Rational Numbers

Q1: Multiply:

i. \(\frac{2}{3}\) by \(\frac{4}{5}\)

Step 1: Multiply the numerators and the denominators \[ \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \] Answer: \(\frac{8}{15}\)

ii. \(\frac{7}{6}\) by \(\frac{9}{2}\)

Step 1: Multiply the numerators and denominators \[ \frac{7}{6} \times \frac{9}{2} = \frac{63}{12} = \frac{21}{4} \] Answer: \(\frac{21}{4}\)

iii. \(\frac{5}{6}\) by 30

Step 1: Convert 30 into a fraction → \(\frac{30}{1}\) \[ \frac{5}{6} \times \frac{30}{1} = \frac{150}{6} = 25 \] Answer: 25

iv. \(\frac{-3}{4}\) by \(\frac{8}{7}\)

Step 1: Multiply the numerators and denominators \[ \frac{-3}{4} \times \frac{8}{7} = \frac{-24}{28} = \frac{-6}{7} \] Answer: \(\frac{-6}{7}\)

v. \(\frac{-16}{9}\) by \(\frac{12}{-5}\)

Step 1: Multiply the signs and simplify \[ \frac{-16}{9} \times \frac{12}{-5} = \frac{192}{45} = \frac{64}{15} \] Answer: \(\frac{64}{15}\)

vi. \(\frac{35}{-8}\) by \(\frac{12}{-5}\)

Step 1: Multiply and handle negatives \[ \frac{35}{-8} \times \frac{12}{-5} = \frac{420}{40} = \frac{21}{2} \] Answer: \(\frac{21}{2}\)

vii. \(\frac{-3}{10}\) by \(\frac{-40}{9}\)

Step 1: Multiply the numbers and cancel out \[ \frac{-3}{10} \times \frac{-40}{9} = \frac{120}{90} = \frac{4}{3} \] Answer: \(\frac{4}{3}\)

viii. \(\frac{-32}{5}\) by \(\frac{15}{-16}\)

Step 1: Multiply and simplify signs \[ \frac{-32}{5} \times \frac{15}{-16} = \frac{480}{80} = 6 \] Answer: 6

ix. \(\frac{-8}{15}\) by \(\frac{-25}{32}\)

Step 1: Multiply numerators and denominators \[ \frac{-8}{15} \times \frac{-25}{32} = \frac{200}{480} = \frac{5}{12} \] Answer: \(\frac{5}{12}\)

Q2: Simplify:

i. \(\frac{7}{15} \times \frac{5}{6}\)

Step 1: Multiply the numerators: \[ 7 \times 5 = 35 \] Step 2: Multiply the denominators: \[ 15 \times 6 = 90 \] Step 3: Write the fraction: \[ \frac{35}{90} \] Step 4: Simplify by dividing numerator and denominator by 5: \[ \frac{35 \div 5}{90 \div 5} = \frac{7}{18} \] Answer: \(\frac{7}{18}\)

ii. \(\frac{-5}{24} \times \frac{6}{25}\)

Step 1: Multiply numerators: \[ -5 \times 6 = -30 \] Step 2: Multiply denominators: \[ 24 \times 25 = 600 \] Step 3: Write the fraction: \[ \frac{-30}{600} \] Step 4: Simplify by dividing numerator and denominator by 30: \[ \frac{-30 \div 30}{600 \div 30} = \frac{-1}{20} \] Answer: \(\frac{-1}{20}\)

iii. \(\frac{7}{-18} \times \frac{-9}{14}\)

Step 1: Multiply numerators: \[ 7 \times -9 = -63 \] Step 2: Multiply denominators: \[ -18 \times 14 = -252 \] Step 3: Write the fraction: \[ \frac{-63}{-252} \] Step 4: Negative signs cancel each other: \[ \frac{63}{252} \] Step 5: Simplify by dividing numerator and denominator by 63: \[ \frac{63 \div 63}{252 \div 63} = \frac{1}{4} \] Answer: \(\frac{1}{4}\)

iv. \(\frac{-9}{5} \times \frac{-10}{3}\)

Step 1: Multiply numerators: \[ -9 \times -10 = 90 \] Step 2: Multiply denominators: \[ 5 \times 3 = 15 \] Step 3: Write the fraction: \[ \frac{90}{15} \] Step 4: Simplify by dividing numerator and denominator by 15: \[ \frac{90 \div 15}{15 \div 15} = \frac{6}{1} = 6 \] Answer: 6

v. \(-28 \times \frac{-8}{7}\)

Step 1: Express -28 as a fraction: \[ -28 = \frac{-28}{1} \] Step 2: Multiply numerators: \[ -28 \times -8 = 224 \] Step 3: Multiply denominators: \[ 1 \times 7 = 7 \] Step 4: Write the fraction: \[ \frac{224}{7} \] Step 5: Simplify by dividing numerator and denominator by 7: \[ \frac{224 \div 7}{7 \div 7} = \frac{32}{1} = 32 \] Answer: 32

vi. \(\frac{8}{-21} \times \frac{-14}{3}\)

Step 1: Multiply numerators: \[ 8 \times -14 = -112 \] Step 2: Multiply denominators: \[ -21 \times 3 = -63 \] Step 3: Write the fraction: \[ \frac{-112}{-63} \] Step 4: Negative signs cancel out: \[ \frac{112}{63} \] Step 5: Simplify by dividing numerator and denominator by 7: \[ \frac{112 \div 7}{63 \div 7} = \frac{16}{9} \] Answer: \(\frac{16}{9}\)

Q3: Simplify:

i. \(\frac{5}{12} \times (-36)\)

Step 1: Express -36 as a fraction: \[ -36 = \frac{-36}{1} \] Step 2: Multiply numerators: \[ 5 \times (-36) = -180 \] Step 3: Multiply denominators: \[ 12 \times 1 = 12 \] Step 4: Write the fraction: \[ \frac{-180}{12} \] Step 5: Simplify by dividing numerator and denominator by 12: \[ \frac{-180 \div 12}{12 \div 12} = \frac{-15}{1} = -15 \] Answer: -15

ii. \(\frac{-17}{18} \times 12\)

Step 1: Express 12 as a fraction: \[ 12 = \frac{12}{1} \] Step 2: Multiply numerators: \[ -17 \times 12 = -204 \] Step 3: Multiply denominators: \[ 18 \times 1 = 18 \] Step 4: Write the fraction: \[ \frac{-204}{18} \] Step 5: Simplify by dividing numerator and denominator by 6: \[ \frac{-204 \div 6}{18 \div 6} = \frac{-34}{3} \] Answer: \(\frac{-34}{3}\)

iii. \(\frac{-5}{6} \times \frac{6}{5}\)

Step 1: Multiply numerators: \[ -5 \times 6 = -30 \] Step 2: Multiply denominators: \[ 6 \times 5 = 30 \] Step 3: Write the fraction: \[ \frac{-30}{30} \] Step 4: Simplify the fraction: \[ \frac{-30}{30} = -1 \] Answer: -1

iv. \(-14 \times \frac{9}{28}\)

Step 1: Express -14 as a fraction: \[ -14 = \frac{-14}{1} \] Step 2: Multiply numerators: \[ -14 \times 9 = -126 \] Step 3: Multiply denominators: \[ 1 \times 28 = 28 \] Step 4: Write the fraction: \[ \frac{-126}{28} \] Step 5: Simplify by dividing numerator and denominator by 14: \[ \frac{-126 \div 14}{28 \div 14} = \frac{-9}{2} \] Answer: \(\frac{-9}{2}\)

v. \(-4\frac{4}{5} \times \left(-7\frac{1}{2}\right)\)

Step 1: Convert mixed numbers to improper fractions: \[ -4\frac{4}{5} = \frac{-24}{5}, \quad -7\frac{1}{2} = \frac{-15}{2} \] Step 2: Multiply numerators: \[ -24 \times -15 = 360 \] Step 3: Multiply denominators: \[ 5 \times 2 = 10 \] Step 4: Write the fraction: \[ \frac{360}{10} \] Step 5: Simplify by dividing numerator and denominator by 10: \[ \frac{360 \div 10}{10 \div 10} = \frac{36}{1} = 36 \] Answer: 36

vi. \(\frac{-8}{15} \times \frac{-25}{32}\)

Step 1: Multiply numerators: \[ -8 \times -25 = 200 \] Step 2: Multiply denominators: \[ 15 \times 32 = 480 \] Step 3: Write the fraction: \[ \frac{200}{480} \] Step 4: Simplify by dividing numerator and denominator by 40: \[ \frac{200 \div 40}{480 \div 40} = \frac{5}{12} \] Answer: \(\frac{5}{12}\)

Q4: Simplify:

i. \(\left(\frac{2}{5} \times \frac{5}{8}\right) + \left(\frac{-3}{7} \times \frac{14}{-15}\right)\)

Step 1: Multiply the first pair: \[ \frac{2}{5} \times \frac{5}{8} = \frac{2 \times 5}{5 \times 8} = \frac{10}{40} = \frac{1}{4} \]Step 2: Multiply the second pair: \[ \frac{-3}{7} \times \frac{14}{-15} = \frac{-3 \times 14}{7 \times -15} = \frac{-42}{-105} = \frac{42}{105} \] Simplify \(\frac{42}{105}\): divide numerator and denominator by 21: \[ \frac{42 \div 21}{105 \div 21} = \frac{2}{5} \]Step 3: Add the two results: \[ \frac{1}{4} + \frac{2}{5} = \frac{1 \times 5}{4 \times 5} + \frac{2 \times 4}{5 \times 4} = \frac{5}{20} + \frac{8}{20} = \frac{13}{20} \]Answer: \(\frac{13}{20}\)

ii. \(\left(\frac{-14}{3} \times \frac{-12}{7}\right) + \left(\frac{-6}{25} \times \frac{15}{8}\right)\)

Step 1: Multiply the first pair: \[ \frac{-14}{3} \times \frac{-12}{7} = \frac{-14 \times -12}{3 \times 7} = \frac{168}{21} = 8 \]Step 2: Multiply the second pair: \[ \frac{-6}{25} \times \frac{15}{8} = \frac{-6 \times 15}{25 \times 8} = \frac{-90}{200} \] Simplify \(\frac{-90}{200}\) dividing numerator and denominator by 10: \[ \frac{-90 \div 10}{200 \div 10} = \frac{-9}{20} \]Step 3: Add the two results: \[ 8 + \left(\frac{-9}{20}\right) = \frac{8 \times 20}{20} – \frac{9}{20} = \frac{160}{20} – \frac{9}{20} = \frac{151}{20} \]Answer: \(\frac{151}{20}\)

iii. \(\left(\frac{6}{25} \times \frac{-15}{8}\right) – \left(\frac{13}{100} \times \frac{-25}{26}\right)\)

Step 1: Multiply the first pair: \[ \frac{6}{25} \times \frac{-15}{8} = \frac{6 \times -15}{25 \times 8} = \frac{-90}{200} \] Simplify by dividing numerator and denominator by 10: \[ \frac{-9}{20} \]Step 2: Multiply the second pair: \[ \frac{13}{100} \times \frac{-25}{26} = \frac{13 \times -25}{100 \times 26} = \frac{-325}{2600} \] Simplify by dividing numerator and denominator by 25: \[ \frac{-13}{104} \]Step 3: Subtract: \[ \frac{-9}{20} – \left(\frac{-13}{104}\right) = \frac{-9}{20} + \frac{13}{104} \]Find LCM of 20 and 104, which is 520: \[ \frac{-9 \times 26}{520} + \frac{13 \times 5}{520} = \frac{-234}{520} + \frac{65}{520} = \frac{-169}{520} \]Simplify numerator and denominator by 13: \[ \frac{-169 \div 13}{520 \div 13} = \frac{-13}{40} \]Answer: \(\frac{-13}{40}\)

iv. \(\left(\frac{-14}{5} \times \frac{-10}{7}\right) – \left(\frac{-8}{9} \times \frac{3}{16}\right)\)

Step 1: Multiply the first pair: \[ \frac{-14}{5} \times \frac{-10}{7} = \frac{-14 \times -10}{5 \times 7} = \frac{140}{35} = 4 \]Step 2: Multiply the second pair: \[ \frac{-8}{9} \times \frac{3}{16} = \frac{-8 \times 3}{9 \times 16} = \frac{-24}{144} = \frac{-1}{6} \]Step 3: Subtract: \[ 4 – \left(\frac{-1}{6}\right) = 4 + \frac{1}{6} = \frac{24}{6} + \frac{1}{6} = \frac{25}{6} \]Answer: \(\frac{25}{6}\)

Q5: Find the cost of \(3\frac{1}{3}\) kg of rice at ₹\(40\frac{1}{2}\) per kg.

Step 1: Convert mixed fractions into improper fractions.\[ 3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \\ 40\frac{1}{2} = \frac{40 \times 2 + 1}{2} = \frac{81}{2} \]Step 2: Multiply the quantity of rice by the price per kg to find the total cost.\[ \text{Cost} = \frac{10}{3} \times \frac{81}{2} = \frac{10 \times 81}{3 \times 2} = \frac{810}{6} \]Step 3: Simplify the fraction.\[ \frac{810}{6} = 135 \]Answer: ₹135

Q6: Find the distance covered by a car in \(2\frac{2}{5}\) hours at a speed of \(46\frac{2}{3}\) km per hour.

Step 1: Convert the mixed fractions into improper fractions.\[ 2\frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{12}{5} \\ 46\frac{2}{3} = \frac{46 \times 3 + 2}{3} = \frac{140}{3} \]Step 2: Use the formula for distance:\[ \text{Distance} = \text{Speed} \times \text{Time} \]Substitute the values:\[ \text{Distance} = \frac{140}{3} \times \frac{12}{5} = \frac{140 \times 12}{3 \times 5} = \frac{1680}{15} \]Step 3: Simplify the fraction.\[ \frac{1680}{15} = 112 \]Answer: 112 km

Q7: Write the multiplicative inverse of:

i. \(\frac{5}{6}\)

Step 1: The multiplicative inverse of a rational number \(\frac{a}{b}\) (where \(a \neq 0\)) is \(\frac{b}{a}\).\[ \text{Multiplicative inverse of } \frac{5}{6} = \frac{6}{5} \]Answer: \(\frac{6}{5}\)

ii. \(\frac{-3}{7}\)

\[ \text{Multiplicative inverse of } \frac{-3}{7} = \frac{7}{-3} = -\frac{7}{3} \]Answer: \(-\frac{7}{3}\)

iii. \(-8\)

\[ \text{Multiplicative inverse of } -8 = \frac{1}{-8} = -\frac{1}{8} \]Answer: \(-\frac{1}{8}\)

iv. \(\frac{-11}{3}\)

\[ \text{Multiplicative inverse of } \frac{-11}{3} = \frac{3}{-11} = -\frac{3}{11} \]Answer: \(-\frac{3}{11}\)

v. \(\frac{-1}{8}\)

\[ \text{Multiplicative inverse of } \frac{-1}{8} = \frac{8}{-1} = -8 \]Answer: \(-8\)