CBSE Class 7 Mathematics Rational Numbers Assignment Set 10

Mathematics Assignment No. 03 - Rational Numbers

Question 1. Add the following rational numbers.
(a) \(\frac{2}{3} + \frac{5}{6}\)
Answer: Find the LCM of denominators 3 and 6, which is 6.
Convert each fraction to have the denominator 6:
\(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)
\(\frac{5}{6} = \frac{5}{6}\)
Add the numerators: \(\frac{4 + 5}{6} = \frac{9}{6} = \frac{3}{2}\)
In simple words: Make the bottom numbers the same by finding a common multiple, then add the top numbers.

Exam Tip: Always reduce the final fraction to its simplest form.

Question 1 (b). Add the following rational numbers.
\(\frac{7}{3} + \frac{20}{3}\)
Answer: Since the denominators are already the same, add the numerators directly:
\(\frac{7 + 20}{3} = \frac{27}{3} = 9\)
In simple words: When the bottom numbers match, simply add the top numbers together.

Exam Tip: Check if the resulting fraction can be divided evenly into a whole number.

Question 2. Add the following rational numbers.
(a) \(\frac{1}{2} + \frac{1}{4} + \frac{1}{6}\)
Answer: Find the LCM of denominators 2, 4, and 6, which is 12.
Convert each fraction:
\(\frac{1 \times 6}{2 \times 6} = \frac{6}{12}\)
\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)
\(\frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)
Add the converted fractions: \(\frac{6 + 3 + 2}{12} = \frac{11}{12}\)
In simple words: Find a common denominator for all three fractions before adding their top numbers.

Question 2 (b). Add the following rational numbers.
\(\frac{2}{3} + \frac{4}{6} + \frac{1}{5}\)
Answer: First simplify \(\frac{4}{6}\) to \(\frac{2}{3}\).
Now expression is: \(\frac{2}{3} + \frac{2}{3} + \frac{1}{5} = \frac{4}{3} + \frac{1}{5}\)
Find the LCM of 3 and 5, which is 15.
Convert fractions: \(\frac{4 \times 5}{3 \times 5} = \frac{20}{15}\) and \(\frac{1 \times 3}{5 \times 3} = \frac{3}{15}\)
Add them: \(\frac{20 + 3}{15} = \frac{23}{15}\)
In simple words: Simplify fractions first to make calculations easier, then use a common denominator.

Exam Tip: Always look for fractions that can be reduced before finding the LCM.

Question 3. Add the following rational numbers.
(a) \(\frac{-5}{9} + \frac{-5}{6}\)
Answer: Find the LCM of 9 and 6, which is 18.
Convert fractions:
\(\frac{-5 \times 2}{9 \times 2} = \frac{-10}{18}\)
\(\frac{-5 \times 3}{6 \times 3} = \frac{-15}{18}\)
Add the numerators: \(\frac{-10 + (-15)}{18} = \frac{-25}{18}\)
In simple words: Combine negative numerators carefully after making denominators equal.

Exam Tip: Pay close attention to negative signs when adding numerators.

Question 3 (b). Add the following rational numbers.
\(\frac{-3}{8} + \frac{-1}{7}\)
Answer: Find the LCM of 8 and 7, which is 56.
Convert fractions:
\(\frac{-3 \times 7}{8 \times 7} = \frac{-21}{56}\)
\(\frac{-1 \times 8}{7 \times 8} = \frac{-8}{56}\)
Add the numerators: \(\frac{-21 + (-8)}{56} = \frac{-29}{56}\)
In simple words: Since 8 and 7 share no common factors, their LCM is simply their product, 56.

Exam Tip: Coprime denominators can be multiplied directly to find the common denominator.

Question 4. Find the sum of \(\frac{5}{-9}\), \(\frac{-7}{12}\) and \(\frac{11}{18}\).
Answer: Express all fractions with positive denominators:
\(\frac{-5}{9} + \frac{-7}{12} + \frac{11}{18}\)
Find the LCM of 9, 12, and 18, which is 36.
Convert each fraction:
\(\frac{-5 \times 4}{9 \times 4} = \frac{-20}{36}\)
\(\frac{-7 \times 3}{12 \times 3} = \frac{-21}{36}\)
\(\frac{11 \times 2}{18 \times 2} = \frac{22}{36}\)
Add the numerators: \(\frac{-20 - 21 + 22}{36} = \frac{-41 + 22}{36} = \frac{-19}{36}\)
In simple words: Shift any negative sign from the denominator to the numerator first, then proceed with the LCM method.

Exam Tip: Standard form requires the denominator of a rational number to be positive.

Question 5. Add \(2\frac{1}{3}\) and \(5\frac{1}{4}\).
Answer: Convert mixed fractions into improper fractions:
\(2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}\)
\(5\frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{21}{4}\)
Find the LCM of 3 and 4, which is 12.
Convert fractions:
\(\frac{7 \times 4}{3 \times 4} = \frac{28}{12}\)
\(\frac{21 \times 3}{4 \times 3} = \frac{63}{12}\)
Add the numerators: \(\frac{28 + 63}{12} = \frac{91}{12}\)
Convert back to mixed fraction: \(7\frac{7}{12}\)
In simple words: Convert mixed numbers to improper fractions before finding a common denominator.

Exam Tip: If the question gives mixed fractions, present the final answer as a mixed fraction unless instructed otherwise.

Question 6. Evaluate: \(\frac{3}{5} + \frac{7}{3} + \frac{-11}{5} + \frac{-2}{3}\)
Answer: Group terms with common denominators together:
\(\left(\frac{3}{5} + \frac{-11}{5}\right) + \left(\frac{7}{3} + \frac{-2}{3}\right)\)
Simplify each group:
\(\frac{3 - 11}{5} + \frac{7 - 2}{3} = \frac{-8}{5} + \frac{5}{3}\)
Find the LCM of 5 and 3, which is 15.
Convert fractions:
\(\frac{-8 \times 3}{5 \times 3} = \frac{-24}{15}\)
\(\frac{5 \times 5}{3 \times 5} = \frac{25}{15}\)
Add the numerators: \(\frac{-24 + 25}{15} = \frac{1}{15}\)
In simple words: Rearrange terms to group fractions with the same bottom numbers together to make addition faster.

Exam Tip: Use commutative and associative properties to rearrange terms for easier calculation.

Question 7. Simplify: \(\frac{4}{7} + \frac{-8}{9} + \frac{-5}{21} + \frac{1}{3}\)
Answer: Find the LCM of denominators 7, 9, 21, and 3, which is 63.
Convert each fraction to have denominator 63:
\(\frac{4 \times 9}{7 \times 9} = \frac{36}{63}\)
\(\frac{-8 \times 7}{9 \times 7} = \frac{-56}{63}\)
\(\frac{-5 \times 3}{21 \times 3} = \frac{-15}{63}\)
\(\frac{1 \times 21}{3 \times 21} = \frac{21}{63}\)
Combine all numerators over the common denominator:
\(\frac{36 - 56 - 15 + 21}{63} = \frac{-20 - 15 + 21}{63} = \frac{-35 + 21}{63} = \frac{-14}{63}\)
Simplify the fraction by dividing numerator and denominator by 7:
\(\frac{-14 \div 7}{63 \div 7} = \frac{-2}{9}\)
In simple words: Find one large LCM for all denominators, combine all numerators, and reduce the final answer.

Exam Tip: Always reduce the final fraction by finding the highest common factor (HCF) of the numerator and denominator.

Question 8. A city received \(\frac{5}{12}\) cm rainfall on Saturday and \(\frac{14}{3}\) cm on Friday. How much rainfall did it receive in two days.
Answer: Total rainfall = Rainfall on Saturday + Rainfall on Friday
\(\text{Total rainfall} = \frac{5}{12} + \frac{14}{3}\)
Find the LCM of 12 and 3, which is 12.
Convert the second fraction:
\(\frac{14 \times 4}{3 \times 4} = \frac{56}{12}\)
Add the fractions: \(\frac{5 + 56}{12} = \frac{61}{12} \text{ cm}\)
In simple words: Add the two given amounts of rainfall using a common denominator.

Exam Tip: Include units (cm) in your final answer for word problems.

Question 9. Arun spent \(\frac{3}{5}\) of his pocket money during lunch break and \(\frac{3}{8}\) after school. He also spent \(\frac{1}{6}\) of the pocket money for his younger brother. What part of money he spent in all.
Answer: Total money spent = \(\frac{3}{5} + \frac{3}{8} + \frac{1}{6}\)
Find the LCM of denominators 5, 8, and 6, which is 120.
Convert each fraction:
\(\frac{3 \times 24}{5 \times 24} = \frac{72}{120}\)
\(\frac{3 \times 15}{8 \times 15} = \frac{45}{120}\)
\(\frac{1 \times 20}{6 \times 20} = \frac{20}{120}\)
Add the numerators: \(\frac{72 + 45 + 20}{120} = \frac{137}{120}\)
Convert to mixed fraction: \(1\frac{17}{120}\)
In simple words: Add all the separate fractions representing money spent to find the total fraction spent.

Exam Tip: Word problems involving fractions often require finding the LCM of larger numbers.

Question 10. The sum of two rational numbers is \(\frac{-1}{3}\). If one of the numbers is \(\frac{7}{8}\). Find the other.
Answer: Let the other number be \(x\).
\(x + \frac{7}{8} = \frac{-1}{3}\)
\(x = \frac{-1}{3} - \frac{7}{8}\)
Find the LCM of 3 and 8, which is 24.
Convert fractions:
\(\frac{-1 \times 8}{3 \times 8} = \frac{-8}{24}\)
\(\frac{7 \times 3}{8 \times 3} = \frac{21}{24}\)
Subtract: \(x = \frac{-8 - 21}{24} = \frac{-29}{24}\)
In simple words: Subtract the known number from the total sum to find the missing number.

Exam Tip: Moving a term across the equals sign changes its sign from positive to negative.

Question 11. What number should be added to \(\frac{-6}{5}\) to get \(\frac{3}{4}\)?
Answer: Let the required number be \(x\).
\(\frac{-6}{5} + x = \frac{3}{4}\)
\(x = \frac{3}{4} - \left(\frac{-6}{5}\right)\)
\(x = \frac{3}{4} + \frac{6}{5}\)
Find the LCM of 4 and 5, which is 20.
Convert fractions:
\(\frac{3 \times 5}{4 \times 5} = \frac{15}{20}\)
\(\frac{6 \times 4}{5 \times 4} = \frac{24}{20}\)
Add: \(x = \frac{15 + 24}{20} = \frac{39}{20}\)
In simple words: Set up an equation and isolate the unknown variable by subtraction.

Exam Tip: Subtracting a negative number is the same as adding its positive counterpart.

Question 12. Subtract the sum of \(\frac{-36}{11}\) and \(\frac{49}{22}\) from the sum of \(\frac{33}{8}\) and \(\frac{-17}{4}\).
Answer: Step 1: Find the sum of \(\frac{-36}{11}\) and \(\frac{49}{22}\).
LCM of 11 and 22 is 22.
\(\frac{-36 \times 2}{11 \times 2} + \frac{49}{22} = \frac{-72 + 49}{22} = \frac{-23}{22}\)
Step 2: Find the sum of \(\frac{33}{8}\) and \(\frac{-17}{4}\).
LCM of 8 and 4 is 8.
\(\frac{33}{8} + \frac{-17 \times 2}{4 \times 2} = \frac{33 - 34}{8} = \frac{-1}{8}\)
Step 3: Subtract the result of Step 1 from the result of Step 2.
\(\frac{-1}{8} - \left(\frac{-23}{22}\right) = \frac{-1}{8} + \frac{23}{22}\)
Find the LCM of 8 and 22, which is 88.
Convert fractions:
\(\frac{-1 \times 11}{8 \times 11} = \frac{-11}{88}\)
\(\frac{23 \times 4}{22 \times 4} = \frac{92}{88}\)
Subtract: \(\frac{-11 + 92}{88} = \frac{81}{88}\)
In simple words: Calculate the two sums separately first, then subtract the first sum from the second sum.

Exam Tip: Break complex multi-step word problems into numbered sequential steps to avoid calculation errors.