Rational Numbers

Q1: Find the multiplicative inverse (or reciprocal) of each of the following rational numbers:

i. \(\frac{6}{25}\)

Step 1: The multiplicative inverse of \(\frac{6}{25}\) is \(\frac{25}{6}\).
Answer: \(\frac{25}{6}\)

ii. \(-2\frac{3}{11}\)

Step 1: Convert mixed fraction \(-2\frac{3}{11}\) to improper fraction: \[ -2\frac{3}{11} = -\frac{25}{11} \] Step 2: Find reciprocal: \[ \text{Reciprocal} = \frac{11}{-25} = -\frac{11}{25} \]Answer: \(-\frac{11}{25}\)

iii. \(\frac{-8}{9}\)

\[ \text{Reciprocal of } \frac{-8}{9} = \frac{9}{-8} = -\frac{9}{8} \]Answer: \(-\frac{9}{8}\)

iv. \(\frac{-23}{16}\)

\[ \text{Reciprocal of } \frac{-23}{16} = \frac{16}{-23} = -\frac{16}{23} \]Answer: \(-\frac{16}{23}\)

v. \(12\)

\[ 12 = \frac{12}{1} \\ \text{Reciprocal} = \frac{1}{12} \]Answer: \(\frac{1}{12}\)

vi. \(\frac{1}{10}\)

\[ \text{Reciprocal of } \frac{1}{10} = \frac{10}{1} = 10 \]Answer: 10

vii. \(-6\)

\[ -6 = \frac{-6}{1} \\ \text{Reciprocal} = \frac{1}{-6} = -\frac{1}{6} \]Answer: \(-\frac{1}{6}\)

viii. \(-1\)

\[ \text{Reciprocal of } -1 = -1 \]Answer: -1

ix. \(\frac{-1}{5}\)

\[ \text{Reciprocal of } \frac{-1}{5} = \frac{5}{-1} = -5 \]Answer: -5

x. \(\frac{-7}{-9}\)

\[ \frac{-7}{-9} = \frac{7}{9} \\ \text{Reciprocal} = \frac{9}{7} \]Answer: \(\frac{9}{7}\)

Q2: Evaluate:

i. \(\frac{7}{12}\div\frac{-4}{3}\)

Step 1: Division of fractions means multiplying by the reciprocal: \[ \frac{7}{12} \div \frac{-4}{3} = \frac{7}{12} \times \frac{3}{-4} \]Step 2: Multiply numerators and denominators: \[ = \frac{7 \times 3}{12 \times -4} = \frac{21}{-48} = -\frac{21}{48} \]Step 3: Simplify fraction: \[ -\frac{21}{48} = -\frac{7}{16} \]Answer: \(-\frac{7}{16}\)

ii. \(\frac{-12}{25}\div\frac{-5}{6}\)

\[ \frac{-12}{25} \div \frac{-5}{6} = \frac{-12}{25} \times \frac{6}{-5} \]Multiply numerators and denominators: \[ = \frac{-12 \times 6}{25 \times -5} = \frac{-72}{-125} = \frac{72}{125} \]Answer: \(\frac{72}{125}\)

iii. \(\frac{-27}{32}\div\frac{-9}{16}\)

\[ \frac{-27}{32} \div \frac{-9}{16} = \frac{-27}{32} \times \frac{16}{-9} \]Multiply: \[ = \frac{-27 \times 16}{32 \times -9} = \frac{-432}{-288} = \frac{432}{288} \]Simplify: \[ = \frac{3}{2} \]Answer: \(\frac{3}{2}\)

iv. \(-2\frac{4}{7}\div\frac{6}{35}\)

Convert mixed fraction: \[ -2\frac{4}{7} = -\frac{18}{7} \]Division: \[ -\frac{18}{7} \div \frac{6}{35} = -\frac{18}{7} \times \frac{35}{6} \]Multiply: \[ = \frac{-18 \times 35}{7 \times 6} = \frac{-630}{42} \]Simplify: \[ = -15 \]Answer: -15

v. \(26\div\frac{-1}{13}\)

\[ 26 \div \frac{-1}{13} = 26 \times \frac{13}{-1} = \frac{26 \times 13}{-1} = \frac{338}{-1} = -338 \]Answer: -338

vi. \(\frac{1}{25}\div-5\)

\[ \frac{1}{25} \div -5 = \frac{1}{25} \times \frac{1}{-5} = \frac{1}{-125} = -\frac{1}{125} \]Answer: \(-\frac{1}{125}\)

Q3: The product of two rational numbers is \(\frac{2}{5}\). If one of them is \(\frac{-8}{25}\), find the other.

Step 1: Let the other rational number be \(x\).Given: \[ x \times \frac{-8}{25} = \frac{2}{5} \]Step 2: To find \(x\), divide both sides by \(\frac{-8}{25}\), or multiply by its reciprocal: \[ x = \frac{2}{5} \div \frac{-8}{25} = \frac{2}{5} \times \frac{25}{-8} \]Step 3: Multiply numerators and denominators: \[ x = \frac{2 \times 25}{5 \times -8} = \frac{50}{-40} = -\frac{50}{40} \]Step 4: Simplify the fraction: \[ -\frac{50}{40} = -\frac{5}{4} \]Answer: \(-\frac{5}{4}\)

Q4: The product of two rational numbers is \(\frac{-2}{3}\). If one of them is \(\frac{16}{39}\), find the other.

Step 1: Let the other rational number be \(x\).Given: \[ x \times \frac{16}{39} = \frac{-2}{3} \]Step 2: To find \(x\), divide both sides by \(\frac{16}{39}\), or multiply by its reciprocal: \[ x = \frac{-2}{3} \div \frac{16}{39} = \frac{-2}{3} \times \frac{39}{16} \]Step 3: Multiply numerators and denominators: \[ x = \frac{-2 \times 39}{3 \times 16} = \frac{-78}{48} \]Step 4: Simplify the fraction: \[ \frac{-78}{48} = \frac{-13}{8} \]Answer: \(\displaystyle -\frac{13}{8}\)

Q5: By what rational number should \(\frac{-9}{35}\) be multiplied to get \(\frac{3}{5}\)?

Step 1: Let the required rational number be \(x\).Given: \[ x \times \frac{-9}{35} = \frac{3}{5} \]Step 2: To find \(x\), divide both sides by \(\frac{-9}{35}\) or multiply by its reciprocal: \[ x = \frac{3}{5} \div \frac{-9}{35} = \frac{3}{5} \times \frac{35}{-9} \]Step 3: Multiply numerators and denominators: \[ x = \frac{3 \times 35}{5 \times -9} = \frac{105}{-45} \]Step 4: Simplify the fraction: \[ \frac{105}{-45} = \frac{7}{-3} = -\frac{7}{3} \]Answer: \(\displaystyle -\frac{7}{3}\)

Q6: By what rational should \(\frac{25}{8}\) multiplied to get \(\frac{-20}{7}\)?

Step 1: Let the required rational number be \(x\).Given: \[ x \times \frac{25}{8} = \frac{-20}{7} \]Step 2: To find \(x\), divide both sides by \(\frac{25}{8}\) or multiply by its reciprocal: \[ x = \frac{-20}{7} \div \frac{25}{8} = \frac{-20}{7} \times \frac{8}{25} \]Step 3: Multiply numerators and denominators: \[ x = \frac{-20 \times 8}{7 \times 25} = \frac{-160}{175} \]Step 4: Simplify the fraction by dividing numerator and denominator by 5: \[ x = \frac{-160 \div 5}{175 \div 5} = \frac{-32}{35} \]Answer: \(\displaystyle -\frac{32}{35}\)

Q7: The cost of 17 pencils is ₹\(59\frac{1}{2}\). Find the cost of each pencil.

Step 1: Convert the mixed fraction ₹\(59\frac{1}{2}\) into an improper fraction: \[ 59\frac{1}{2} = \frac{(59 \times 2) + 1}{2} = \frac{118 + 1}{2} = \frac{119}{2} \]Step 2: Let the cost of each pencil be \(x\). Since 17 pencils cost ₹\(\frac{119}{2}\), the cost of one pencil is: \[ x = \frac{\frac{119}{2}}{17} = \frac{119}{2} \times \frac{1}{17} = \frac{119}{34} \]Step 3: Simplify the fraction if possible: \[ \frac{119}{34} \] Since 119 and 34 have no common factors other than 1, the fraction is in simplest form.Step 4: Convert to a mixed fraction to express the cost in rupees and paise: \[ \frac{119}{34} = 3 \frac{17}{34} = 3 \frac{1}{2} \]Answer: The cost of each pencil is ₹\(3\frac{1}{2}\) or ₹3.50.

Q8: The cost of 20 metres of ribbon is ₹335. Find the cost of each metre of it.

Step 1: Let the cost of each metre of ribbon be \(x\).Step 2: Total cost of 20 metres = ₹335. So, \(20 \times x = 335\).Step 3: To find \(x\), divide both sides by 20: \[ x = \frac{335}{20} \]Step 4: Simplify the fraction: \[ x = \frac{335}{20} = \frac{67 \times 5}{4 \times 5} = \frac{67}{4} = 16 \frac{3}{4} \]Answer: The cost of each metre of ribbon is ₹\(16\frac{3}{4}\) or ₹16.75.

Q9: How many pieces, each of length \(2\frac{3}{4}\) m, can be cut from a rope of length 66 m?

Step 1: Convert the mixed fraction \(2\frac{3}{4}\) into an improper fraction. \[ 2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} \]Step 2: Total length of rope = 66 m.Step 3: Number of pieces = \(\frac{\text{Total length of rope}}{\text{Length of each piece}}\) \[ = \frac{66}{\frac{11}{4}} \]Step 4: Dividing by a fraction is the same as multiplying by its reciprocal: \[ = 66 \times \frac{4}{11} \]Step 5: Simplify the multiplication: \[ = \frac{66}{11} \times 4 = 6 \times 4 = 24 \]Answer: 24 pieces of length \(2\frac{3}{4}\) m each can be cut from the rope of length 66 m.

Q10: Fill in the blanks:

i. \(( \_\_\_\_ ) \div \left(\frac{-5}{6}\right) = -30\)

Step 1: Let the unknown number be \(x\). So, \[ x \div \left(\frac{-5}{6}\right) = -30 \]Step 2: Dividing by a fraction means multiplying by its reciprocal: \[ x \times \left(\frac{6}{-5}\right) = -30 \]Step 3: Solve for \(x\): \[ x = -30 \times \frac{-5}{6} \]Step 4: Simplify: \[ x = -30 \times \left(\frac{-5}{6}\right) = \frac{-30 \times -5}{6} = \frac{150}{6} = 25 \]Answer: \(x = 25\)

ii. \(( \_\_\_\_ ) \div (-8) = \frac{-3}{4}\)

Step 1: Let the unknown number be \(x\). So, \[ x \div (-8) = \frac{-3}{4} \]Step 2: Multiply both sides by \(-8\): \[ x = \frac{-3}{4} \times (-8) \]Step 3: Simplify: \[ x = \frac{-3 \times -8}{4} = \frac{24}{4} = 6 \]Answer: \(x = 6\)

iii. \(\left(\frac{-15}{14}\right) \div ( \_\_\_\_ ) = \frac{5}{2}\)

Step 1: Let the unknown number be \(x\). So, \[ \frac{-15}{14} \div x = \frac{5}{2} \]Step 2: Dividing by \(x\) means multiplying by \(\frac{1}{x}\), so: \[ \frac{-15}{14} \times \frac{1}{z} = \frac{5}{2} \]Step 3: Solve for \(x\): \[ \frac{-15}{14x} = \frac{5}{2} \\ x = \frac{-15}{14} \div \frac{5}{2} = \frac{-15}{14} \times \frac{2}{5} = \frac{-30}{70} = \frac{-3}{7} \]Answer: \(z = \frac{-3}{7}\)

iv. \(-16 \div ( \_\_\_\_ ) = 6\)

Step 1: Let the unknown number be \(x\). So, \[ -16 \div x = 6 \]Step 2: Multiply both sides by \(x\): \[ -16 = 6x \]Step 3: Solve for \(x\): \[ x = \frac{-16}{6} = \frac{-8}{3} \]Answer: \(x = \frac{-8}{3}\)