( \frac{5}{9} \) equal?
Answer:
To check if fractions are equal, we simplify them to their lowest terms or find a common denominator.

(i) Compare \( \frac{12}{15} \) and \( \frac{15}{30} \).
Simplify \( \frac{12}{15} \): Divide both by 3, so \( \frac{12 \div 3}{15 \div 3} = \frac{4}{5} \).
Simplify \( \frac{15}{30} \): Divide both by 15, so \( \frac{15 \div 15}{30 \div 15} = \frac{1}{2} \).
Since \( \frac{4}{5} \neq \frac{1}{2} \), these fractions are not equal.

(ii) Compare \( \frac{4}{5} \) and \( \frac{5}{6} \).
These fractions are already in their simplest form.
To compare them, find the L.C.M. of 5 and 6, which is 30.
\( \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \)
\( \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \)
Since \( \frac{24}{30} \neq \frac{25}{30} \), these fractions are not equal.

(iii) Compare \( \frac{3}{5} \) and \( \frac{9}{15} \).
The fraction \( \frac{3}{5} \) is in its simplest form.
Simplify \( \frac{9}{15} \): Divide both by 3, so \( \frac{9 \div 3}{15 \div 3} = \frac{3}{5} \).
Since \( \frac{3}{5} = \frac{3}{5} \), these fractions are equal.

(iv) Compare \( \frac{9}{16} \) and \( \frac{5}{9} \).
These fractions are already in their simplest form.
To compare them, find the L.C.M. of 16 and 9, which is 144.
\( \frac{9}{16} = \frac{9 \times 9}{16 \times 9} = \frac{81}{144} \)
\( \frac{5}{9} = \frac{5 \times 16}{9 \times 16} = \frac{80}{144} \)
Since \( \frac{81}{144} \neq \frac{80}{144} \), these fractions are not equal.
The simplest way to check equality is to reduce both fractions to their lowest terms and then compare.
In simple words: To see if two fractions are the same, you can either simplify both fractions until they cannot be simplified any further and then compare them, or you can make the bottom numbers (denominators) of both fractions the same and then compare their top numbers (numerators). If the numbers match after doing this, the fractions are equal.

🎯 Exam Tip: To determine if two fractions are equal, always simplify both fractions to their lowest terms. If their lowest terms are identical, the fractions are equal.

Question 5. 20 students passed with first division in class A of 25 students. In class B, 24 students passed with first division out of 30 students. From which class more part of students passed with first division?
Answer:
For Class A:
Total students = 25
Students passed with first division = 20
Fraction of students passed with first division = \( \frac{20}{25} \)
Simplified form: Divide both by 5, \( \frac{20 \div 5}{25 \div 5} = \frac{4}{5} \).

For Class B:
Total students = 30
Students passed with first division = 24
Fraction of students passed with first division = \( \frac{24}{30} \)
Simplified form: Divide both by 6, \( \frac{24 \div 6}{30 \div 6} = \frac{4}{5} \).

Since the simplified form of both fractions is \( \frac{4}{5} \), an equal part of students passed with first division in both classes. This means the performance was the same proportionally.
In simple words: We write down the number of students who passed in first division as a fraction of the total students for each class. Then we simplify these fractions. Since both classes get the same simplified fraction, it means the same proportion of students passed in both classes.

🎯 Exam Tip: To compare proportions from different total numbers, always convert them into simplified fractions. This allows for a fair comparison of relative amounts.

Question 6. Rohit eats 4 chapatis out of total 8. Rohini eats \( \frac{1}{4} \) of total 8 chapatis. Who ate less?
Answer:
Total chapatis = 8

Rohit eats:
He ate 4 chapatis out of 8.
Fraction Rohit ate = \( \frac{4}{8} \).
Simplified fraction = \( \frac{1}{2} \).

Rohini eats:
She ate \( \frac{1}{4} \) of total 8 chapatis.
Number of chapatis Rohini ate = \( \frac{1}{4} \times 8 = 2 \) chapatis.
Fraction Rohini ate = \( \frac{2}{8} \).
Simplified fraction = \( \frac{1}{4} \).

Now compare the simplified fractions: \( \frac{1}{2} \) (Rohit) and \( \frac{1}{4} \) (Rohini).
We know that \( \frac{1}{4} < \frac{1}{2} \). A quarter is smaller than a half.
Therefore, Rohini ate less chapatis. It's often easier to compare when you convert fractions to the same denominator or to actual quantities.
In simple words: First, find out how much Rohit ate as a fraction of the total chapatis. Then, calculate how many chapatis Rohini ate from her given fraction. Once you have both amounts as simple fractions or actual numbers, you can easily tell who ate less.

🎯 Exam Tip: When comparing amounts, especially with fractions, it's helpful to convert them to a common unit, either by finding a common denominator for fractions or by converting fractions into whole numbers where possible.