Word Problems With Fractions

Word problems involvong fractions are presented along with their detailed solutions.

problems


  1. A farmer wants to sell his corn harvest to 3 buyers. The first would buy 2/7 of it, the second would buy 2/5 of it, and the third would buy one-third of it. Can he satisfy all the buyers?

  2. To make a rabbit pate, half of the rabbit meat's weight is mixed with chicken liver, and a quarter of its weight is mixed with veal. Calculate the ratio of the rabbit meat's weight to the weight of the pate.

  3. Three people buy a barrel of juics together. The first takes 2/5, and the second takes 3/10. What will be the share of the third person?

  4. Rapeseed contains about 48% of its weight in oil, but only 2/3 of this oil can be extracted by pressing. What FRACTION of the total weight of the rapeseed does the extracted oil represent?

  5. 2/9 of a post is painted white, and 4/7 of the remaining part is painted red. What FRACTION of the post is left unpainted?

  6. A farmer takes three days to plow his field. The first day, he plows 1/3 of it; the second day, he plows 3/8 of the remainder. What FRACTION of the field's area remains to be plowed?

  7. A wood merchant first sells one-third of his merchandise. Then, he sells 2/5 of the remaining amount. What FRACTION of his merchandise does he have left?

  8. A plane makes a 6.5-hour journey. What FRACTION of the journey has it completed after 1.5 hours of flight?

  9. What FRACTION of the day has passed when it is 8 o'clock in the morning?

Solutions

Let's work through each problem step by step.


  1. Can the farmer satisfy all the buyers?
    The farmer wants to sell his corn harvest to three buyers:
    The first buyer would purchase \( \dfrac{2}{7} \) of the harvest.
    The second buyer would purchase \( \dfrac{2}{5} \) of the harvest.
    The third buyer would purchase \( \dfrac{1}{3} \) of the harvest.
    To determine if he can satisfy all buyers, we need to find out if the sum of these fractions is less than or equal to 1 (the total harvest).
    First, let's add the fractions. To do so, we'll find a common denominator.
    The common denominator for 7, 5, and 3 is 105.
    Convert each fraction to have this common denominator:

    \( \dfrac{2}{7} = \dfrac{2 \times 15}{7 \times 15} = \dfrac{30}{105} \)
    \( \dfrac{2}{5} = \dfrac{2 \times 21}{5 \times 21} = \dfrac{42}{105} \)
    \( \dfrac{1}{3} = \dfrac{1 \times 35}{3 \times 35} = \dfrac{35}{105} \)
    Now, add the fractions: \[ \dfrac{30}{105} + \dfrac{42}{105} + \dfrac{35}{105} = \dfrac{107}{105} \] Since \( \dfrac{107}{105} \) is greater than 1, the farmer cannot satisfy all the buyers.


  2. To make a rabbit pate:
    Half of the rabbit meat's weight is mixed with chicken liver.
    A quarter of the rabbit meat's weight is mixed with veal.
    Let the weight of the rabbit meat be \( x \).
    The total weight of the pate: \[ \text{Total weight} = x + \dfrac{1}{2}x + \dfrac{1}{4}x \] Add the terms: \[ \text{Total weight} = x\left(1 + \dfrac{1}{2} + \dfrac{1}{4}\right) = x\left(\dfrac{4}{4} + \dfrac{2}{4} + \dfrac{1}{4}\right) = x\left(\dfrac{7}{4}\right) \] The ratio of rabbit meat to the total weight of the pate: \[ \text{Ratio} = \dfrac{x}{\dfrac{7}{4}x} = \dfrac{4}{7} \] So, the ratio of the rabbit meat's weight to the weight of the pate is \( 4:7 \).

  3. Three people buy a barrel of juics together:
    The first person takes \( \dfrac{2}{5} \).
    The second person takes \( \dfrac{3}{10} \).
    We need to find the share of the third person.
    First, let's find the combined share of the first two people: \[ \dfrac{2}{5} + \dfrac{3}{10} \] Convert \( \dfrac{2}{5} \) to have a denominator of 10: \[ \dfrac{2}{5} = \dfrac{4}{10} \] Now, add: \[ \dfrac{4}{10} + \dfrac{3}{10} = \dfrac{7}{10} \] The remaining share for the third person: \[ 1 - \dfrac{7}{10} = \dfrac{10}{10} - \dfrac{7}{10} = \dfrac{3}{10} \] So, the third person gets \( \dfrac{3}{10} \) of the barrel.

  4. Fraction of total weight of rapeseed oil obtained after pressing Rapeseed contains 48% oil by weight, but only \( \dfrac{2}{3} \) of this oil can be extracted. The fraction of the total weight of the rapeseed represented by the extracted oil: \[ \text{Fraction} = 48\% \times \dfrac{2}{3} = \dfrac{48}{100} \times \dfrac{2}{3} = \dfrac{48 \times 2}{100 \times 3} = \dfrac{96}{300} = \dfrac{32}{100} = 0.32 \] So, the extracted oil represents \( \dfrac{32}{100} \) or \(32\% \) of the total weight of the rapeseed.

  5. \( \dfrac{2}{9} \) of the post is painted white.
    \( \dfrac{4}{7} \) of the remaining part is painted red.
    First, find the remaining part after the white paint: \[ \text{Remaining after white paint} = 1 - \dfrac{2}{9} = \dfrac{9}{9} - \dfrac{2}{9} = \dfrac{7}{9} \] Now, find the part painted red: \[ \text{Red painted part} = \dfrac{4}{7} \times \dfrac{7}{9} = \dfrac{28}{63} = \dfrac{4}{9} \] The unpainted part: \[ \text{Unpainted part} = \dfrac{7}{9} - \dfrac{4}{9} = \dfrac{3}{9} = \dfrac{1}{3} \] So, \( \dfrac{1}{3} \) of the post is left unpainted.

  6. Fraction of the field left to plow - The farmer plows \( \dfrac{1}{3} \) of the field on the first day. - On the second day, he plows \( \dfrac{3}{8} \) of the remaining part. First, calculate the remaining part after the first day: \[ \text{Remaining after the first day} = 1 - \dfrac{1}{3} = \dfrac{3}{3} - \dfrac{1}{3} = \dfrac{2}{3} \] Now, calculate the part plowed on the second day: \[ \text{Second day's plowed part} = \dfrac{3}{8} \times \dfrac{2}{3} = \dfrac{6}{24} = \dfrac{1}{4} \] Finally, calculate the remaining part of the field: \[ \text{Remaining part} = \dfrac{2}{3} - \dfrac{1}{4} \] Find a common denominator (12): \[ \dfrac{2}{3} = \dfrac{8}{12}, \quad \dfrac{1}{4} = \dfrac{3}{12} \] Subtract: \[ \text{Remaining part} = \dfrac{8}{12} - \dfrac{3}{12} = \dfrac{5}{12} \] So, \( \dfrac{5}{12} \) of the field is left to plow.

  7. Fraction of the merchant's merchandise left - The merchant first sells \( \dfrac{1}{3} \) of his merchandise. - Then he sells \( \dfrac{2}{5} \) of the remaining part. First, calculate the remaining part after the first sale: \[ \text{Remaining after first sale} = 1 - \dfrac{1}{3} = \dfrac{2}{3} \] Now, calculate the part sold in the second sale: \[ \text{Second sale part} = \dfrac{2}{5} \times \dfrac{2}{3} = \dfrac{4}{15} \] Finally, calculate the remaining part of the merchandise: \[ \text{Remaining part} = \dfrac{2}{3} - \dfrac{4}{15} \] Find a common denominator (15): \[ \dfrac{2}{3} = \dfrac{10}{15} \] Subtract: \[ \text{Remaining part} = \dfrac{10}{15} - \dfrac{4}{15} = \dfrac{6}{15} = \dfrac{2}{5} \] So, \( \dfrac{2}{5} \) of the merchandise is left.

  8. Fraction of the journey completed - The total journey time is 6.5 hours. - The plane has flown for 1.5 hours. The fraction of the journey completed: \[ \text{Fraction} = \dfrac{1.5}{6.5} = \dfrac{15}{65} = \dfrac{3}{13} \] So, the plane has completed \( \dfrac{3}{13} \) of the journey.

  9. Fraction of the day elapsed by 8 AM - A day has 24 hours. - 8 hours have passed by 8 AM. The fraction of the day that has passed: \[ \text{Fraction} = \dfrac{8}{24} = \dfrac{1}{3} \] So, \( \dfrac{1}{3} \) of the day has passed by 8 AM.



More References and links

  1. Fractions Questions and Problems with Solutions .
  2. Fractions and Mixed Numbers- Grade 7 Maths Questions and Problems With Answers .
  3. Fractions and Mixed Numbers- Grade 6 Math Questions and Problems With Answers .
  4. Fractions - Grade 5 Maths Questions With Solutions .
  5. Fractions - Grade 4 Maths Questions With Answers .