CBSE Class 6 Mathematics Fractions Assignment Set 07

VI - Mathematics Assignment No-04 - Fractions

Q1. Arrange the following fractions in ascending order. \(\frac{1}{9}, \frac{5}{18}, \frac{3}{4}, \frac{13}{24}\)
Answer: To arrange fractions in ascending order, first find the LCM of denominators 9, 18, 4, and 24.
Prime factorizations:
\(9 = 3^2\)
\(18 = 2 \times 3^2\)
\(4 = 2^2\)
\(24 = 2^3 \times 3\)
LCM = \(2^3 \times 3^2 = 8 \times 9 = 72\)
Now, convert each fraction to an equivalent fraction with 72 as the denominator:
\(\frac{1}{9} = \frac{1 \times 8}{9 \times 8} = \frac{8}{72}\)
\(\frac{5}{18} = \frac{5 \times 4}{18 \times 4} = \frac{20}{72}\)
\(\frac{3}{4} = \frac{3 \times 18}{4 \times 18} = \frac{54}{72}\)
\(\frac{13}{24} = \frac{13 \times 3}{24 \times 3} = \frac{39}{72}\)
Comparing the numerators: \(8 < 20 < 39 < 54\)
Therefore, the ascending order is: \(\frac{1}{9}, \frac{5}{18}, \frac{13}{24}, \frac{3}{4}\)
In simple words: Make the bottom numbers the same using LCM, then arrange by the top numbers from smallest to largest.

Exam Tip: Always find the LCM of denominators correctly before comparing fractions.

Q2. Sunita read 35 pages of a book having 100 pages. Ramila read \(\frac{1}{2}\) of the same book. Who read more pages?
Answer: Total pages in the book = 100
Pages read by Sunita = 35
Pages read by Ramila = \(\frac{1}{2}\) of 100 = \(\frac{1}{2} \times 100 = 50\) pages
Comparing the pages read: \(50 > 35\)
Therefore, Ramila read more pages.
In simple words: Calculate half of 100 pages to find Ramila's read pages, then compare with Sunita's 35 pages.

Exam Tip: Convert all quantities into the same unit or absolute values before comparing.

Q3. Arun plays for \(\frac{3}{4}\) of an hour and Anil plays for \(\frac{2}{5}\) of an hour. Who plays more time and by how much?
Answer: Time played by Arun = \(\frac{3}{4}\) hour
Time played by Anil = \(\frac{2}{5}\) hour
To compare, find the LCM of denominators 4 and 5, which is 20.
\(\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\) hour
\(\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\) hour
Since \(\frac{15}{20} > \frac{8}{20}\), Arun plays more time.
Difference in time = \(\frac{15}{20} - \frac{8}{20} = \frac{7}{20}\) hour
Converting into minutes: \(\frac{7}{20} \times 60 \text{ minutes} = 7 \times 3 = 21\) minutes.
Therefore, Arun plays more time by 21 minutes.
In simple words: Compare the two fractions by making denominators equal, find who is greater, and subtract to find the difference.

Exam Tip: When comparing fractions with different denominators, always convert them to like fractions first.

Q4. Add \(\frac{1}{2}, \frac{1}{4}, \frac{7}{8}\) and express the result in a mixed fraction.
Answer: Sum = \(\frac{1}{2} + \frac{1}{4} + \frac{7}{8}\)
LCM of denominators 2, 4, and 8 is 8.
\(\frac{1}{2} = \frac{4}{8}\)
\(\frac{1}{4} = \frac{2}{8}\)
\(\frac{7}{8} = \frac{7}{8}\)
Sum = \(\frac{4 + 2 + 7}{8} = \frac{13}{8}\)
Converting \(\frac{13}{8}\) into a mixed fraction: \(1\frac{5}{8}\)
In simple words: Find the common denominator, add the numerators, and convert the improper fraction into a mixed fraction.

Exam Tip: Always check if the final fraction can be simplified or needs conversion to a mixed fraction as requested.

Q5. Add (i) \(\frac{13}{15} + \frac{1}{2} + \frac{2}{3}\)
Answer: LCM of denominators 15, 2, and 3 is 30.
\(\frac{13}{15} = \frac{13 \times 2}{15 \times 2} = \frac{26}{30}\)
\(\frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30}\)
\(\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}\)
Sum = \(\frac{26 + 15 + 20}{30} = \frac{61}{30} = 2\frac{1}{30}\)
In simple words: Make all denominators 30, add the top numbers, and write as a mixed fraction.

Exam Tip: Double check your LCM calculation to avoid arithmetic errors.

Q5. (ii) \(\frac{4}{5} + \frac{6}{7} + \frac{1}{2}\)
Answer: LCM of denominators 5, 7, and 2 is \(5 \times 7 \times 2 = 70\)
\(\frac{4}{5} = \frac{4 \times 14}{5 \times 14} = \frac{56}{70}\)
\(\frac{6}{7} = \frac{6 \times 10}{7 \times 10} = \frac{60}{70}\)
\(\frac{1}{2} = \frac{1 \times 35}{2 \times 35} = \frac{35}{70}\)
Sum = \(\frac{56 + 60 + 35}{70} = \frac{151}{70} = 2\frac{11}{70}\)
In simple words: Multiply each fraction to have a denominator of 70, then add the numerators.

Exam Tip: For prime denominators, their product is always the LCM.

Q6. Add (i) \(2\frac{1}{3} + 1\frac{5}{6} + 1\frac{3}{8}\)
Answer: Convert mixed fractions to improper fractions:
\(2\frac{1}{3} = \frac{7}{3}\)
\(1\frac{5}{6} = \frac{11}{6}\)
\(1\frac{3}{8} = \frac{11}{8}\)
LCM of denominators 3, 6, and 8 is 24.
\(\frac{7}{3} = \frac{7 \times 8}{3 \times 8} = \frac{56}{24}\)
\(\frac{11}{6} = \frac{11 \times 4}{6 \times 4} = \frac{44}{24}\)
\(\frac{11}{8} = \frac{11 \times 3}{8 \times 3} = \frac{33}{24}\)
Sum = \(\frac{56 + 44 + 33}{24} = \frac{133}{24} = 5\frac{13}{24}\) (Note: Printed answer key mentions \(4\frac{13}{24}\) which is a typo in the original sheet; exact calculation gives \(5\frac{13}{24}\)).
In simple words: Change mixed numbers to improper fractions first, find the common denominator, and add.

Exam Tip: Always convert mixed fractions to improper fractions before performing addition or subtraction.

Q6. (ii) \(2\frac{2}{3} + 1\frac{4}{7} + 4\frac{2}{3}\)
Answer: Convert mixed fractions to improper fractions:
\(2\frac{2}{3} = \frac{8}{3}\)
\(1\frac{4}{7} = \frac{11}{7}\)
\(4\frac{2}{3} = \frac{14}{3}\)
Alternative method (grouping whole numbers and fractions):
Whole numbers sum = \(2 + 1 + 4 = 7\)
Fraction sum = \(\frac{2}{3} + \frac{4}{7} + \frac{2}{3} = \frac{4}{3} + \frac{4}{7}\)
LCM of 3 and 7 is 21.
\(\frac{4}{3} = \frac{28}{21}\)
\(\frac{4}{7} = \frac{12}{21}\)
Fraction sum = \(\frac{28 + 12}{21} = \frac{40}{21} = 1\frac{19}{21}\)
Total sum = \(7 + 1\frac{19}{21} = 8\frac{19}{21}\)
In simple words: Add whole numbers and fractions separately to make calculation easier.

Exam Tip: Grouping whole numbers and fractions separately saves time when numbers repeat.

Q7. Subtract (i) \(2\frac{3}{4}\) from \(6\frac{1}{6}\)
Answer: \(6\frac{1}{6} - 2\frac{3}{4}\)
Convert to improper fractions:
\(6\frac{1}{6} = \frac{37}{6}\)
\(2\frac{3}{4} = \frac{11}{4}\)
LCM of 6 and 4 is 12.
\(\frac{37}{6} = \frac{37 \times 2}{6 \times 2} = \frac{74}{12}\)
\(\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}\)
Difference = \(\frac{74 - 33}{12} = \frac{41}{12} = 3\frac{5}{12}\)
In simple words: Convert both to improper fractions, find a common denominator, and subtract.

Exam Tip: Note the order of subtraction: "from" means the second number comes first.

Q7. (ii) \(2\frac{2}{3}\) from 4
Answer: \(4 - 2\frac{2}{3}\)
Rewrite 4 as \(3\frac{3}{3}\):
\(3\frac{3}{3} - 2\frac{2}{3} = (3 - 2) + \left(\frac{3}{3} - \frac{2}{3}\right) = 1 + \frac{1}{3} = 1\frac{1}{3}\)
In simple words: Borrow 1 from 4 and write it as \(\frac{3}{3}\) to easily subtract the fraction.

Q8. Simplify \(2 - \frac{2}{3} + 1\frac{1}{4} - 1\frac{5}{6}\)
Answer: Convert mixed fractions to improper fractions:
\(1\frac{1}{4} = \frac{5}{4}\)
\(1\frac{5}{6} = \frac{11}{6}\)
Expression: \(2 - \frac{2}{3} + \frac{5}{4} - \frac{11}{6}\)
LCM of denominators 3, 4, and 6 is 12.
\(2 = \frac{24}{12}\)
\(-\frac{2}{3} = -\frac{8}{12}\)
\(\frac{5}{4} = \frac{15}{12}\)
\(-\frac{11}{6} = -\frac{22}{12}\)
Combine terms: \(\frac{24 - 8 + 15 - 22}{12} = \frac{31 - 30}{12} = \frac{1}{12}\) (Note: Answer key gives \(\frac{3}{4}\), recalculating with exact values gives \(\frac{1}{12}\)).
In simple words: Convert all terms to fractions with a common denominator and combine numerators following BODMAS.

Exam Tip: Be careful with signs of positive and negative terms during combination.

Q9. Multiply (i) \(\frac{3}{4}\) by \(\frac{12}{15}\)
Answer: \(\frac{3}{4} \times \frac{12}{15}\)
Cancel common factors:
\(\frac{3}{15} = \frac{1}{5}\)
\(\frac{12}{4} = 3\)
Product = \(\frac{1 \times 3}{1 \times 5} = \frac{3}{5}\)
In simple words: Simplify top and bottom numbers across fractions before multiplying straight across.

Exam Tip: Always cancel common factors diagonally or vertically before multiplying to keep numbers small.

Q9. (ii) \(\frac{6}{7}\) by \(\frac{21}{18}\)
Answer: \(\frac{6}{7} \times \frac{21}{18}\)
Cancel common factors:
\(\frac{21}{7} = 3\)
\(\frac{6}{18} = \frac{1}{3}\)
Product = \(3 \times \frac{1}{3} = 1\)
In simple words: Reduce fractions by canceling common numbers to get 1.

Exam Tip: Cross-canceling makes multiplication much faster and cleaner.

Q10. Divide (i) \(13\frac{4}{5}\) by \(\frac{23}{10}\)
Answer: Convert \(13\frac{4}{5}\) to improper fraction: \(\frac{13 \times 5 + 4}{5} = \frac{69}{5}\)
Division: \(\frac{69}{5} \div \frac{23}{10}\)
Multiply by the reciprocal of the divisor:
\(= \frac{69}{5} \times \frac{10}{23}\)
Cancel terms: \(\frac{69}{23} = 3\) and \(\frac{10}{5} = 2\)
Result = \(3 \times 2 = 6\)
In simple words: Turn division into multiplication by flipping the second fraction, then simplify.

Exam Tip: Remember the rule for dividing fractions: "Keep-Change-Flip".

Q10. (ii) \(5\frac{1}{2}\) by \(\frac{33}{8}\)
Answer: Convert \(5\frac{1}{2}\) to improper fraction: \(\frac{11}{2}\)
Division: \(\frac{11}{2} \div \frac{33}{8}\)
Multiply by reciprocal:
\(= \frac{11}{2} \times \frac{8}{33}\)
Cancel terms: \(\frac{11}{33} = \frac{1}{3}\) and \(\frac{8}{2} = 4\)
Result = \(\frac{4}{3} = 1\frac{1}{3}\)
In simple words: Convert to improper fraction, flip the divisor, multiply, and simplify.

Exam Tip: Always express improper fraction answers as mixed fractions if the question started with mixed fractions.

Q11. Simplify: \(2\frac{1}{3} + 5\frac{5}{7} - 1\frac{1}{2}\)
Answer: Convert mixed fractions to improper fractions:
\(2\frac{1}{3} = \frac{7}{3}\)
\(5\frac{5}{7} = \frac{40}{7}\)
\(1\frac{1}{2} = \frac{3}{2}\)
Expression: \(\frac{7}{3} + \frac{40}{7} - \frac{3}{2}\)
LCM of 3, 7, and 2 is \(3 \times 7 \times 2 = 42\)
\(\frac{7}{3} = \frac{7 \times 14}{42} = \frac{98}{42}\)
\(\frac{40}{7} = \frac{40 \times 6}{42} = \frac{240}{42}\)
\(-\frac{3}{2} = -\frac{3 \times 21}{42} = -\frac{63}{42}\)
Combine numerators: \(\frac{98 + 240 - 63}{42} = \frac{338 - 63}{42} = \frac{275}{42} = 6\frac{23}{42}\) (Note: Answer key lists \(6\frac{29}{42}\), exact calculation gives \(6\frac{23}{42}\)).
In simple words: Convert mixed numbers to improper fractions, find LCM 42, add and subtract.

Exam Tip: Be meticulous with addition and subtraction signs across fractions.

Q12. Simplify: \(\left[\frac{3}{5} - \frac{7}{10} - \frac{3}{15} + \frac{7}{9}\right] \div \frac{43}{90}\)
Answer: First, simplify the expression inside the brackets: \(\frac{3}{5} - \frac{7}{10} - \frac{3}{15} + \frac{7}{9}\)
Simplify individual fractions where possible: \(\frac{3}{15} = \frac{1}{5}\)
Expression becomes: \(\left(\frac{3}{5} - \frac{1}{5}\right) - \frac{7}{10} + \frac{7}{9} = \frac{2}{5} - \frac{7}{10} + \frac{7}{9}\)
LCM of denominators 5, 10, and 9 is 90.
\(\frac{2}{5} = \frac{2 \times 18}{90} = \frac{36}{90}\)
\(-\frac{7}{10} = -\frac{7 \times 9}{90} = -\frac{63}{90}\)
\(\frac{7}{9} = \frac{7 \times 10}{90} = \frac{70}{90}\)
Bracket sum = \(\frac{36 - 63 + 70}{90} = \frac{43}{90}\)
Now, perform the division: \(\frac{43}{90} \div \frac{43}{90} = 1\)
In simple words: Solve the brackets first using LCM, then divide the resulting fraction by itself to get 1.

Exam Tip: Always follow BODMAS order: solve brackets completely before doing division.