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Which is greater: a) 2/7 of 3/4 or 3/5 of 5/8? b) 1/2 of 6/7 or 2/3 of 3/7?

Mathematics

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Here are the solutions to the problems.

4. Which is greater:

  • a) 27\frac{2}{7} of 34\frac{3}{4} or 35\frac{3}{5} of 58\frac{5}{8}
  1. Calculate the first product.

    $$
    \frac{2}{7} \text{ of } \frac{3}{4} = \frac{2}{7} \times \frac{3}{4} = \frac{2 \times 3}{7 \times 4} = \frac{6}{28} = \frac{3}{14}
    $$
    
  2. Calculate the second product.

    $$
    \frac{3}{5} \text{ of } \frac{5}{8} = \frac{3}{5} \times \frac{5}{8} = \frac{3 \times 5}{5 \times 8} = \frac{15}{40} = \frac{3}{8}
    $$
    
  3. Compare the two fractions 314\frac{3}{14} and 38\frac{3}{8}.

    To compare, find a common denominator, which is 56. 314=3×414×4=1256 \frac{3}{14} = \frac{3 \times 4}{14 \times 4} = \frac{12}{56} 38=3×78×7=2156 \frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56} Since 2156>1256\frac{21}{56} > \frac{12}{56}, then 38\frac{3}{8} is greater.
    Therefore, 35\frac{3}{5} of 58\frac{5}{8} is greater. 35 of 58 \boxed{\frac{3}{5} \text{ of } \frac{5}{8}}

    • b) 12\frac{1}{2} of 67\frac{6}{7} or 23\frac{2}{3} of 37\frac{3}{7}
  4. Calculate the first product.

    $$
    \frac{1}{2} \text{ of } \frac{6}{7} = \frac{1}{2} \times \frac{6}{7} = \frac{1 \times 6}{2 \times 7} = \frac{6}{14} = \frac{3}{7}
    $$
    
  5. Calculate the second product.

    $$
    \frac{2}{3} \text{ of } \frac{3}{7} = \frac{2}{3} \times \frac{3}{7} = \frac{2 \times 3}{3 \times 7} = \frac{6}{21} = \frac{2}{7}
    $$
    
  6. Compare the two fractions 37\frac{3}{7} and 27\frac{2}{7}.

    Since the denominators are the same, compare the numerators. $3 > 2$.  
    Therefore, $\frac{3}{7}$ is greater.
    $$
    \boxed{\frac{1}{2} \text{ of } \frac{6}{7}}
    $$
    

    5. Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent saplings is 34\frac{3}{4} m. Find the distance between the first and the last sapling.

  7. Determine the number of gaps between the saplings.

    For 4 saplings planted in a row, there are 4−1=34 - 1 = 3 gaps.

  8. Calculate the total distance.

    The distance between the first and the last sapling is the number of gaps multiplied by the distance between adjacent saplings.

    Total distance=3×34 m\text{Total distance} = 3 \times \frac{3}{4} \text{ m}
  9. Perform the multiplication.

    3×34=3×34=94 m3 \times \frac{3}{4} = \frac{3 \times 3}{4} = \frac{9}{4} \text{ m}
  10. Convert the improper fraction to a mixed number.

    94=214 m\frac{9}{4} = 2\frac{1}{4} \text{ m}

    The distance between the first and the last sapling is 214 m\boxed{2\frac{1}{4} \text{ m}}.

    6. Lipika reads a book for 1341\frac{3}{4} hours everyday. She reads the entire book in 6 days. How many hours in all were required by her to read the book?

  11. Convert the daily reading time from a mixed number to an improper fraction.

    134 hours=(1×4)+34 hours=4+34 hours=74 hours1\frac{3}{4} \text{ hours} = \frac{(1 \times 4) + 3}{4} \text{ hours} = \frac{4 + 3}{4} \text{ hours} = \frac{7}{4} \text{ hours}
  12. Calculate the total hours required to read the book.

    Total hours = (Hours read per day) ×\times (Number of days)

    Total hours=74×6\text{Total hours} = \frac{7}{4} \times 6
  13. Perform the multiplication.

    Total hours=7×64=424 hours\text{Total hours} = \frac{7 \times 6}{4} = \frac{42}{4} \text{ hours}
  14. Simplify the fraction and convert to a mixed number.

    424=212 hours=1012 hours\frac{42}{4} = \frac{21}{2} \text{ hours} = 10\frac{1}{2} \text{ hours}

    Lipika required 1012 hours\boxed{10\frac{1}{2} \text{ hours}} to read the book.

    7. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 2342\frac{3}{4} litres of petrol.

  15. Convert the amount of petrol from a mixed number to an improper fraction.

    234 litres=(2×4)+34 litres=8+34 litres=114 litres2\frac{3}{4} \text{ litres} = \frac{(2 \times 4) + 3}{4} \text{ litres} = \frac{8 + 3}{4} \text{ litres} = \frac{11}{4} \text{ litres}
  16. Calculate the total distance covered.

    Distance covered = (Distance per litre) ×\times (Total litres of petrol)

    Distance covered=16 km/litre×114 litres\text{Distance covered} = 16 \text{ km/litre} \times \frac{11}{4} \text{ litres}
  17. Perform the multiplication.

    Distance covered=16×114 km\text{Distance covered} = \frac{16 \times 11}{4} \text{ km}
  18. Simplify the expression.

    Distance covered=4×11 km=44 km\text{Distance covered} = 4 \times 11 \text{ km} = 44 \text{ km}

    The car will cover 44 km\boxed{44 \text{ km}}.

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