Working with Fractions

Chess Puzzles — Non-attacking Queens

Chess is a popular 2-player strategy game. This game has its origins in India. It is played on an $8 \times 8$ chequered grid. There are 2 sets of pieces — black and white — one set for each player. Find out how each piece should move and the rules of the game.

Here is a famous chess-based puzzle. From its current position, a Queen piece can move along the horizontal, vertical or diagonal. Place 4 Queens such that no 2 queens attack each other. For example, the arrangement below is not valid as the queens are in the line of attack of each other.

Now, place 8 queens on this $8 \times 8$ grid so that no 2 queens attack each other!

Tenzin drinks $\frac{1}{2}$ 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?

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.

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?

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

Multiply and then convert it into a mixed fraction:

(a) $7 \times \frac{3}{5}$

(b) $4 \times \frac{1}{3}$

(c) $\frac{9}{7} \times 6$

(d) $\frac{13}{11} \times 6$

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

(a) $\frac{1}{3} \times \frac{1}{5}$

(b) $\frac{1}{4} \times \frac{1}{3}$

(c) $\frac{1}{5} \times \frac{1}{2}$

(d) $\frac{1}{6} \times \frac{1}{5}$

Now, find $\frac{1}{12} \times \frac{1}{18}$.

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

(a) $\frac{2}{3} \times \frac{4}{5}$

(b) $\frac{1}{4} \times \frac{2}{3}$

(c) $\frac{3}{5} \times \frac{1}{2}$

(d) $\frac{4}{6} \times \frac{3}{5}$

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

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

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

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

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

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

The government has taken $\frac{1}{6}$ 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 $\frac{1}{3}$ 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?

Find the area of a rectangle of sides $3\frac{3}{4}\text{ ft}$ and $9\frac{3}{5}\text{ ft}$.

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

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

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.

Evaluate the following:

(a) $3 \div \frac{7}{9}$

(b) $\frac{14}{4} \div 2$

(c) $\frac{2}{3} \div \frac{2}{3}$

(d) $\frac{14}{6} \div \frac{7}{3}$

(e) $\frac{4}{3} \div \frac{3}{4}$

(f) $\frac{7}{4} \div \frac{1}{7}$

(g) $\frac{8}{2} \div \frac{4}{15}$

(h) $\frac{1}{5} \div \frac{1}{9}$

(i) $\frac{1}{6} \div \frac{11}{12}$

(j) $3\frac{2}{3} \div 1\frac{3}{8}$

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

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

(i) $8 \times \frac{1}{4}$

(ii) $\frac{1}{8} \times \frac{1}{4}$

(iii) $8 \div \frac{1}{4}$

(iv) $\frac{1}{4} \div 8$

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

(i) $8 \times \frac{1}{2}$

(ii) $\frac{1}{2} \cdot \frac{1}{8}$

(iii) $8 \div \frac{1}{2}$

(iv) $\frac{1}{2} \div 8$

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

(i) $5 \times \frac{1}{6}$

(ii) $\frac{1}{6} \div 5$

(iii) $5 \div \frac{1}{6}$

(iv) $5 \times 6$

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

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 $1 \div \frac{1}{6}$, $1 \div \frac{1}{10}$, $1 \div \frac{1}{13}$, $1 \div \frac{1}{9}$, and $1 \div \frac{1}{2}$". What should the friend say?

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

A car runs 16 km using 1 litre of petrol. How far will it go using $2 \frac{3}{4}$ litres of petrol?

Amritpal decides on a destination for his vacation. If he takes a train, it will take him $5 \frac{1}{6}$ hours to get there. If he takes a plane, it will take him $\frac{1}{2}$ hour. How many hours does the plane save?

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

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

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

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

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

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

(e) $> 1$

(f) $< 1$

What fraction of the whole square is shaded?

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?

What is $1 - \frac{1}{2}$?

$\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right)$?

$\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right)$?

$\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right) \times \left(1 - \frac{1}{6}\right) \times \left(1 - \frac{1}{7}\right) \times \left(1 - \frac{1}{8}\right) \times \left(1 - \frac{1}{9}\right) \times \left(1 - \frac{1}{10}\right)$?

Make a general statement and explain.

Context: In Fig. 8.3, the length and breadth of the shaded rectangle are $\frac{1}{2}$ unit and $\frac{1}{4}$ unit, and its area is $\frac{1}{8}$ square units.

Q. Do you see any relation between the area and the product of length and breadth?

In each of the division problems above, observe how we found the answer. Can we frame a rule that tells us how to divide two fractions?

When do you think the quotient is less than the dividend and when is it greater than the dividend?

Is there a similar relationship between the divisor and the quotient?

In each of the figures given below, find the fraction of the big square that the shaded region occupies.

Context: If we assume 1 gold dinar = 12 silver drammas, 1 silver dramma = 4 copper panas, 1 copper pana = 6 mashakas, and 1 pana = 30 cowrie shells,

Given: 1 copper pana = $\frac{1}{48}$ gold dinar $\left(\frac{1}{12} \times \frac{1}{4}\right)$

Fill in the blanks:

(i) 1 cowrie shell = ______ copper panas

(ii) 1 cowrie shell = ______ gold dinar.