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: $\frac{5}{14}$

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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: $\frac{1}{4}$

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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: $24\frac{1}{2}$

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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: $1$

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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: $\frac{3}{5}$

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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: $8$

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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: $1\frac{2}{3}$

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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: $1\frac{1}{3}$

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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: $28$

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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: $36\frac{1}{4}$

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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

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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

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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 ₹$5133\frac{3}{5}$

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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

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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

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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

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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 $374.42\ m^2$

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