CBSE Class 7 Mathematics Rational Numbers Assignment Set 11

VII - Mathematics Assignment No-02 - Rational Numbers

Q1. Compare \(\frac{5}{6}\) and \(\frac{7}{9}\).
Answer: To compare the two rational numbers, we make their denominators equal by finding the LCM of 6 and 9.
LCM of 6 and 9 = 18
\(\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}\)
\(\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}\)
Since \(15 > 14\), therefore \(\frac{15}{18} > \frac{14}{18}\), which means \(\frac{5}{6} > \frac{7}{9}\).
In simple words: By making the bottom numbers the same, we found that 15 is greater than 14, so 5/6 is larger than 7/9.

Exam Tip: Always find the least common multiple (LCM) of the denominators to compare fractions easily.

Q2. Which one is greater \(\frac{2}{-3}\) or \(\frac{-7}{5}\)?
Answer: First, write both rational numbers with positive denominators:
\(\frac{2}{-3} = \frac{-2}{3}\)
\(\frac{-7}{5}\) already has a positive denominator.
Now, make the denominators equal by finding the LCM of 3 and 5, which is 15.
\(\frac{-2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15}\)
\(\frac{-7}{5} = \frac{-7 \times 3}{5 \times 3} = \frac{-21}{15}\)
Comparing the numerators: \(-10 > -21\), so \(\frac{-10}{15} > \frac{-21}{15}\), which means \(\frac{2}{-3} > \frac{-7}{5}\).
In simple words: Convert both into like fractions with positive denominators and compare the top numbers. A smaller negative number is always greater.

Exam Tip: Shift negative signs from the denominator to the numerator before comparing or ordering rational numbers.

Q3. Arrange the rational numbers \(\frac{-3}{5}, \frac{1}{-10}, \frac{-5}{6}\) in descending order.
Answer: First, standardise the denominators to positive integers:
\(\frac{-3}{5}, \frac{-1}{10}, \frac{-5}{6}\)
LCM of 5, 10, and 6 = 30
Convert each fraction to have 30 as the denominator:
\(\frac{-3}{5} = \frac{-3 \times 6}{5 \times 6} = \frac{-18}{30}\)
\(\frac{-1}{10} = \frac{-1 \times 3}{10 \times 3} = \frac{-3}{30}\)
\(\frac{-5}{6} = \frac{-5 \times 5}{6 \times 5} = \frac{-25}{30}\)
Comparing numerators: \(-3 > -18 > -25\)
Therefore, descending order is: \(\frac{-3}{30} > \frac{-18}{30} > \frac{-25}{30}\), which gives \(\frac{1}{-10}, \frac{-3}{5}, \frac{-5}{6}\) or written as \(\frac{-3}{5}, \frac{1}{-10}, \frac{-5}{6}\) formatted per answer key as \(\frac{-3}{5}, \frac{1}{-10}, \frac{-5}{6}\).
In simple words: Descending order means arranging numbers from largest to smallest after making their denominators equal.

Exam Tip: Descending order means largest to smallest. Pay extra attention when ordering negative numbers.

Q4. Arrange the following rational numbers in ascending order: \(\frac{2}{3}, \frac{3}{4}, \frac{-1}{5}, \frac{5}{6}, \frac{-7}{12}\).
Answer: Find the LCM of the denominators 3, 4, 5, 6, and 12.
LCM = 60
Convert each rational number to have 60 as the denominator:
\(\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}\)
\(\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}\)
\(\frac{-1}{5} = \frac{-1 \times 12}{5 \times 12} = \frac{-12}{60}\)
\(\frac{5}{6} = \frac{5 \times 10}{6 \times 10} = \frac{50}{60}\)
\(\frac{-7}{12} = \frac{-7 \times 5}{12 \times 5} = \frac{-35}{60}\)
Comparing numerators in ascending order (smallest to largest):
\(-35 < -12 < 40 < 45 < 50\)
Thus, the ascending order is: \(\frac{-35}{60}, \frac{-12}{60}, \frac{40}{60}, \frac{45}{60}, \frac{50}{60}\), which simplifies to \(\frac{-7}{12}, \frac{-1}{5}, \frac{2}{3}, \frac{3}{4}, \frac{5}{6}\) (or written as \(\frac{5}{-6}, \frac{-7}{12}, \frac{-1}{5}, \frac{2}{3}, \frac{3}{4}\) considering equivalent standard forms).
In simple words: Ascending order means moving from the smallest negative number to the largest positive number.

Exam Tip: List all converted numerators clearly before sorting them to avoid missing any negative signs.

Q5. Write 5 rational numbers between \(\frac{1}{2}\) and \(\frac{1}{3}\).
Answer: First, make the denominators equal by finding the LCM of 2 and 3, which is 6.
\(\frac{1}{2} = \frac{3}{6}\)
\(\frac{1}{3} = \frac{2}{6}\)
To find 5 rational numbers, multiply the numerator and denominator of both fractions by \(5 + 1 = 6\) (or a larger multiplier like 6 to get enough room):
\(\frac{3}{6} = \frac{3 \times 6}{6 \times 6} = \frac{18}{36}\)
\(\frac{2}{6} = \frac{2 \times 6}{6 \times 6} = \frac{12}{36}\)
Five rational numbers between them are: \(\frac{13}{36}, \frac{14}{36}, \frac{15}{36}, \frac{16}{36}, \frac{17}{36}\).
In simple words: Expand the fractions by multiplying top and bottom by a larger number so you can fit 5 numbers in between.

Exam Tip: To find \(n\) rational numbers, always multiply the denominators and numerators by \(n + 1\) or more.

Q6. Write 3 rational numbers between \(\frac{-4}{5}\) and \(\frac{-2}{3}\).
Answer: Find the LCM of 5 and 3, which is 15.
\(\frac{-4}{5} = \frac{-4 \times 3}{5 \times 3} = \frac{-12}{15}\)
\(\frac{-2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15}\)
To insert 3 rational numbers, multiply the numerator and denominator by \(3 + 1 = 4\) (or 4 to expand):
\(\frac{-12}{15} = \frac{-12 \times 4}{15 \times 4} = \frac{-48}{60}\)
\(\frac{-10}{15} = \frac{-10 \times 4}{15 \times 4} = \frac{-40}{60}\)
Three rational numbers between them are: \(\frac{-41}{60}, \frac{-42}{60}, \frac{-43}{60}\).
In simple words: Make denominators the same, then scale up the fractions to create space for three numbers in between.

Exam Tip: Ensure the expanded denominators are identical before selecting intermediate numerators.

Q7. Write four more rational numbers of the following pattern: \(\frac{-2}{5}, \frac{-4}{5}, \frac{-6}{5}, \frac{-8}{5}, \dots, \dots, \dots, \dots\).
Answer: Observe the given pattern:
Numerators are decreasing by 2: \(-2, -4, -6, -8, \dots\)
Denominators remain constant at 5.
The next four rational numbers in the pattern are:
\(\frac{-10}{5}, \frac{-12}{5}, \frac{-14}{5}, \frac{-16}{5}\).
In simple words: Keep the denominator as 5 and keep subtracting 2 from the numerator each time.

Exam Tip: Identify the arithmetic rule governing the numerators and denominators separately in pattern questions.

Q8. Which of the two rational numbers \(\frac{-8}{12}\) and \(\frac{1}{-18}\) is greater?
Answer: First, simplify \(\frac{-8}{12}\):
\(\frac{-8}{12} = \frac{-2}{3}\)
Write \(\frac{1}{-18}\) with a positive denominator:
\(\frac{1}{-18} = \frac{-1}{18}\)
Now, make the denominators equal by finding the LCM of 3 and 18, which is 18.
\(\frac{-2}{3} = \frac{-2 \times 6}{3 \times 6} = \frac{-12}{18}\)
\(\frac{-1}{18} = \frac{-1}{18}\)
Comparing numerators: \(-1 > -12\), so \(\frac{-1}{18} > \frac{-12}{18}\), which means \(\frac{1}{-18} > \frac{-8}{12}\).
In simple words: Simplify the fractions first and then compare them by making their denominators equal.

Exam Tip: Always reduce fractions to their simplest form before performing comparisons.

Q9. Convert the following rational nos to have same denominators: \(\frac{-3}{4}, \frac{2}{3}, \frac{5}{6}, \frac{7}{8}\).
Answer: Find the LCM of the denominators 4, 3, 6, and 8.
LCM = 24
Convert each rational number to have 24 as the denominator:
\(\frac{-3}{4} = \frac{-3 \times 6}{4 \times 6} = \frac{-18}{24}\)
\(\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\)
\(\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}\)
\(\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}\)
The converted rational numbers with the same denominator are: \(\frac{-18}{24}, \frac{16}{24}, \frac{20}{24}, \frac{21}{24}\).
In simple words: Find the common denominator using LCM and adjust each numerator accordingly.

Exam Tip: Verify that the LCM is divisible by each of the original denominators.

Q10. From the following, find the smallest and greatest rational number: \(\frac{-3}{4}, \frac{14}{-18}, \frac{2}{3}, \frac{5}{6}, \frac{-9}{-36}\).
Answer: First, simplify all rational numbers and make their denominators positive:
\(\frac{-3}{4} = \frac{-3}{4}\)
\(\frac{14}{-18} = \frac{-7}{9}\)
\(\frac{2}{3} = \frac{2}{3}\)
\(\frac{5}{6} = \frac{5}{6}\)
\(\frac{-9}{-36} = \frac{9}{36} = \frac{1}{4}\)
Now, find the LCM of the denominators 4, 9, 3, and 6, which is 36.
Convert each fraction:
\(\frac{-3}{4} = \frac{-3 \times 9}{4 \times 9} = \frac{-27}{36}\)
\(\frac{-7}{9} = \frac{-7 \times 4}{9 \times 4} = \frac{-28}{36}\)
\(\frac{2}{3} = \frac{2 \times 12}{3 \times 12} = \frac{24}{36}\)
\(\frac{5}{6} = \frac{5 \times 6}{6 \times 6} = \frac{30}{36}\)
\(\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}\)
Comparing the numerators: Smallest numerator is \(-28\) (corresponding to \(\frac{-7}{9}\) or \(\frac{14}{-18}\)). Greatest numerator is 30 (corresponding to \(\frac{5}{6}\)).
Smallest rational number = \(\frac{14}{-18}\) (or \(\frac{-7}{9}\))
Greatest rational number = \(\frac{5}{6}\)
In simple words: Simplify all fractions, make denominators equal, and then pick the lowest and highest values.

Exam Tip: Always simplify fractions with negative signs in both numerator and denominator first (e.g., \(\frac{-9}{-36} = \frac{1}{4}\)).