Working with Fractions | FIO

Question 14

Evaluate the following:

(a) 3÷793 \div \frac{7}{9}

(b) 144÷2\frac{14}{4} \div 2

(c) 23÷23\frac{2}{3} \div \frac{2}{3}

(d) 146÷73\frac{14}{6} \div \frac{7}{3}

(e) 43÷34\frac{4}{3} \div \frac{3}{4}

(f) 74÷17\frac{7}{4} \div \frac{1}{7}

(g) 82÷415\frac{8}{2} \div \frac{4}{15}

(h) 15÷19\frac{1}{5} \div \frac{1}{9}

(i) 16÷1112\frac{1}{6} \div \frac{11}{12}

(j) 323÷1383\frac{2}{3} \div 1\frac{3}{8}

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Solution

To divide by a fraction, we multiply by its reciprocal.

Step 1 — Evaluate (a)

We want to divide 3 by 79\frac{7}{9}. First, we write the whole number 3 as a fraction. It becomes 31\frac{3}{1}. Then, we find the reciprocal of the second fraction. The reciprocal of 79\frac{7}{9} is 97\frac{9}{7}. Now, we multiply the first fraction by this reciprocal.

3÷793 \div \frac{7}{9}

=31×97= \frac{3}{1} \times \frac{9}{7}

=3×91×7= \frac{3 \times 9}{1 \times 7}

=277= \frac{27}{7}

277\boxed{\frac{27}{7}}

Step 2 — Evaluate (b)

We want to divide 144\frac{14}{4} by 2. First, we write the whole number 2 as a fraction. It becomes 21\frac{2}{1}. Then, we find the reciprocal of the second fraction. The reciprocal of 21\frac{2}{1} is 12\frac{1}{2}. Now, we multiply the first fraction by this reciprocal.

144÷2\frac{14}{4} \div 2

=144×12= \frac{14}{4} \times \frac{1}{2}

=14×14×2= \frac{14 \times 1}{4 \times 2}

=148= \frac{14}{8}

We can simplify this fraction. We divide both the numerator and denominator by 2.

=14÷28÷2= \frac{14 \div 2}{8 \div 2}

=74= \frac{7}{4}

74\boxed{\frac{7}{4}}

Step 3 — Evaluate (c)

We want to divide 23\frac{2}{3} by 23\frac{2}{3}. First, we find the reciprocal of the second fraction. The reciprocal of 23\frac{2}{3} is 32\frac{3}{2}. Now, we multiply the first fraction by this reciprocal.

23÷23\frac{2}{3} \div \frac{2}{3}

=23×32= \frac{2}{3} \times \frac{3}{2}

=2×33×2= \frac{2 \times 3}{3 \times 2}

=66= \frac{6}{6}

=1= 1

1\boxed{1}

Step 4 — Evaluate (d)

We want to divide 146\frac{14}{6} by 73\frac{7}{3}. First, we find the reciprocal of the second fraction. The reciprocal of 73\frac{7}{3} is 37\frac{3}{7}. Now, we multiply the first fraction by this reciprocal.

146÷73\frac{14}{6} \div \frac{7}{3}

=146×37= \frac{14}{6} \times \frac{3}{7}

=14×36×7= \frac{14 \times 3}{6 \times 7}

=4242= \frac{42}{42}

=1= 1

1\boxed{1}

Step 5 — Evaluate (e)

We want to divide 43\frac{4}{3} by 34\frac{3}{4}. First, we find the reciprocal of the second fraction. The reciprocal of 34\frac{3}{4} is 43\frac{4}{3}. Now, we multiply the first fraction by this reciprocal.

43÷34\frac{4}{3} \div \frac{3}{4}

=43×43= \frac{4}{3} \times \frac{4}{3}

=4×43×3= \frac{4 \times 4}{3 \times 3}

=169= \frac{16}{9}

169\boxed{\frac{16}{9}}

Step 6 — Evaluate (f)

We want to divide 74\frac{7}{4} by 17\frac{1}{7}. First, we find the reciprocal of the second fraction. The reciprocal of 17\frac{1}{7} is 71\frac{7}{1}. Now, we multiply the first fraction by this reciprocal.

74÷17\frac{7}{4} \div \frac{1}{7}

=74×71= \frac{7}{4} \times \frac{7}{1}

=7×74×1= \frac{7 \times 7}{4 \times 1}

=494= \frac{49}{4}

494\boxed{\frac{49}{4}}

Step 7 — Evaluate (g)

We want to divide 82\frac{8}{2} by 415\frac{4}{15}. First, we can simplify the first fraction 82\frac{8}{2}. 82\frac{8}{2} is equal to 4. So, the problem becomes 4÷4154 \div \frac{4}{15}. We write the whole number 4 as a fraction. It becomes 41\frac{4}{1}. Then, we find the reciprocal of the second fraction. The reciprocal of 415\frac{4}{15} is 154\frac{15}{4}. Now, we multiply the first fraction by this reciprocal.

82÷415\frac{8}{2} \div \frac{4}{15}

=4÷415= 4 \div \frac{4}{15}

=41×154= \frac{4}{1} \times \frac{15}{4}

=4×151×4= \frac{4 \times 15}{1 \times 4}

=604= \frac{60}{4}

=15= 15

15\boxed{15}

Step 8 — Evaluate (h)

We want to divide 15\frac{1}{5} by 19\frac{1}{9}. First, we find the reciprocal of the second fraction. The reciprocal of 19\frac{1}{9} is 91\frac{9}{1}. Now, we multiply the first fraction by this reciprocal.

15÷19\frac{1}{5} \div \frac{1}{9}

=15×91= \frac{1}{5} \times \frac{9}{1}

=1×95×1= \frac{1 \times 9}{5 \times 1}

=95= \frac{9}{5}

95\boxed{\frac{9}{5}}

Step 9 — Evaluate (i)

We want to divide 16\frac{1}{6} by 1112\frac{11}{12}. First, we find the reciprocal of the second fraction. The reciprocal of 1112\frac{11}{12} is 1211\frac{12}{11}. Now, we multiply the first fraction by this reciprocal.

16÷1112\frac{1}{6} \div \frac{11}{12}

=16×1211= \frac{1}{6} \times \frac{12}{11}

=1×126×11= \frac{1 \times 12}{6 \times 11}

=1266= \frac{12}{66}

We can simplify this fraction. We divide both the numerator and denominator by 6.

=12÷666÷6= \frac{12 \div 6}{66 \div 6}

=211= \frac{2}{11}

211\boxed{\frac{2}{11}}

Step 10 — Evaluate (j)

We want to divide 3233\frac{2}{3} by 1381\frac{3}{8}. First, we convert both mixed numbers to improper fractions. For 3233\frac{2}{3}: Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

323=(3×3)+23=9+23=1133\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}

For 1381\frac{3}{8}: Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

138=(1×8)+38=8+38=1181\frac{3}{8} = \frac{(1 \times 8) + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8}

Now, the problem is 113÷118\frac{11}{3} \div \frac{11}{8}. We find the reciprocal of the second fraction. The reciprocal of 118\frac{11}{8} is 811\frac{8}{11}. Now, we multiply the first fraction by this reciprocal.

323÷1383\frac{2}{3} \div 1\frac{3}{8}

=113÷118= \frac{11}{3} \div \frac{11}{8}

=113×811= \frac{11}{3} \times \frac{8}{11}

=11×83×11= \frac{11 \times 8}{3 \times 11}

We can cancel out the common factor of 11.

=83= \frac{8}{3}

83\boxed{\frac{8}{3}}

Answer

(a) 277\frac{27}{7} (b) 74\frac{7}{4} (c) 11 (d) 11 (e) 169\frac{16}{9} (f) 494\frac{49}{4} (g) 1515 (h) 95\frac{9}{5} (i) 211\frac{2}{11} (j) 83\frac{8}{3}

More questions in FIO

Q1

Tenzin drinks 12\frac{1}{2} glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

Q2

A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make ___ km of the water canal. If they work 5 days a week, they can make ___ km of the water canal in a week.

Q3

Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

Q4

Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets 56\frac{5}{6} hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

Q5

Multiply and then convert it into a mixed fraction:

(a) 7×357 \times \frac{3}{5}

(b) 4×134 \times \frac{1}{3}

(c) 97×6\frac{9}{7} \times 6

(d) 1311×6\frac{13}{11} \times 6

Q6

Find the following products. Use a unit square as a whole for representing the fractions:

(a) 13×15\frac{1}{3} \times \frac{1}{5}

(b) 14×13\frac{1}{4} \times \frac{1}{3}

(c) 15×12\frac{1}{5} \times \frac{1}{2}

(d) 16×15\frac{1}{6} \times \frac{1}{5}

Now, find 112×118\frac{1}{12} \times \frac{1}{18}.

Q7

Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations.

(a) 23×45\frac{2}{3} \times \frac{4}{5}

(b) 14×23\frac{1}{4} \times \frac{2}{3}

(c) 35×12\frac{3}{5} \times \frac{1}{2}

(d) 46×35\frac{4}{6} \times \frac{3}{5}

Q8

A water tank is filled from a tap. If the tap is open for 1 hour, 710\frac{7}{10} of the tank gets filled. How much of the tank is filled if the tap is open for

(a) 13\frac{1}{3} hour

(b) 23\frac{2}{3} hour

(c) 34\frac{3}{4} hour

(d) 710\frac{7}{10} hour

(e) For the tank to be full, how long should the tap be running?

Q9

The government has taken 16\frac{1}{6} of Somu's land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and 13\frac{1}{3} of it to her son Bora. After giving them their shares, she keeps the remaining land for herself.

(a) What part of the original land did Krishna get?

(b) What part of the original land did Bora get?

(c) What part of the original land did Somu keep for herself?

Q10

Find the area of a rectangle of sides 334 ft3\frac{3}{4}\text{ ft} and 935 ft9\frac{3}{5}\text{ ft}.

Q11

Tsewang plants four saplings in a row in his garden. The distance between two saplings is 34 m\frac{3}{4}\text{ m}. Find the distance between the first and last sapling. [Hint: Draw a rough diagram with four saplings with distance between two saplings as 34 m\frac{3}{4}\text{ m}]

Q12

Which is heavier: 1215\frac{12}{15} of 500 grams or 320\frac{3}{20} of 4 kg?

Q13

What can you conclude about the relationship between the numbers multiplied and the product? Fill in the blanks:

  • When one of the numbers being multiplied is between 0 and 1, the product is ________ (greater/less) than the other number.
  • When one of the numbers being multiplied is greater than 1, the product is ________ (greater/less) than the other number.
Q14

Evaluate the following:

(a) 3÷793 \div \frac{7}{9}

(b) 144÷2\frac{14}{4} \div 2

(c) 23÷23\frac{2}{3} \div \frac{2}{3}

(d) 146÷73\frac{14}{6} \div \frac{7}{3}

(e) 43÷34\frac{4}{3} \div \frac{3}{4}

(f) 74÷17\frac{7}{4} \div \frac{1}{7}

(g) 82÷415\frac{8}{2} \div \frac{4}{15}

(h) 15÷19\frac{1}{5} \div \frac{1}{9}

(i) 16÷1112\frac{1}{6} \div \frac{11}{12}

(j) 323÷1383\frac{2}{3} \div 1\frac{3}{8}

Q15

For each of the questions below, choose the expression that describes the solution. Then simplify it.

(a) Maria bought 8 m8\text{ m} of lace to decorate the bags she made for school. She used 14 m\frac{1}{4}\text{ m} for each bag and finished the lace. How many bags did she decorate?

(i) 8×148 \times \frac{1}{4}

(ii) 18×14\frac{1}{8} \times \frac{1}{4}

(iii) 8÷148 \div \frac{1}{4}

(iv) 14÷8\frac{1}{4} \div 8

(b) 12\frac{1}{2} meter of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge?

(i) 8×128 \times \frac{1}{2}

(ii) 12⋅18\frac{1}{2} \cdot \frac{1}{8}

(iii) 8÷128 \div \frac{1}{2}

(iv) 12÷8\frac{1}{2} \div 8

(c) A baker needs 16 kg\frac{1}{6}\text{ kg} of flour to make one loaf of bread. He has 5 kg5\text{ kg} of flour. How many loaves of bread can he make?

(i) 5×165 \times \frac{1}{6}

(ii) 16÷5\frac{1}{6} \div 5

(iii) 5÷165 \div \frac{1}{6}

(iv) 5×65 \times 6

Q16

If 14\frac{1}{4} kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?

Q17

Pāṭīgaṇita, a book written by Sridharacharya in the 9th century CE, mentions this problem: "Friend, after thinking, what sum will be obtained by adding together 1÷161 \div \frac{1}{6}, 1÷1101 \div \frac{1}{10}, 1÷1131 \div \frac{1}{13}, 1÷191 \div \frac{1}{9}, and 1÷121 \div \frac{1}{2}". What should the friend say?

Q18

Mira is reading a novel that has 400 pages. She read 15\frac{1}{5} of the pages yesterday and 310\frac{3}{10} of the pages today. How many more pages does she need to read to finish the novel?

Q19

A car runs 16 km using 1 litre of petrol. How far will it go using 2342 \frac{3}{4} litres of petrol?

Q20

Amritpal decides on a destination for his vacation. If he takes a train, it will take him 5165 \frac{1}{6} hours to get there. If he takes a plane, it will take him 12\frac{1}{2} hour. How many hours does the plane save?

Q21

Mariam's grandmother baked a cake. Mariam and her cousins finished 45\frac{4}{5} of the cake. The remaining cake was shared equally by Mariam's three friends. How much of the cake did each friend get?

Q22

Choose the option(s) describing the product of (565465×707676)\left(\frac{565}{465} \times \frac{707}{676}\right):

(a) >565465> \frac{565}{465}

(b) <565465< \frac{565}{465}

(c) >707676> \frac{707}{676}

(d) <707676< \frac{707}{676}

(e) >1> 1

(f) <1< 1

Q23

What fraction of the whole square is shaded?

Q24

A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig. 8.7) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?

Q25

What is 1−121 - \frac{1}{2}?

(1−12)×(1−13)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right)?

(1−12)×(1−13)×(1−14)×(1−15)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right)?

(1−12)×(1−13)×(1−14)×(1−15)×(1−16)×(1−17)×(1−18)×(1−19)×(1−110)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right) \times \left(1 - \frac{1}{6}\right) \times \left(1 - \frac{1}{7}\right) \times \left(1 - \frac{1}{8}\right) \times \left(1 - \frac{1}{9}\right) \times \left(1 - \frac{1}{10}\right)?

Make a general statement and explain.

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