CBSE Class 6 Mathematics Fractions Assignment Set 02

Question 1. Write six improper fractions with denominator 7.
Answer: An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.
Six improper fractions with denominator 7 are: \(\frac{8}{7}, \frac{9}{7}, \frac{10}{7}, \frac{11}{7}, \frac{12}{7}, \frac{13}{7}\)
In simple words: Fractions where the top number is bigger than 7 are improper fractions with denominator 7.

Exam Tip: Remember that for an improper fraction, numerator \(\geq\) denominator.

Question 2. Convert each of the following into an improper fraction: 
a) \(9\frac{3}{8}\)      b) \(8\frac{8}{13}\)      c) \(12\frac{7}{15}\)
Answer: To convert a mixed fraction into an improper fraction, multiply the whole number by the denominator and add the numerator, placing the result over the original denominator.
a) \(9\frac{3}{8} = \frac{(9 \times 8) + 3}{8} = \frac{72 + 3}{8} = \frac{75}{8}\)
b) \(8\frac{8}{13} = \frac{(8 \times 13) + 8}{13} = \frac{104 + 8}{13} = \frac{112}{13}\)
c) \(12\frac{7}{15} = \frac{(12 \times 15) + 7}{15} = \frac{180 + 7}{15} = \frac{187}{15}\)
In simple words: Multiply whole number by bottom number, then add top number. Keep the bottom number same.

Exam Tip: Always double check your multiplication before adding the numerator.

Question 3. Convert each of the following into a mixed fraction: 
a) \(\frac{101}{8}\)      b) \(\frac{87}{16}\)      c) \(\frac{95}{13}\)
Answer: To convert an improper fraction into a mixed fraction, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the numerator, and the divisor remains the denominator.
a) \(101 \div 8\): Quotient = 12, Remainder = 5. So, \(\frac{101}{8} = 12\frac{5}{8}\)
b) \(87 \div 16\): Quotient = 5, Remainder = 7. So, \(\frac{87}{16} = 5\frac{7}{16}\)
c) \(95 \div 13\): Quotient = 7, Remainder = 4. So, \(\frac{95}{13} = 7\frac{4}{13}\)
In simple words: Divide top by bottom to get the whole number part and put the leftover on top.

Exam Tip: The denominator of the fractional part in a mixed fraction is always the same as the denominator of the original improper fraction.

Question 4. Neelam has 25 pencils. She gives \(\frac{4}{5}\) of them to Meena. How many pencils does Meena get? How many pencils are left with Neelam?
Answer: Total number of pencils = 25
Pencils given to Meena = \(\frac{4}{5}\) of 25
\(= \frac{4}{5} \times 25 = 4 \times 5 = 20\) pencils
Pencils left with Neelam = Total pencils - Pencils given to Meena
\(= 25 - 20 = 5\) pencils
In simple words: Meena gets 20 pencils, leaving 5 pencils with Neelam.

Exam Tip: The word 'of' in fractions always means multiplication.

Question 5. Represent \(2\frac{2}{5}\) on the number line.
Answer: 1. Draw a number line with integers marked at equal intervals (0, 1, 2, 3, etc.).
2. Since the whole number part is 2, the number lies between 2 and 3.
3. Divide the segment between 2 and 3 into 5 equal parts (matching the denominator 5).
4. Count 2 parts to the right of 2 to mark the position of \(2\frac{2}{5}\) (which is equal to \(\frac{12}{5}\)).
In simple words: Go past 2 on the number line and take 2 small steps out of 5 towards 3.

Exam Tip: Clearly mark each subdivision between integers to show the fractional parts accurately.

Question 6. Find equivalent fraction of \(\frac{6}{11}\) having 
i) denominator 77      ii) numerator 54
Answer:
i) Given denominator = 77. To find what 11 is multiplied by to get 77, divide 77 by 11: \(77 \div 11 = 7\).
Multiply both numerator and denominator by 7:
\(\frac{6 \times 7}{11 \times 7} = \frac{42}{77}\)
ii) Given numerator = 54. Divide 54 by 6: \(54 \div 6 = 9\).
Multiply both numerator and denominator by 9:
\(\frac{6 \times 9}{11 \times 9} = \frac{54}{99}\)
In simple words: Multiply the top and bottom by the same number to keep the fraction equivalent.

Exam Tip: Find the multiplier first by dividing the target number by the given number.

Question 7. Find equivalent fraction of \(\frac{56}{70}\) with 
i) numerator 4      ii) denominator 10
Answer:
i) Given numerator = 4. Divide 56 by 4: \(56 \div 4 = 14\).
Divide both numerator and denominator by 14:
\(\frac{56 \div 14}{70 \div 14} = \frac{4}{5}\)
ii) Given denominator = 10. Divide 70 by 10: \(70 \div 10 = 7\).
Divide both numerator and denominator by 7:
\(\frac{56 \div 7}{70 \div 7} = \frac{8}{10}\)
In simple words: Divide top and bottom numbers by the same common factor.

Exam Tip: Reducing or scaling fractions requires performing the exact same operation on both numerator and denominator.

Question 8. Reduce the fraction \(\frac{84}{98}\) into its simplest form.
Answer: To reduce a fraction to its simplest form, divide the numerator and denominator by their HCF.
HCF of 84 and 98 is 14.
Divide both numerator and denominator by 14:
\(\frac{84 \div 14}{98 \div 14} = \frac{6}{7}\)
In simple words: Cancel out common factors until top and bottom numbers have no common factor other than 1.

Exam Tip: If you cannot find the HCF directly, cancel common prime factors like 2, 3, or 7 step by step.

Question 9. Arrange the following fractions in ascending & descending order: \(\frac{3}{4}, \frac{5}{8}, \frac{11}{12}, \frac{17}{24}\)
Answer: First, find the LCM of the denominators (4, 8, 12, 24).
LCM = 24.
Convert each fraction to an equivalent fraction with denominator 24:
\(\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}\)
\(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\)
\(\frac{11}{12} = \frac{11 \times 2}{12 \times 2} = \frac{22}{24}\)
\(\frac{17}{24} = \frac{17}{24}\)
Comparing the numerators (15, 17, 18, 22):
Ascending order: \(\frac{15}{24}, \frac{17}{24}, \frac{18}{24}, \frac{22}{24}\), which means \(\frac{5}{8}, \frac{17}{24}, \frac{3}{4}, \frac{11}{12}\)
Descending order: \(\frac{22}{24}, \frac{18}{24}, \frac{17}{24}, \frac{15}{24}\), which means \(\frac{11}{12}, \frac{3}{4}, \frac{17}{24}, \frac{5}{8}\)
In simple words: Make all denominators equal using LCM, then compare the top numbers.

Exam Tip: Always write back the final ordered list using the original fractions given in the question.

Question 10. The weight of an empty gas cylinder is \(16\frac{4}{5}\) kg and it contains \(14\frac{2}{3}\) kg of gas. What is the weight of cylinder filled with gas?
Answer: Weight of empty cylinder = \(16\frac{4}{5}\) kg = \(\frac{84}{5}\) kg
Weight of gas = \(14\frac{2}{3}\) kg = \(\frac{44}{3}\) kg
Total weight = Weight of cylinder + Weight of gas
\(= 16\frac{4}{5} + 14\frac{2}{3}\)
\(= (16 + 14) + \left(\frac{4}{5} + \frac{2}{3}\right)\)
\(= 30 + \left(\frac{12 + 10}{15}\right)\)
\(= 30 + \frac{22}{15}\)
Since \(\frac{22}{15} = 1\frac{7}{15}\):
\(= 30 + 1\frac{7}{15} = 31\frac{7}{15}\) kg
In simple words: Add the weight of the empty container and the weight of the gas inside.

Exam Tip: You can either add whole numbers and fractions separately or convert mixed fractions to improper fractions before adding.

Question 11. Simplify: \(5\frac{1}{6} - 3\frac{1}{4} + 3\frac{1}{3} + 4\)
Answer: Convert all mixed fractions to improper fractions:
\(5\frac{1}{6} = \frac{31}{6}\)
\(3\frac{1}{4} = \frac{13}{4}\)
\(3\frac{1}{3} = \frac{10}{3}\)
\(4 = \frac{4}{1}\)
Expression: \(\frac{31}{6} - \frac{13}{4} + \frac{10}{3} + \frac{4}{1}\)
LCM of denominators (6, 4, 3, 1) = 12.
Convert each fraction to have denominator 12:
\(\frac{31 \times 2}{6 \times 2} = \frac{62}{12}\)
\(\frac{13 \times 3}{4 \times 3} = \frac{39}{12}\)
\(\frac{10 \times 4}{3 \times 4} = \frac{40}{12}\)
\(\frac{4 \times 12}{1 \times 12} = \frac{48}{12}\)
Combine over the common denominator:
\(= \frac{62 - 39 + 40 + 48}{12}\)
\(= \frac{23 + 40 + 48}{12} = \frac{63 + 48}{12} = \frac{111}{12}\)
Simplifying by dividing numerator and denominator by 3:
\(= \frac{37}{4} = 9\frac{1}{4}\)
In simple words: Convert to improper fractions, find LCM to make denominators same, then add and subtract numerators.

Exam Tip: Be very careful with the plus and minus signs when combining numerators.

Question 12. Write natural numbers from 1 to 50. What fraction of them are prime numbers?
Answer: Total natural numbers from 1 to 50 = 50.
Prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Total number of prime numbers = 15.
Fraction of prime numbers = \(\frac{15}{50}\)
Simplifying by dividing by 5: \(\frac{3}{10}\)
In simple words: There are 15 prime numbers out of 50, which simplifies to \(\frac{3}{10}\).

Exam Tip: Remember that 1 is neither prime nor composite.

Question 13. Rama had Rs 40, Sheelu had Rs 100 & Jina had Rs 100. In school canteen, Rama spent Rs 20, Sheelu spent Rs 25 & Jina spent Rs 40. What fraction of money did each spend? Arrange the fraction of the expenditure in ascending order.
Answer:
- Rama: Spent Rs 20 out of Rs 40. Fraction = \(\frac{20}{40} = \frac{1}{2}\)
- Sheelu: Spent Rs 25 out of Rs 100. Fraction = \(\frac{25}{100} = \frac{1}{4}\)
- Jina: Spent Rs 40 out of Rs 100. Fraction = \(\frac{40}{100} = \frac{2}{5}\)
Fractions are: Rama = \(\frac{1}{2}\), Sheelu = \(\frac{1}{4}\), Jina = \(\frac{2}{5}\).
To arrange in ascending order, find LCM of denominators (2, 4, 5) = 20.
- Rama: \(\frac{1}{2} = \frac{10}{20}\)
- Sheelu: \(\frac{1}{4} = \frac{5}{20}\)
- Jina: \(\frac{2}{5} = \frac{8}{20}\)
Comparing numerators (5, 8, 10):
Ascending order of expenditure: \(\frac{5}{20}, \frac{8}{20}, \frac{10}{20}\), which is Sheelu (\(\frac{1}{4}\)), Jina (\(\frac{2}{5}\)), Rama (\(\frac{1}{2}\)).
In simple words: Find each person's spent fraction, make denominators equal, and compare.

Exam Tip: Always write the fraction with respect to the initial money each person had, not the total combined money.

Question 14. Shaleen was able to solve \(\frac{3}{7}\) of the equations given for maths homework. What fraction of homework was left incomplete?
Answer: Total homework considered as a whole = 1
Fraction of homework solved = \(\frac{3}{7}\)
Fraction of homework left incomplete = \(1 - \frac{3}{7}\)
\(= \frac{7}{7} - \frac{3}{7} = \frac{4}{7}\)
In simple words: Subtract the solved fraction from 1 to find the remaining unfinished homework.

Exam Tip: Whole is always represented as 1, where numerator equals denominator.

Question 15. What type of fraction can be converted into mixed fraction?
Answer: Improper fractions can be converted into mixed fractions.
In simple words: Only improper fractions (where top number is bigger than or equal to bottom number) can become mixed fractions.

Question 16. What is a fraction?
Answer: A fraction is a number representing a part of a whole. It is written in the form \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator (\(b \neq 0\)).
In simple words: A fraction shows equal parts of a whole object or collection.
Exam Tip: Never forget that the denominator can never be zero.

Question 17. What should be added to \(3\frac{7}{15}\) to get \(8\frac{1}{5}\)?
Answer: Let the required number be \(x\).
\(3\frac{7}{15} + x = 8\frac{1}{5}\)
Convert mixed fractions to improper fractions:
\(3\frac{7}{15} = \frac{52}{15}\)
\(8\frac{1}{5} = \frac{41}{5}\)
\(x = \frac{41}{5} - \frac{52}{15}\)
LCM of 5 and 15 is 15.
\(x = \frac{41 \times 3}{5 \times 3} - \frac{52}{15}\)
\(x = \frac{123}{15} - \frac{52}{15} = \frac{71}{15}\)
Converting into a mixed fraction: \(4\frac{11}{15}\)
In simple words: Subtract the starting number from the target number to find the missing part.

Exam Tip: Set up a linear equation with a variable to clearly understand whether to add or subtract.

Question 18. What fraction of a week is 3 days?
Answer: Total days in a week = 7 days.
Given days = 3 days.
Fraction = \(\frac{3}{7}\)
In simple words: Put the given number of days over the total days in a week.

Exam Tip: Both quantities must be in the same unit before writing as a fraction.

Question 19. Evaluate the following: \(\frac{1}{2} + \frac{3}{4} + \frac{1}{8} - \frac{1}{12}\)
Answer: Find the LCM of denominators (2, 4, 8, 12).
LCM = 24.
Convert each fraction to have denominator 24:
\(\frac{1}{2} = \frac{1 \times 12}{2 \times 12} = \frac{12}{24}\)
\(\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}\)
\(\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}\)
\(\frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24}\)
Combine over the common denominator:
\(= \frac{12 + 18 + 3 - 2}{24}\)
\(= \frac{30 + 3 - 2}{24} = \frac{33 - 2}{24} = \frac{31}{24} = 1\frac{7}{24}\)
In simple words: Make all denominators the same using LCM, then combine the top numbers.

Exam Tip: Follow the order of operations and pay close attention to subtraction signs.

Question 20. Define Like, Unlike and Unit fraction.
Answer:
- Like fractions: Fractions with the same denominators (e.g., \(\frac{2}{7}, \frac{5}{7}, \frac{4}{7}\)).
- Unlike fractions: Fractions with different denominators (e.g., \(\frac{1}{3}, \frac{2}{5}\)).
- Unit fraction: A fraction with 1 as its numerator (e.g., \(\frac{1}{4}, \frac{1}{9}\)).
In simple words: Like fractions share bottom numbers, unlike fractions have different bottom numbers, and unit fractions always have 1 on top.

Exam Tip: Learn definitions along with clear numerical examples to score full marks.

Question 21. What fraction is: 
a) 6 hours of a day 
b) 70 mm of a km 
c) 290 ml of a liter
Answer:
a) 1 day = 24 hours. Fraction = \(\frac{6}{24} = \frac{1}{4}\)
b) 1 km = 1,000,000 mm. Fraction = \(\frac{70}{1,000,000} = \frac{7}{100,000}\)
c) 1 liter = 1000 ml. Fraction = \(\frac{290}{1000} = \frac{29}{100}\)
In simple words: Convert both quantities to the same unit, then write as a fraction and simplify.

Question 22. Rohit and Rahul have study tables of the same size. Rohit used \(\frac{5}{6}\) of its space and Rahul used \(\frac{2}{5}\) of its space. Whose study table is more occupied and by what fraction?
Answer: Rohit's table used = \(\frac{5}{6}\)
Rahul's table used = \(\frac{2}{5}\)
To compare, find the LCM of denominators (6 and 5) = 30.
\(\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}\)
\(\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}\)
Since \(\frac{25}{30} > \frac{12}{30}\), Rohit's table is more occupied.
Difference in fraction = \(\frac{25}{30} - \frac{12}{30} = \frac{13}{30}\)
In simple words: Rohit used more space by \(\frac{13}{30}\) compared to Rahul.
Exam Tip: To find 'by what fraction', always subtract the smaller fraction from the larger one.

Question 23. 
a) Is \(\frac{4}{5}\) equal to \(\frac{5}{9}\)? 
b) Is \(\frac{8}{60}\) equal to \(\frac{1}{15}\)?
Answer:
a) By cross-multiplication: \(4 \times 9 = 36\) and \(5 \times 5 = 25\). Since \(36 \neq 25\), \(\frac{4}{5}\) is not equal to \(\frac{5}{9}\).
b) Simplify \(\frac{8}{60}\) by dividing numerator and denominator by 4: \(\frac{8 \div 4}{60 \div 4} = \frac{2}{15}\). Since \(\frac{2}{15} \neq \frac{1}{15}\), they are not equal.
In simple words: Use cross-multiplication or simplify fractions to check if they represent the same amount.

Exam Tip: Cross multiplication (\(a \times d = b \times c\)) is the fastest way to check equality of two fractions.

Question 24. Write three equivalent fractions of: 
a) \(\frac{2}{5}\)      b) \(\frac{1}{7}\)      c) \(\frac{5}{11}\)      d) \(\frac{6}{23}\)
Answer: Multiply the numerator and denominator by 2, 3, and 4 respectively:
a) \(\frac{2}{5}\): \(\frac{4}{10}, \frac{6}{15}, \frac{8}{20}\)
b) \(\frac{1}{7}\): \(\frac{2}{14}, \frac{3}{21}, \frac{4}{28}\)
c) \(\frac{5}{11}\): \(\frac{10}{22}, \frac{15}{33}, \frac{20}{44}\)
d) \(\frac{6}{23}\): \(\frac{12}{46}, \frac{18}{69}, \frac{24}{92}\)
In simple words: Multiply top and bottom numbers by 2, 3, and 4 to get three equivalent fractions.

Exam Tip: You can multiply by any non-zero whole number to generate equivalent fractions.

Question 25. Mika was given \(\frac{2}{9}\) of a pack of toffees. What fraction of toffees was left in the pack?
Answer: Total pack of toffees = 1
Fraction given to Mika = \(\frac{2}{9}\)
Fraction left in the pack = \(1 - \frac{2}{9} = \frac{9}{9} - \frac{2}{9} = \frac{7}{9}\)
In simple words: Subtract Mika's share from 1 to find the remaining fraction.

Exam Tip: Always write 1 as a fraction with the same denominator as the given fraction for easy subtraction.

Question 26. Compute: 
a) \(\frac{23}{67} - \frac{21}{67}\) 
b) \(\frac{11}{23} - \frac{9}{23} + \frac{13}{23}\)
Answer:
a) Since denominators are same:
\(\frac{23 - 21}{67} = \frac{2}{67}\)
b) Since denominators are same:
\(\frac{11 - 9 + 13}{23} = \frac{2 + 13}{23} = \frac{15}{23}\)
In simple words: When denominators are identical, just operate directly on the top numbers.

Question 27. Nitin bought a piece of rope \(\frac{11}{8}\) m in length. He uses \(\frac{1}{4}\) of it. What is the length of the piece left?
Answer: Total length of rope = \(\frac{11}{8}\) m
Length of rope used = \(\frac{1}{4}\) of \(\frac{11}{8}\)
\(= \frac{1}{4} \times \frac{11}{8} = \frac{11}{32}\) m
Length of rope left = Total length - Length used
\(= \frac{11}{8} - \frac{11}{32}\)
LCM of 8 and 32 is 32.
\(= \frac{11 \times 4}{8 \times 4} - \frac{11}{32}\)
\(= \frac{44}{32} - \frac{11}{32} = \frac{33}{32}\) m (or \(1\frac{1}{32}\) m)
In simple words: Find how much was used by multiplying, then subtract it from the total length.
Exam Tip: Read carefully whether the question asks for a fraction 'of it' (multiplication) or a fixed quantity.