CBSE Class 7 Mathematics Rational Numbers Assignment Set 01

Question. Express each of the following as a rational number with positive denominator.
(i) (-15/-28)
(ii) (6/-9)
Solution:
(i) Given (-15/-28)
Multiplying both numerator and denominator we can rational number with positive denominator.
(-15/-28) = (-15/-28) × -1
= (15/28)

(ii) Given (6/-9)
Multiplying both numerator and denominator we can rational number with positive denominator.
(6/-9) = (6/-9) × -1
= (-6/9)

Question. Express (3/5) as a rational number with numerator:
(i) 6
(ii) -15
Solution:
(i) Given (3/5)
To get numerator 6 we have to multiply both numerator and denominator by 2
Then we get, (3/5) × (2/2) = (6/10)
Therefore (3/5) as a rational number with numerator 6 is (6/10)

(ii) Given (3/5)
To get numerator -15 we have to multiply both numerator and denominator by -5
Then we get, (3/5) × (-5/-5)
= (-15/-25)
Therefore (3/5) as a rational number with numerator -15 is (-15/-25)

Question. Express (5/7) as a rational number with denominator:
(i) -14
(ii) 70
Solution:
(i) Given (5/7)
To get denominator -14 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (-2/-2)
= (-10/-14)
Therefore (5/7) as a rational number with denominator -14 is (-10/-14)

(ii) Given (5/7)
To get denominator 70 we have to multiply both numerator and denominator by -2
Then we get, (5/7) × (10/10)
= (50/70)
Therefore (5/7) as a rational number with denominator 70 is (50/70)

Question. Express (3/4) as a rational number with denominator:
(i) 20
(ii) 36
Solution:
(i) Given (3/4)
To get denominator 20 we have to multiply both numerator and denominator by 5
Then we get, (3/4) × (5/5)
= (15/20)
Therefore (3/4) as a rational number with denominator 20 is (15/20)

(ii) Given (3/4)
To get denominator 36 we have to multiply both numerator and denominator by 9
Then we get, (3/4) × (9/9)
= (27/36)
Therefore (3/4) as a rational number with denominator 36 is (27/36)

Question. Express (2/5) as a rational number with numerator:
(i) -56
(ii) 154
Solution:
(i) Given (2/5)
To get numerator -56 we have to multiply both numerator and denominator by -28
Then we get, (2/5) × (-28/-28)
= (-56/-140)
Therefore (2/5) as a rational number with numerator -56 is (-56/-150)

(ii) Given (2/5)
To get numerator 154 we have to multiply both numerator and denominator by 77
Then we get, (2/5) × (77/77)
= (154/385)
Therefore (2/5) as a rational number with numerator 154 is (154/385)

Question. Express (-192/108) as a rational number with numerator:
(i) 64
(ii) -16
Solution:
(i) Given (-192/108)
To get numerator 64 we have to divide both numerator and denominator by -3
Then we get, (-192/108) ÷ (-3/-3)
= (64/-36)
Therefore (-192/108) as a rational number with numerator 64 is (64/-36)

(ii) Given (-192/108)
To get numerator -16 we have to divide both numerator and denominator by 12
Then we get, (-192/108) ÷ (12/12)
= (-16/9)
Therefore (-192/108) as a rational number with numerator -16 is (-16/9)

Question. Express (169/-294) as a rational number with denominator:
(i) 14
(ii) -7
Solution:
(i) Given (169/-294)
To get denominator 14 we have to divide both numerator and denominator by -21
Then we get, (169/-294) ÷ (-21/-21)
= (-8/14)
Therefore (169/-294) as a rational number with denominator 14 is (-8/14)

(ii) Given (169/-294)
To get denominator -7 we have to divide both numerator and denominator by 42
Then we get, (169/-294) ÷ (42/42)
= (4/-7)
Therefore (169/-294) as a rational number with denominator -7 is (4/-7)

Question. Write (-14/42) in a form so that the numerator is equal to:
(i) -2
(ii) 7
Solution:
(i) Given (-14/42)
To get numerator -2 we have to divide both numerator and denominator by 7
Then we get, (-14/42) ÷ (7/7)
= (-2/6)
Therefore (-14/42) as a rational number with numerator -2 is (-2/6)

(ii) Given (-14/42)
To get numerator 7 we have to divide both numerator and denominator by -2
Then we get, (-14/42) ÷ (-2/-2)
= (7/-21)
Therefore (-14/42) as a rational number with numerator -14 is (-14/21)

Question. Select those rational numbers which can be written as a rational number with numerator 6:
(1/22), (2/3), (3/4), (4/-5), (5/6), (-6/7), (-7/8)
Solution:
Given rational numbers that can be written as a rational number with numerator 6 are:
Consider (1/22)
On multiplying by 6, (1/22) can be written as
(1/22) = (6/132)
Consider (2/3)
On multiplying by 3, (2/3) can be written as
(2/3) = (6/9)
Consider (3/4)
On multiplying by 2, (3/4) can be written as
(3/4) = (6/8)
Consider (-6/7)
On multiplying by -1, (-6/7) can be written as
(-6/7) = (6/-7)
Therefore rational numbers that can be written as a rational number with numerator 6 are (1/22), (2/3), (3/4) and (-6/7)

Question. Select those rational numbers which can be written as rational number with denominator 4:
(7/8), (64/16), (36/-12), (-16/17), (5/-4), (140/28)
Solution:
Given rational numbers that can be written as a rational number with denominator 4 are:
(7/8) = (3.5/4) (On dividing both denominator and denominator by 2)
(64/16) = (16/4) (On dividing both denominator and numerator by 4)
(36/-12) = (-12/4) (On dividing both denominator and numerator by -3)
(5/- 4) = (- 5/4) (On multiplying both denominator and numerator by -1)
(140/28) = (20/4) (On dividing both numerator and denominator by 7)

Question.In each of the following, find an equivalent form of the rational number having a common denominator:
(i) (3/4) and (5/12)
(ii) (2/3), (7/6) and (11/12)
Solution:
(i) Given (3/4) and (5/12)
On multiplying both numerator and denominator by 3
(3/4) = (3 × 3)/ (4 × 3) = (9/12)
Equivalent forms with same denominators are (9/12) and (5/12)
(ii) Given (2/3), (7/6) and (11/12)
On multiplying both numerator and denominator by 4
(2/3) = (2 × 4)/ (3 × 4) = (8/12)
And (7/6) = (7 × 2)/ (6 × 2) = (14/12)
Equivalent forms are (8/12), (14/12) and (11/12) having same denominators

Question 1. List five rational numbers between:
(a) -1 and 0
(b) -2 and -1
(c) \(-\frac{4}{5}\) and \(\frac{-2}{3}\)
(d) \(-\frac{1}{2}\) and \(\frac{2}{3}\)
(e) 2/3 and 3/4
Answer:
To find rational numbers between two given numbers, make their denominators equal.
(a) For -1 and 0: Write them with denominator 6.
\(-1 = -\frac{6}{6}\) and \(0 = \frac{0}{6}\)
Five rational numbers between them are: \(-\frac{1}{6}, -\frac{2}{6}, -\frac{3}{6}, -\frac{4}{6}, -\frac{5}{6}\)
(b) For -2 and -1: Write them with denominator 6.
\(-2 = -\frac{12}{6}\) and \(-1 = -\frac{6}{6}\)
Five rational numbers between them are: \(-\frac{7}{6}, -\frac{8}{6}, -\frac{9}{6}, -\frac{10}{6}, -\frac{11}{6}\)
(c) For \(-\frac{4}{5}\) and \(-\frac{2}{3}\): LCM of 5 and 3 is 15.
\(-\frac{4}{5} = -\frac{12}{15}\) and \(-\frac{2}{3} = -\frac{10}{15}\)
Multiply numerator and denominator by 3 to get enough space: \(-\frac{36}{45}\) and \(-\frac{30}{45}\)
Five rational numbers are: \(-\frac{31}{45}, -\frac{32}{45}, -\frac{33}{45}, -\frac{34}{45}, -\frac{35}{45}\)
(d) For \(-\frac{1}{2}\) and \(\frac{2}{3}\): LCM of 2 and 3 is 6.
\(-\frac{1}{2} = -\frac{3}{6}\) and \(\frac{2}{3} = \frac{4}{6}\)
Five rational numbers are: \(-\frac{2}{6}, -\frac{1}{6}, 0, \frac{1}{6}, \frac{2}{6}\)
(e) For \(\frac{2}{3}\) and \(\frac{3}{4}\): LCM of 3 and 4 is 12.
\(\frac{2}{3} = \frac{8}{12}\) and \(\frac{3}{4} = \frac{9}{12}\)
Multiply numerator and denominator by 6: \(\frac{48}{72}\) and \(\frac{54}{72}\)
Five rational numbers are: \(\frac{49}{72}, \frac{50}{72}, \frac{51}{72}, \frac{52}{72}, \frac{53}{72}\)
In simple words: Convert the fractions to have the same bottom number, then pick numbers that fit in between.

Exam Tip: There are infinitely many rational numbers between any two numbers; any valid set with common denominators is correct.

Question 2. Write four more rational numbers in each of the following patterns:
(i) \(\frac{-3}{5}, \frac{-6}{10}, \frac{-9}{15}, \frac{-12}{20}, \dots\)
(ii) \(\frac{-1}{2}, \frac{-2}{4}, \frac{-3}{6}, \dots\)
Answer:
(i) Observing the pattern, numerators decrease by 3 (multiples of -3) and denominators increase by 5 (multiples of 5):
Next four numbers are: \(\frac{-15}{25}, \frac{-18}{30}, \frac{-21}{35}, \frac{-24}{40}\)
(ii) Observing the pattern, numerators decrease by 1 and denominators increase by 2 (equivalent fractions of \(-\frac{1}{2}\)):
Next four numbers are: \(\frac{-4}{8}, \frac{-5}{10}, \frac{-6}{12}, \frac{-7}{14}\)
In simple words: Continue multiplying both the top and bottom numbers by increasing whole integers.

Exam Tip: Identify the multiplier for each successive term before writing the next sequence elements.

Question 3. Give four rational numbers equivalent to
(i) \(\frac{-2}{7}\)
(ii) \(\frac{5}{-3}\)
(iii) \(\frac{-4}{9}\)
Answer:
Multiply the numerator and denominator by 2, 3, 4, and 5 respectively:
(i) For \(\frac{-2}{7}\):
\(\frac{-2 \times 2}{7 \times 2} = \frac{-4}{14}\)
\(\frac{-2 \times 3}{7 \times 3} = \frac{-6}{21}\)
\(\frac{-2 \times 4}{7 \times 4} = \frac{-8}{28}\)
\(\frac{-2 \times 5}{7 \times 5} = \frac{-10}{35}\)
(ii) For \(\frac{5}{-3}\):
\(\frac{5 \times 2}{-3 \times 2} = \frac{10}{-6}\)
\(\frac{5 \times 3}{-3 \times 3} = \frac{15}{-9}\)
\(\frac{5 \times 4}{-3 \times 4} = \frac{20}{-12}\)
\(\frac{5 \times 5}{-3 \times 5} = \frac{25}{-15}\)
(iii) For \(\frac{-4}{9}\):
\(\frac{-4 \times 2}{9 \times 2} = \frac{-8}{18}\)
\(\frac{-4 \times 3}{9 \times 3} = \frac{-12}{27}\)
\(\frac{-4 \times 4}{9 \times 4} = \frac{-16}{36}\)
\(\frac{-4 \times 5}{9 \times 5} = \frac{-20}{45}\)
In simple words: Multiply top and bottom by the same number to get equivalent fractions.

Exam Tip: Always multiply both numerator and denominator by the exact same non-zero integer.

Question 4. Draw the number line and represent the following rational numbers on it:
(i) \(\frac{3}{4}\)
(ii) \(\frac{-5}{8}\)
(iii) \(\frac{-7}{4}\)
(iv) \(\frac{7}{8}\)
Answer:
Steps to represent on a number line:
1. Draw a straight horizontal line with arrows on both ends and mark an origin point 0.
2. Mark equal distances on the right side for positive numbers and left side for negative numbers.
3. For denominator 4, divide each unit length into 4 equal parts. For denominator 8, divide each unit into 8 equal parts.
(i) \(\frac{3}{4}\) lies between 0 and 1 on the positive side (at the 3rd subdivision).
(ii) \(\frac{-5}{8}\) lies between 0 and -1 on the negative side (at the 5th subdivision left of 0).
(iii) \(\frac{-7}{4}\) lies between -1 and -2 on the negative side (at the 7th subdivision left of 0).
(iv) \(\frac{7}{8}\) lies between 0 and 1 on the positive side (at the 7th subdivision right of 0).
In simple words: Divide the space between integers into equal parts according to the denominator, then count the steps using the numerator.

Exam Tip: Ensure spacing between consecutive integer markings on your drawn number line is strictly uniform.

Question 5. Which of the following rational numbers represent the same rational numbers?
(i) \(\frac{-7}{21}\) and \(\frac{3}{9}\)
(ii) \(\frac{-16}{20}\) and \(\frac{20}{-25}\)
(iii) \(\frac{-2}{-3}\) and \(\frac{2}{3}\)
(iv) \(\frac{-3}{5}\) and \(\frac{-12}{20}\)
(v) \(\frac{8}{-5}\) and \(\frac{-24}{15}\)
(vi) \(\frac{1}{3}\) and \(\frac{-1}{9}\)
(vii) \(\frac{-5}{9}\) and \(\frac{-5}{-9}\)
Answer:
Reduce each rational number to its simplest form to check equality:
(i) \(\frac{-7}{21} = -\frac{1}{3}\) and \(\frac{3}{9} = \frac{1}{3}\). Since \(-\frac{1}{3} \neq \frac{1}{3}\), they do not represent the same number.
(ii) \(\frac{-16}{20} = -\frac{4}{5}\) and \(\frac{20}{-25} = -\frac{4}{5}\). They are the same.
(iii) \(\frac{-2}{-3} = \frac{2}{3}\) and \(\frac{2}{3} = \frac{2}{3}\). They are the same.
(iv) \(\frac{-3}{5}\) and \(\frac{-12}{20} = -\frac{3}{5}\). They are the same.
(v) \(\frac{8}{-5} = -\frac{8}{5}\) and \(\frac{-24}{15} = -\frac{8}{5}\). They are the same.
(vi) \(\frac{1}{3}\) and \(\frac{-1}{9}\). Since \(\frac{1}{3} \neq -\frac{1}{9}\), they do not represent the same number.
(vii) \(\frac{-5}{9}\) and \(\frac{-5}{-9} = \frac{5}{9}\). Since \(-\frac{5}{9} \neq \frac{5}{9}\), they do not represent the same number.
In simple words: Simplify both fractions to lowest terms; if they match completely, they represent the same number.

Exam Tip: Alternatively, cross-multiply the two fractions; if products are equal (\(ad = bc\)), they are equivalent.

Question 6. Rewrite the following rational numbers in simplest form:
(i) \(\frac{-8}{9}\)
(ii) \(\frac{25}{45}\)
(iii) \(\frac{-44}{72}\)
(iv) \(\frac{-8}{10}\)
Answer:
Divide numerator and denominator by their Highest Common Factor (HCF):
(i) \(\frac{-8}{9}\): HCF of 8 and 9 is 1. Simplest form is \(\mathbf{-\frac{8}{9}}\).
(ii) \(\frac{25}{45}\): HCF of 25 and 45 is 5.
\(\frac{25 \div 5}{45 \div 5} = \mathbf{\frac{5}{9}}\)
(iii) \(\frac{-44}{72}\): HCF of 44 and 72 is 4.
\(\frac{-44 \div 4}{72 \div 4} = \mathbf{-\frac{11}{18}}\)
(iv) \(\frac{-8}{10}\): HCF of 8 and 10 is 2.
\(\frac{-8 \div 2}{10 \div 2} = \mathbf{-\frac{4}{5}}\)
In simple words: Cancel out any common factors from both the top and bottom numbers until no more common factors remain.

Exam Tip: Always verify that the HCF of the numerator and denominator in your final answer is 1.

Question 7. Solve:
a) \(1\frac{2}{4} + \frac{3}{8}\)
b) \(\frac{4}{9} - \frac{-2}{8}\)
c) \(\frac{3}{16} \div \left(-\frac{7}{20}\right)\)
Answer:
a) Convert mixed fraction to improper fraction: \(1\frac{2}{4} = \frac{6}{4} = \frac{3}{2}\)
\(\frac{3}{2} + \frac{3}{8}\)
LCM of 2 and 8 is 8.
\(= \frac{3 \times 4}{2 \times 4} + \frac{3}{8}\)
\(= \frac{12}{8} + \frac{3}{8} = \mathbf{\frac{15}{8}} \text{ or } 1\frac{7}{8}\)
b) \(\frac{4}{9} - \left(-\frac{2}{8}\right) = \frac{4}{9} + \frac{2}{8}\)
Simplify \(\frac{2}{8} = \frac{1}{4}\)
\(= \frac{4}{9} + \frac{1}{4}\)
LCM of 9 and 4 is 36.
\(= \frac{4 \times 4}{9 \times 4} + \frac{1 \times 9}{4 \times 9}\)
\(= \frac{16}{36} + \frac{9}{36} = \mathbf{\frac{25}{36}}\)
c) \(\frac{3}{16} \div \left(-\frac{7}{20}\right) = \frac{3}{16} \times \left(-\frac{20}{7}\right)\)
\(= \frac{3 \times (-20)}{16 \times 7} = \frac{-60}{112}\)
Divide by HCF (4):
\(= \mathbf{-\frac{15}{28}}\)
In simple words: Find common denominators for addition/subtraction, and multiply by the reciprocal for division.

Exam Tip: Remember that subtracting a negative number is equivalent to adding its positive counterpart.

Question 8. Salma finished her homework in \(\frac{7}{12}\) hour. Vijay finished the same homework in \(\frac{3}{4}\) hour. Who worked longer? By what fraction was it longer?
Answer:
Time taken by Salma = \(\frac{7}{12}\) hour
Time taken by Vijay = \(\frac{3}{4}\) hour
To compare, make denominators equal (LCM of 12 and 4 is 12):
Vijay's time = \(\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\) hour
Since \(\frac{9}{12} > \frac{7}{12}\), Vijay worked longer.
Difference in time = \(\frac{9}{12} - \frac{7}{12} = \frac{2}{12} = \mathbf{\frac{1}{6}}\) hour.
In simple words: Convert both times to fractions with the same bottom number to easily see who took more time, then subtract to find the difference.

Question 9. Find:
a) \(\frac{-4}{9} \div 5\)
b) \(-2\frac{1}{3} + \frac{3}{5}\)
c) \(-2\frac{1}{3} \times 3\)
Answer:
a) \(\frac{-4}{9} \div 5 = \frac{-4}{9} \times \frac{1}{5} = \mathbf{-\frac{4}{45}}\)
b) \(-2\frac{1}{3} + \frac{3}{5} = -\frac{7}{3} + \frac{3}{5}\)
LCM of 3 and 5 is 15.
\(= \frac{-7 \times 5}{3 \times 5} + \frac{3 \times 3}{5 \times 3}\)
\(= \frac{-35}{15} + \frac{9}{15} = \mathbf{-\frac{26}{15}} \text{ or } -1\frac{11}{15}\)
c) \(-2\frac{1}{3} \times 3 = -\frac{7}{3} \times 3 = \frac{-7 \times 3}{3} = \mathbf{-7}\)
In simple words: Convert mixed numbers to improper fractions first, then perform the operation.
Exam Tip: Do not forget to carry the negative sign correctly through multiplication and addition steps.

Question 10. Arrange the following rational numbers in ascending order:
(i) \(-\frac{3}{5}, \frac{-2}{5}, -\frac{1}{5}\) (Note: original OCR lists elements, let us sort standard set: \(-\frac{3}{5}, \frac{1}{3}, -\frac{2}{5}, \frac{2}{3}, -\frac{4}{5}\))
(ii) \(\frac{1}{3}, -\frac{2}{9}, \frac{3}{4}, -\frac{5}{16}\)
(iii) \(\frac{2}{3}, -\frac{3}{4}, \frac{2}{5}\)
Answer:
(i) Given numbers: \(-\frac{3}{5}, \frac{1}{3}, -\frac{2}{5}, \frac{2}{3}, -\frac{4}{5}\)
LCM of denominators 5 and 3 is 15.
\(-\frac{3}{5} = -\frac{9}{15}\)
\(\frac{1}{3} = \frac{5}{15}\)
\(-\frac{2}{5} = -\frac{6}{15}\)
\(\frac{2}{3} = \frac{10}{15}\)
\(-\frac{4}{5} = -\frac{12}{15}\)
Comparing numerators: \(-12 < -9 < -6 < 5 < 10\)
Ascending order: \(\mathbf{-\frac{4}{5}, -\frac{3}{5}, -\frac{2}{5}, \frac{1}{3}, \frac{2}{3}}\)

(ii) Given numbers: \(\frac{1}{3}, -\frac{2}{9}, \frac{3}{4}, -\frac{5}{16}\)
LCM of 3, 9, 4, 16 (144):
\(\frac{1}{3} = \frac{48}{144}\)
\(-\frac{2}{9} = -\frac{32}{144}\)
\(\frac{3}{4} = \frac{108}{144}\)
\(-\frac{5}{16} = -\frac{45}{144}\)
Comparing numerators: \(-45 < -32 < 48 < 108\)
Ascending order: \(\mathbf{-\frac{5}{16}, -\frac{2}{9}, \frac{1}{3}, \frac{3}{4}}\)

(iii) Given numbers: \(\frac{2}{3}, -\frac{3}{4}, \frac{2}{5}\)
LCM of 3, 4, 5 is 60.
\(\frac{2}{3} = \frac{40}{60}\)
\(-\frac{3}{4} = -\frac{45}{60}\)
\(\frac{2}{5} = \frac{24}{60}\)
Comparing numerators: \(-45 < 24 < 40\)
Ascending order: \(\mathbf{-\frac{3}{4}, \frac{2}{5}, \frac{2}{3}}\)
In simple words: Make all denominators the same, then arrange from smallest negative to largest positive number based on the numerators.

Exam Tip: For negative fractions, the number with the larger absolute value is actually smaller.

Question 11. Simplify:
(i) \(\frac{-2}{3} + \frac{5}{6} - \frac{-7}{9}\)
(ii) \(7\frac{1}{2} - 4\frac{3}{8} + 2\frac{3}{12}\)
Answer:
(i) \(\frac{-2}{3} + \frac{5}{6} + \frac{7}{9}\) (since minus of minus is plus)
LCM of 3, 6, and 9 is 18.
\(= \frac{-2 \times 6}{3 \times 6} + \frac{5 \times 3}{6 \times 3} + \frac{7 \times 2}{9 \times 2}\)
\(= \frac{-12}{18} + \frac{15}{18} + \frac{14}{18}\)
\(= \frac{-12 + 15 + 14}{18} = \mathbf{\frac{17}{18}}\)
(ii) Convert mixed fractions:
\(7\frac{1}{2} = \frac{15}{2}\)
\(4\frac{3}{8} = \frac{35}{8}\)
\(2\frac{3}{12} = \frac{27}{12} = \frac{9}{4}\)
Expression: \(\frac{15}{2} - \frac{35}{8} + \frac{9}{4}\)
LCM of 2, 8, and 4 is 8.
\(= \frac{15 \times 4}{2 \times 4} - \frac{35}{8} + \frac{9 \times 2}{4 \times 2}\)
\(= \frac{60}{8} - \frac{35}{8} + \frac{18}{8}\)
\(= \frac{60 - 35 + 18}{8} = \mathbf{\frac{43}{8}} \text{ or } 5\frac{3}{8}\)
In simple words: Find the common denominator for all fractions, combine numerators, and simplify.

Exam Tip: Simplify individual mixed fractions before finding the LCM to make calculations easier.

Question 12. Solve:
(i) \(\frac{-7}{11} \times \frac{3}{5}\)
(ii) \(\frac{-5}{9} \times 25\)
(iii) \(-3\frac{1}{5} \times 24\)
(iv) \(-3\frac{1}{8} \times 4\frac{6}{11}\)
(v) \(\frac{-6}{10} \div \frac{-8}{3}\)
(vi) \(-4\frac{1}{5} \div \frac{-3}{5}\)
(vii) \(-5\frac{3}{7} \div 1\frac{3}{10}\)
(viii) \(-4 \div 2\frac{3}{5}\)
(ix) \(-6 \div \frac{-2}{3}\)
Answer:
(i) \(\frac{-7 \times 3}{11 \times 5} = \mathbf{-\frac{7}{55}}\)
(ii) \(\frac{-5 \times 25}{9} = \mathbf{-\frac{125}{9}}\)
(iii) \(-\frac{16}{5} \times 24 = \frac{-16 \times 24}{5} = \mathbf{-\frac{384}{5}}\)
(iv) \(-\frac{25}{8} \times \frac{50}{11} = \frac{-1250}{88} = \mathbf{-\frac{625}{44}}\)
(v) \(\frac{-6}{10} \times \left(-\frac{3}{8}\right) = \frac{-6 \times (-3)}{10 \times 8} = \frac{18}{80} = \mathbf{\frac{9}{40}}\)
(vi) \(-\frac{21}{5} \div \left(-\frac{3}{5}\right) = -\frac{21}{5} \times \left(-\frac{5}{3}\right) = \frac{-21 \times (-5)}{5 \times 3} = \frac{105}{15} = \mathbf{7}\)
(vii) \(-\frac{38}{7} \div \frac{13}{10} = -\frac{38}{7} \times \frac{10}{13} = \mathbf{-\frac{380}{91}}\)
(viii) \(-4 \div \frac{13}{5} = -4 \times \frac{5}{13} = \mathbf{-\frac{20}{13}}\)
(ix) \(-6 \div \left(-\frac{2}{3}\right) = -6 \times \left(-\frac{3}{2}\right) = \frac{-6 \times (-3)}{2} = \frac{18}{2} = \mathbf{9}\)
In simple words: Multiply straight across for multiplication, and flip the second fraction upside down and multiply for division.

Exam Tip: Cancel common factors between numerators and denominators before multiplying to save time.

Question 13. Divide:
(i) \(\frac{-3}{8}\) by 4
(ii) \(\frac{-9}{16}\) by 6
(iii) -9 by \(\frac{3}{16}\)
(iv) -10 by \(\frac{100}{3}\)
Answer:
(i) \(\frac{-3}{8} \div 4 = \frac{-3}{8} \times \frac{1}{4} = \mathbf{-\frac{3}{32}}\)
(ii) \(\frac{-9}{16} \div 6 = \frac{-9}{16} \times \frac{1}{6} = \frac{-9}{96} = \mathbf{-\frac{3}{32}}\)
(iii) \(-9 \div \frac{3}{16} = -9 \times \frac{16}{3} = \frac{-9 \times 16}{3} = -3 \times 16 = \mathbf{-48}\)
(iv) \(-10 \div \frac{100}{3} = -10 \times \frac{3}{100} = \frac{-30}{100} = \mathbf{-\frac{3}{10}}\)
In simple words: Change division into multiplication by writing the reciprocal of the second number.

Exam Tip: Always reduce fractional answers to their simplest form.

Question 14. Satpal walks \(\frac{2}{3}\) km from place P towards east and then from there \(\frac{12}{7}\) km towards west. Where will he be now from P?
Answer:
Let position P be at 0 on the number line.
Distance walked towards east = \(+\frac{2}{3}\) km
Distance walked towards west from that point = \(-\frac{12}{7}\) km
Total position from P = \(\frac{2}{3} - \frac{12}{7}\)
LCM of 3 and 7 is 21.
\(= \frac{2 \times 7}{3 \times 7} - \frac{12 \times 3}{7 \times 3}\)
\(= \frac{14}{21} - \frac{36}{21}\)
\(= \frac{14 - 36}{21} = \mathbf{-\frac{22}{21}} \text{ km}\) (or \(-1\frac{1}{21}\) km)
Since the result is negative, Satpal is \(\mathbf{1\frac{1}{21} \text{ km}}\) to the west of P.
In simple words: Take east as positive and west as negative, then add the two movements together.

Exam Tip: Clearly state the final direction along with the distance when dealing with displacement problems.

Question 15. Multiply \(\frac{-6}{13}\) by the reciprocal of \(\frac{-7}{6}\)
Answer:
Reciprocal of \(\frac{-7}{6} = \frac{6}{-7}\) or \(-\frac{6}{7}\)
Multiply \(\frac{-6}{13}\) by \(-\frac{6}{7}\):
\(= \frac{-6 \times (-6)}{13 \times 7}\)
\(= \mathbf{\frac{36}{91}}\)
In simple words: Flip the second fraction upside down, then multiply the two top numbers together and the two bottom numbers together.

Exam Tip: The product of two negative numbers always results in a positive number.