Fractions

Find the product:

Q1: \(\frac{5}{6} \times \frac{3}{7}\)

Step 1:
Write the multiplication of two fractions: \[ \frac{5}{6} \times \frac{3}{7} \] Step 2:
Multiply the numerators and the denominators: \[ = \frac{5\times3}{6\times7} = \frac{15}{42} \] Step 3:
Simplify the fraction by dividing numerator and denominator by 3: \[ = \frac{15\div3}{42\div3} = \frac{5}{14} \] Answer: \(\frac{5}{14}\)

Q2: \(\frac{7}{18} \times \frac{9}{14}\)

Step 1:
Write the multiplication of two fractions: \[ \frac{7}{18} \times \frac{9}{14} \] Step 2:
Multiply the numerators and the denominators: \[ = \frac{7\times9}{18\times14} = \frac{63}{252} \] Step 3:
Simplify the fraction by dividing numerator and denominator by 9: \[ = \frac{63\div9}{252\div9} = \frac{7}{28} \]Step 4:
Simplify further by dividing by 7: \[ = \frac{7\div7}{28\div7} = \frac{1}{4} \]Answer: \(\frac{1}{4}\)

Q3: \(28 \times \frac{7}{8}\)

Step 1:
Write the multiplication: \[ 28 \times \frac{7}{8} \]Step 2:
Convert 28 into fraction form: \[ = \frac{28}{1} \times \frac{7}{8} \]Step 3:
Multiply the numerators and denominators: \[ = \frac{28\times7}{1\times8} = \frac{196}{8} \]Step 4:
Simplify by dividing numerator and denominator by 4: \[ = \frac{196\div4}{8\div4} = \frac{49}{2} \]Step 5:
Convert into mixed fraction: \[ = 24\frac{1}{2} \]Answer: \(24\frac{1}{2}\)

Q4: \(7 \times \frac{1}{7}\)

Step 1:
Write the multiplication: \[ 7 \times \frac{1}{7} \]Step 2:
Multiply numerator and denominator: \[ = \frac{7\times1}{7} = \frac{7}{7} \]Step 3:
Simplify: \[ = 1 \]Answer: \(1\)

Q5: \(2\frac{1}{25} \times \frac{5}{17}\)

Step 1:
Convert mixed fraction to improper fraction: \[ 2\frac{1}{25} = \frac{(2\times25)+1}{25} = \frac{51}{25} \]Step 2:
Multiply the two fractions: \[ \frac{51}{25} \times \frac{5}{17} \]Step 3:
Multiply numerators and denominators: \[ = \frac{51\times5}{25\times17} = \frac{255}{425} \]Step 4:
Simplify the fraction by dividing by 85: \[ = \frac{255\div85}{425\div85} = \frac{3}{5} \]Answer: \(\frac{3}{5}\)

Q6: \(1\frac{1}{13} \times 7\frac{3}{7}\)

Step 1:
Convert mixed fractions to improper fractions: \[ 1\frac{1}{13} = \frac{14}{13}, \quad 7\frac{3}{7} = \frac{52}{7} \]Step 2:
Multiply the two fractions: \[ \frac{14}{13} \times \frac{52}{7} \]Step 3:
Multiply numerators and denominators: \[ = \frac{14\times52}{13\times7} = \frac{728}{91} \]Step 4:
Convert into mixed fraction:
Divide 728 by 91: \[ 728 \div 91 = 8 \text{ remainder } 0 \] So, \[ = 8 \]Answer: \(8\)

Q7: \(\frac{4}{17} \times 7\frac{1}{12}\)

Step 1:
First, convert the mixed number into an improper fraction: \[ 7\frac{1}{12} = \frac{7\times12 + 1}{12} = \frac{85}{12} \]Step 2:
Now, multiply the two fractions: \[ \frac{4}{17} \times \frac{85}{12} \]Step 3:
Multiply the numerators and denominators: \[ = \frac{4\times85}{17\times12} = \frac{340}{204} \]Step 4:
Simplify the fraction:
Divide numerator and denominator by 68: \[ = \frac{5}{3} \]Step 5:
Convert to mixed fraction if needed: \[ \frac{5}{3} = 1\frac{2}{3} \]Answer: \(1\frac{2}{3}\)

Q8: \(7\frac{1}{4} \times \frac{7}{58} \times 1\frac{11}{21}\)

Step 1:
Convert mixed fractions to improper fractions: \[ 7\frac{1}{4} = \frac{29}{4}, \quad 1\frac{11}{21} = \frac{32}{21} \]Step 2:
Multiply all three fractions: \[ \frac{29}{4} \times \frac{7}{58} \times \frac{32}{21} \]Step 3:
Simplify before multiplying:
\(29\) and \(58\) have common factor 29: \[ = \frac{1}{2} \] Thus, \[ \frac{1}{4} \times \frac{7}{2} \times \frac{32}{21} \]Step 4:
Multiply numerators and denominators: \[ = \frac{1\times7\times32}{4\times2\times21} = \frac{224}{168} \]Step 5:
Simplify by dividing by 56: \[ = \frac{224\div56}{168\div56} = \frac{4}{3} \]Step 6:
Convert into mixed fraction: \[ = 1\frac{1}{3} \]Answer: \(1\frac{1}{3}\)

Q9: \(\frac{1}{19} \times 91 \times 5\frac{11}{13}\)

Step 1:
Convert mixed fraction to improper fraction: \[ 5\frac{11}{13} = \frac{76}{13} \]Step 2:
Multiply all numbers:
First simplify \(\frac{1}{19} \times 91\): \[ \frac{1}{19} \times 91 = \frac{91}{19} = 4\frac{15}{19} \]Step 3:
Now multiply: \[ 4\frac{15}{19} \times \frac{76}{13} \]Convert \(4\frac{15}{19}\) into improper fraction: \[ = \frac{(4\times19)+15}{19} = \frac{91}{19} \]Step 4:
Thus, \[ \frac{91}{19} \times \frac{76}{13} = \frac{91\times76}{19\times13} = \frac{6916}{247} \]Step 5:
Convert to mixed fraction: \[ 6916 \div 247 = 28 \text{ remainder } 0 \] Thus, \[ = 28 \]Answer: \(28\)

Q10: \(7\frac{1}{4} \times 2\frac{3}{16} \times 2\frac{2}{7}\)

Step 1:
Convert mixed fractions to improper fractions: \[ 7\frac{1}{4} = \frac{29}{4}, \quad 2\frac{3}{16} = \frac{35}{16}, \quad 2\frac{2}{7} = \frac{16}{7} \]Step 2:
Multiply all three fractions: \[ \frac{29}{4} \times \frac{35}{16} \times \frac{16}{7} \]Step 3:
Simplify before multiplying:
\(16\) cancels with \(16\): \[ = \frac{29}{4} \times \frac{35}{7} \]Now, \(35\div7 = 5\): \[ = \frac{29}{4} \times 5 = \frac{145}{4} \]Step 4:
Convert into mixed fraction: \[ = 36\frac{1}{4} \]Answer: \(36\frac{1}{4}\)

Q11: Find the value of:

(i) \(\frac{3}{4}\) of \(\frac{8}{9}\) \[ = \frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3} \]Answer = \(\frac{2}{3}\)

(ii) \(\frac{1}{2}\) of \(2\frac{2}{3}\)
Convert \(2\frac{2}{3} = \frac{8}{3}\) \[ = \frac{1}{2} \times \frac{8}{3} = \frac{8}{6} = \frac{4}{3} = 1\frac{1}{3} \]Answer = \(1\frac{1}{3}\)

(iii) \(\frac{4}{5}\) of 1 hour
\[ = \frac{4}{5} \times 60\ \text{minutes} = 48\ \text{minutes} \]Answer = \(48\) minutes

(iv) \(\frac{3}{5}\) of ₹1
\[ = \frac{3}{5} \times 1 = ₹0.60 \]Answer = ₹0.60

(v) \(\frac{8}{15}\) of \(1\frac{1}{2}\) metres
Convert \(1\frac{1}{2} = \frac{3}{2}\) \[ = \frac{8}{15} \times \frac{3}{2} = \frac{24}{30} = \frac{4}{5} \]Answer = \(\frac{4}{5}\) metres = 80 cm

(vi) \(\frac{5}{7}\) of \(2\frac{1}{3}\) kg
Convert \(2\frac{1}{3} = \frac{7}{3}\) \[ = \frac{5}{7} \times \frac{7}{3} = \frac{5}{3} = 1\frac{2}{3} \]Answer = \(1\frac{2}{3}\) kg

Q12: A car can travel \(12\frac{1}{2}\) km in 1 litre of petrol. How much distance can it travel in \(42\frac{3}{5}\) litres of petrol?

Step 1:
Convert mixed numbers into improper fractions: \[ 12\frac{1}{2} = \frac{25}{2}, \quad 42\frac{3}{5} = \frac{213}{5} \]Step 2:
Multiply distance per litre by the number of litres: \[ \text{Distance} = \frac{25}{2} \times \frac{213}{5} \]Step 3:
Multiply numerators and denominators: \[ = \frac{25\times213}{2\times5} = \frac{5325}{10} \]Step 4:
Simplify: \[ = 532.5\ \text{km} \]Answer: \(532.5\) km

Q13: A graphic designer charges ₹\(27\frac{3}{5}\) for each diagram. Pind the amount he will charge if he designs 186 diagrams for a book.

Step 1:
Convert mixed fraction to improper fraction: \[ 27\frac{3}{5} = \frac{138}{5} \]Step 2:
Multiply charge per diagram with number of diagrams: \[ \text{Total Amount} = \frac{138}{5} \times 186 \]Step 3:
Multiply: \[ = \frac{138\times186}{5} = \frac{25668}{5} \]Step 4:
Divide: \[ = 5133.6 = 5133\frac{3}{5} \]Answer: ₹5133.60 or ₹\(5133\frac{3}{5}\)

Q14: If a cloth costs ₹\(715\frac{1}{4}\) per metre, find the cost of \(3\frac{2}{5}\) metres of this cloth.

Step 1:
Convert mixed numbers into improper fractions: \[ 715\frac{1}{4} = \frac{2861}{4}, \quad 3\frac{2}{5} = \frac{17}{5} \]Step 2:
Multiply cost per metre by number of metres: \[ \text{Total Cost} = \frac{2861}{4} \times \frac{17}{5} \]Step 3:
Multiply numerators and denominators: \[ = \frac{2861\times17}{4\times5} = \frac{48637}{20} \]Step 4:
Simplify: \[ = 2431\frac{17}{20} = 2431.85 \]Answer: ₹\(2431\frac{17}{20}\) or ₹2431.85

Q15: Advertising in a magazine costs ₹\(1472\frac{2}{5}\) per square inch. Find the cost of an advertisement of \(17\frac{6}{7}\) square inch.

Step 1:
Convert mixed numbers into improper fractions: \[ 1472\frac{2}{5} = \frac{7362}{5}, \quad 17\frac{6}{7} = \frac{125}{7} \]Step 2:
Multiply cost per square inch with area in square inches: \[ \text{Total Cost} = \frac{7362}{5} \times \frac{125}{7} \]Step 3:
Multiply numerators and denominators: \[ = \frac{7362\times125}{5\times7} = \frac{920250}{35} \]Step 4:
Simplify: \[ = 26292\frac{6}{7} = 26292.8571 \]Answer: ₹\(26292\frac{6}{7}\) or ₹26292.86

Q16: A car from city A to city B with a uniform speed of \(52\frac{2}{7}\) km per hour. Find the distance between the two cities, if it took \(4\frac{3}{8}\) hours for the car to reach city B from city A.

Step 1:
Convert mixed numbers into improper fractions: \[ 52\frac{2}{7} = \frac{366}{7}, \quad 4\frac{3}{8} = \frac{35}{8} \]Step 2:
Multiply speed and time: \[ \text{Distance} = \frac{366}{7} \times \frac{35}{8} \]Step 3:
Simplify before multiplying: \[ \frac{35}{7} = 5,\quad \text{thus} \] \[ \text{Distance} = \frac{366\times5}{8} = \frac{1830}{8} \]Step 4:
Simplify: \[ = 228\frac{3}{4} = 228.75\ \text{km} \]Answer: \(228\frac{3}{4}\) km or 228.75 km

Q17: The length of a rectangular plot of land is \(29\frac{3}{7}\) m. If its breadth is \(12\frac{8}{11}\) m, find its area.

Step 1:
Convert mixed numbers into improper fractions: \[ 29\frac{3}{7} = \frac{206}{7}, \quad 12\frac{8}{11} = \frac{140}{11} \]Step 2:
Area of rectangle = Length × Breadth: \[ \text{Area} = \frac{206}{7} \times \frac{140}{11} \]Step 3:
Multiply numerators and denominators: \[ = \frac{206\times140}{7\times11} = \frac{28840}{77} \]Step 4:
Simplify: \[ = 374\frac{6}{11} = 374.4156\ \text{m}^2 \]Answer: = \(374\frac{6}{11}\ m^2\) or \(374.42\ m^2\)