CBSE Class 7 Mathematics Fractions and Decimals Assignment Set 03

VII - Mathematics Assignment No. 08 - Fractions and Decimals

Q1. Divide \(1 \frac{3}{5}\) by \(1 \frac{1}{15}\)
Answer: Convert mixed fractions into improper fractions:
\(1 \frac{3}{5} = \frac{5 \times 1 + 3}{5} = \frac{8}{5}\)
\(1 \frac{1}{15} = \frac{15 \times 1 + 1}{15} = \frac{16}{15}\)
Now, divide the two fractions by multiplying the first fraction by the reciprocal of the second fraction:
\(\frac{8}{5} \div \frac{16}{15} = \frac{8}{5} \times \frac{15}{16}\)
Cancel out common factors:
\(= \frac{8 \div 8}{5 \div 5} \times \frac{15 \div 5}{16 \div 8} = \frac{1}{1} \times \frac{3}{2} = \frac{3}{2} = 1 \frac{1}{2}\)
In simple words: To divide fractions, turn mixed numbers into improper fractions, flip the second fraction upside down, and multiply.

Exam Tip: Always convert mixed fractions to improper fractions before performing multiplication or division operations.

Q2. Divide \(4 \frac{6}{11}\) by \(3 \frac{7}{12}\)
Answer: Convert mixed fractions into improper fractions:
\(4 \frac{6}{11} = \frac{44 + 6}{11} = \frac{50}{11}\)
\(3 \frac{7}{12} = \frac{36 + 7}{12} = \frac{43}{12}\)
Perform division by multiplying by the reciprocal:
\(\frac{50}{11} \div \frac{43}{12} = \frac{50}{11} \times \frac{12}{43}\)
Multiply numerators and denominators:
\(= \frac{50 \times 12}{11 \times 43} = \frac{600}{473}\)
Convert the improper fraction back to a mixed fraction:
\(\frac{600}{473} = 1 \frac{127}{437}\)
In simple words: Convert to improper fractions, multiply by the reciprocal, and convert the final improper fraction back to a mixed number.

Exam Tip: Check if numerators and denominators share any common factors before multiplying large numbers to make calculations simpler.

Q3. Simplify \(2 \frac{1}{2} \div 1 \frac{7}{8} \times \frac{7}{12}\)
Answer: Convert mixed fractions to improper fractions:
\(2 \frac{1}{2} = \frac{5}{2}\)
\(1 \frac{7}{8} = \frac{15}{8}\)
Rewrite the expression following the BODMAS rule (Division before Multiplication from left to right):
\(= \left(\frac{5}{2} \div \frac{15}{8}\right) \times \frac{7}{12}\)
Convert division to multiplication by taking the reciprocal:
\(= \left(\frac{5}{2} \times \frac{8}{15}\right) \times \frac{7}{12}\)
Simplify inside the brackets:
\(= \left(\frac{1}{1} \times \frac{4}{3}\right) \times \frac{7}{12} = \frac{4}{3} \times \frac{7}{12}\)
Multiply across:
\(= \frac{4 \times 7}{3 \times 12} = \frac{28}{36} = \frac{7}{9}\)
In simple words: Follow order of operations from left to right, turning division into multiplication by flipping the fraction after the division sign.

Exam Tip: Cancel common factors diagonally or vertically across fractions to avoid working with large numbers.

Q4. Divide 15 by \(7 \frac{1}{2}\)
Answer: Convert the whole number and mixed fraction into improper fractions:
\(15 = \frac{15}{1}\)
\(7 \frac{1}{2} = \frac{15}{2}\)
Divide the fractions:
\(\frac{15}{1} \div \frac{15}{2} = \frac{15}{1} \times \frac{2}{15}\)
Cancel out 15 from numerator and denominator:
\(= \frac{1}{1} \times \frac{2}{1} = 2\)
In simple words: Write 15 as \(\frac{15}{1}\), convert the mixed number to an improper fraction, and multiply by its reciprocal.

Exam Tip: Any whole number \(n\) can always be written as a fraction with a denominator of 1 (\(\frac{n}{1}\)).

Q5. Divide \(3 \frac{4}{7}\) by the reciprocal of \(2 \frac{3}{4}\)
Answer: First, find the reciprocal of \(2 \frac{3}{4}\):
\(2 \frac{3}{4} = \frac{11}{4}\)
Reciprocal of \(\frac{11}{4}\) is \(\frac{4}{11}\)
Now, convert the first mixed fraction to an improper fraction:
\(3 \frac{4}{7} = \frac{25}{7}\)
Divide \(3 \frac{4}{7}\) by the reciprocal \(\frac{4}{11}\):
\(\frac{25}{7} \div \frac{4}{11} = \frac{25}{7} \times \frac{11}{4}\)
Multiply numerators and denominators:
\(= \frac{25 \times 11}{7 \times 4} = \frac{275}{28}\)
Convert to mixed fraction:
\(\frac{275}{28} = 9 \frac{23}{28}\)
In simple words: Read carefully - first find the flipped version (reciprocal) of the second number, then divide the first number by that new value.

Exam Tip: Pay close attention to wording like "by the reciprocal of" as it changes the second number before you perform the final operation.

Q6. Divide \(21 \frac{1}{3}\) by \(17 \frac{1}{2}\)
Answer: Convert both mixed fractions into improper fractions:
\(21 \frac{1}{3} = \frac{64}{3}\)
\(17 \frac{1}{2} = \frac{35}{2}\)
Perform division by multiplying by the reciprocal:
\(\frac{64}{3} \div \frac{35}{2} = \frac{64}{3} \times \frac{2}{35}\)
Multiply numerators and denominators:
\(= \frac{64 \times 2}{3 \times 35} = \frac{128}{105}\)
Convert the improper fraction to a mixed fraction:
\(\frac{128}{105} = 1 \frac{23}{105}\)
In simple words: Convert both mixed numbers into improper fractions and multiply the first by the reciprocal of the second.

Exam Tip: Always verify if your final fraction can be simplified further before converting it into a mixed number.

Q7. Divide \(\left( \frac{3}{4} \times 1 \frac{1}{4} \right)\) by \(\left( 3 \frac{1}{4} \times \frac{5}{6} \right)\)
Answer: Convert all mixed fractions to improper fractions first:
\(1 \frac{1}{4} = \frac{5}{4}\)
\(3 \frac{1}{4} = \frac{13}{4}\)
Evaluate the first bracket:
\(\frac{3}{4} \times \frac{5}{4} = \frac{15}{16}\)
Evaluate the second bracket:
\(\frac{13}{4} \times \frac{5}{6} = \frac{65}{24}\)
Now, divide the result of the first bracket by the result of the second bracket:
\(\frac{15}{16} \div \frac{65}{24} = \frac{15}{16} \times \frac{24}{65}\)
Simplify by canceling common factors (divide 16 and 24 by 8; divide 15 and 65 by 5):
\(= \frac{3}{2} \times \frac{3}{13}\)
Multiply across:
\(= \frac{3 \times 3}{2 \times 13} = \frac{9}{26}\)
In simple words: Solve each bracket separately into single fractions first, then divide the first bracket value by the second bracket value.

Exam Tip: Use brackets to maintain order of operations and solve sub-expressions completely before performing the final division.

Q8. Simplify \(2 \frac{1}{7} \times 1 \frac{3}{46} \div \frac{18}{23} \times \frac{5}{7}\)
Answer: Convert mixed fractions to improper fractions:
\(2 \frac{1}{7} = \frac{15}{7}\)
\(1 \frac{3}{46} = \frac{49}{46}\)
Substitute back into the expression:
\(= \frac{15}{7} \times \frac{49}{46} \div \frac{18}{23} \times \frac{5}{7}\)
Change division to multiplication by taking the reciprocal of \(\frac{18}{23}\):
\(= \frac{15}{7} \times \frac{49}{46} \times \frac{23}{18} \times \frac{5}{7}\)
Cancel out common factors across numerators and denominators:
- \(7 \times 7 = 49\), which cancels with the numerator 49.
- 23 cancels with 46 leaving 2 in the denominator.
- 15 and 18 share a factor of 3 (leaving 5 and 6).
Rewriting remaining terms:
\(= \frac{5}{1} \times \frac{1}{2} \times \frac{1}{6} \times \frac{5}{1} = \frac{5 \times 1 \times 1 \times 5}{1 \times 2 \times 6 \times 1} = \frac{25}{12} = 2 \frac{1}{12}\)
In simple words: Convert mixed numbers, change division to multiplication, cancel common terms across all fractions, and multiply straight across.

Exam Tip: Look for cross-cancellation across multiple multiplication signs to drastically reduce numbers before multiplying.

Q9. Simplify \(\left( 5 \frac{1}{4} \div 2 \frac{4}{5} \right) \div 1 \frac{1}{8}\)
Answer: Convert all mixed fractions to improper fractions:
\(5 \frac{1}{4} = \frac{21}{4}\)
\(2 \frac{4}{5} = \frac{14}{5}\)
\(1 \frac{1}{8} = \frac{9}{8}\)
Substitute into the expression:
\(= \left( \frac{21}{4} \div \frac{14}{5} \right) \div \frac{9}{8}\)
Solve the inner bracket first by multiplying by the reciprocal:
\(= \left( \frac{21}{4} \times \frac{5}{14} \right) \div \frac{9}{8}\)
Cancel 21 and 14 by factor of 7 (leaving 3 and 2):
\(= \left( \frac{3}{4} \times \frac{5}{2} \right) \div \frac{9}{8} = \frac{15}{8} \div \frac{9}{8}\)
Now perform the second division:
\(= \frac{15}{8} \times \frac{8}{9}\)
Cancel 8 from numerator and denominator, and divide 15 and 9 by 3:
\(= \frac{5}{1} \times \frac{1}{3} = \frac{5}{3} = 1 \frac{2}{3}\)
In simple words: Always simplify operations inside parentheses first, moving strictly from left to right for sequential division.

Exam Tip: When evaluating multiple division signs, process them strictly from left to right.

Q10. Simplify \(3 \frac{1}{4} \times 4 \frac{4}{5} \div 2 \frac{4}{5}\)
Answer: Convert all mixed fractions to improper fractions:
\(3 \frac{1}{4} = \frac{13}{4}\)
\(4 \frac{4}{5} = \frac{24}{5}\)
\(2 \frac{4}{5} = \frac{14}{5}\)
Substitute into the expression following BODMAS (Multiplication and Division have equal priority, evaluate left to right):
\(= \left( \frac{13}{4} \times \frac{24}{5} \right) \div \frac{14}{5}\)
Simplify inside the multiplication part:
\(= \left( \frac{13}{1} \times \frac{6}{5} \right) \div \frac{14}{5} = \frac{78}{5} \div \frac{14}{5}\)
Convert division to multiplication by taking the reciprocal:
\(= \frac{78}{5} \times \frac{5}{14}\)
Cancel 5 from numerator and denominator:
\(= \frac{78}{14}\)
Simplify the fraction by dividing numerator and denominator by 2:
\(= \frac{39}{7} = 5 \frac{4}{7}\)
In simple words: Convert to improper fractions, perform multiplication and division from left to right, and simplify the final fraction.

Exam Tip: Notice that \(\frac{5}{5}\) cancels out directly when dividing by a fraction with the same denominator.