Chapter 8: Working with Fractions (Fractions)

$\frac{2}{5}\times 3$: divide 3 by 5 to get $\frac{3}{5}$, then multiply by 2: $\frac{6}{5}$ km $=1\frac{1}{5}$ km.

Land: $5\times \frac{2}{3}=\frac{10}{3}=3\frac{1}{3}$ acres.

Internet: $1\frac{1}{4}=\frac{5}{4}$, so $\frac{5}{4}\times 8=5\times 2=$ ₹10.

6/5 km; 3 1/3 acres; ₹10

Week: $7\times \frac{1}{2}=\frac{7}{2}=$ $3\frac{1}{2}$ glasses.

January (31 days): $31\times \frac{1}{2}=\frac{31}{2}=$ $15\frac{1}{2}$ glasses.

3 1/2 glasses in a week; 15 1/2 glasses in January.

One day: $1÷8=$ $\frac{1}{8}$ km. One week (5 days): $5\times \frac{1}{8}=$ $\frac{5}{8}$ km.

1/8 km a day; 5/8 km a week.

One week: $5÷3=$ $\frac{5}{3}=1\frac{2}{3}$ litres. Four weeks: $4\times \frac{5}{3}=\frac{20}{3}=$ $6\frac{2}{3}$ litres.

1 2/3 litres a week; 6 2/3 litres in 4 weeks.

From Monday to Thursday is 3 days: $3\times \frac{5}{6}=\frac{15}{6}=2\frac{1}{2}$ hours.

It sets $2\frac{1}{2}$ hours after 10 pm, i.e. at about 12:30 am (just after midnight).

2 1/2 hours later (around 12:30 am).

(a) $\frac{21}{5}=4\frac{1}{5}$ (b) $\frac{4}{3}=1\frac{1}{3}$ (c) $\frac{54}{7}=7\frac{5}{7}$ (d) $\frac{78}{11}=7\frac{1}{11}$

(a) 4 1/5 (b) 1 1/3 (c) 7 5/7 (d) 7 1/11

Divide $\frac{3}{2}$ into 4 equal parts: the unit squares are split into $2\times 4=8$ parts and 3 of them are taken, $\frac{3}{8}$. Multiply by 5: $\frac{5}{4}\times \frac{3}{2}=5\times \frac{3}{8}=\frac{15}{8}=1\frac{7}{8}$.

15/8 = 1 7/8

(a) $\frac{1}{15}$ (b) $\frac{1}{12}$ (c) $\frac{1}{10}$ (d) $\frac{1}{30}$

In each case one part out of (product of the denominators) is hatched: $\frac{1}{b}\times \frac{1}{d}=\frac{1}{b\times d}$. So $\frac{1}{12}\times \frac{1}{18}=\frac{1}{216}$.

(a) 1/15 (b) 1/12 (c) 1/10 (d) 1/30; 1/12 × 1/18 = 1/216

(a) $\frac{2\times 4}{3\times 5}=\frac{8}{15}$ (b) $\frac{2}{12}=\frac{1}{6}$ (c) $\frac{3}{10}$ (d) $\frac{12}{30}=\frac{2}{5}$

(a) 8/15 (b) 1/6 (c) 3/10 (d) 2/5

(a) $\frac{1}{3}\times \frac{7}{10}=\frac{7}{30}$ (b) $\frac{2}{3}\times \frac{7}{10}=\frac{14}{30}=\frac{7}{15}$ (c) $\frac{3}{4}\times \frac{7}{10}=\frac{21}{40}$ (d) $\frac{7}{10}\times \frac{7}{10}=\frac{49}{100}$

(e) Time $\times \frac{7}{10}=1$, so time $=1÷\frac{7}{10}=\frac{10}{7}=$ $1\frac{3}{7}$ hours.

(a) 7/30 (b) 7/15 (c) 21/40 (d) 49/100 (e) 10/7 = 1 3/7 hours

Remaining after the road: $1-\frac{1}{6}=\frac{5}{6}$.

(a) Krishna: $\frac{1}{2}\times \frac{5}{6}=$ $\frac{5}{12}$

(b) Bora: $\frac{1}{3}\times \frac{5}{6}=$ $\frac{5}{18}$

(c) Somu: $\frac{5}{6}-\frac{5}{12}-\frac{5}{18}=\frac{30-15-10}{36}=$ $\frac{5}{36}$

Remaining 5/6; (a) 5/12 (b) 5/18 (c) 5/36

$\frac{15}{4}\times \frac{48}{5}=\frac{15\times 48}{4\times 5}=3\times 12=$ 36 sq ft (cancelling 15 with 5 and 48 with 4).

36 square feet

Four saplings have 3 gaps: $3\times \frac{3}{4}=\frac{9}{4}=$ $2\frac{1}{4}$ m.

2 1/4 m

$\frac{12}{15}\times 500=400$ g; $\frac{3}{20}\times 4000=600$ g. 3/20 of 4 kg is heavier (by 200 g).

3/20 of 4 kg (600 g > 400 g)

• Between 0 and 1 → the product is less than the other number (e.g. $\frac{1}{2}\times 8=4<8$).
• Greater than 1 → the product is greater than the other number (e.g. $3\times \frac{2}{5}=\frac{6}{5}>\frac{2}{5}$).

More examples: $\frac{5}{2}\times \frac{7}{3}=\frac{35}{6}$ (greater than both); $\frac{1}{3}\times \frac{3}{4}=\frac{1}{4}$ (less than both); $\frac{3}{4}\times 6=4\frac{1}{2}$ (between).

Less; greater.

Since dividend = divisor × quotient:

• If the divisor is between 0 and 1, the quotient is greater than the dividend (e.g. $6÷\frac{1}{4}=24$).
• If the divisor is greater than 1, the quotient is less than the dividend (e.g. $6÷3=2$).

Similarly, looking at quotient × divisor = dividend: if the quotient is greater than 1, the divisor is less than the dividend, and if the quotient is between 0 and 1, the divisor is greater than the dividend. So the quotient is greater than 1 exactly when the divisor is smaller than the dividend (e.g. $\frac{1}{8}÷\frac{1}{4}=\frac{1}{2}$, since $\frac{1}{4}>\frac{1}{8}$).

Divisor < 1 → quotient > dividend; divisor > 1 → quotient < dividend. The quotient is more than 1 exactly when the divisor is smaller than the dividend.

Second: $1÷\frac{1}{2}=$ 2; third: $1÷\frac{1}{4}=$ 4; fourth: $1÷\frac{1}{5}=$ 5.

Together: $1+2+4+5=12$ cisterns a day, so together they fill one in $\frac{1}{12}$ day (2 hours).

2, 4, 5; 1 + 2 + 4 + 5 = 12, so 1/12 day.

Figure (a): take the big square as 2 × 2 (area 4). The shaded strip lies between the main diagonal and the parallel line joining the midpoints of the left and bottom sides. Area $=$ half the square $-$ the small corner triangle $=2-\frac{1}{2}=\frac{3}{2}$, which is $\frac{3}{2}÷4=$ $\frac{3}{8}$ of the square.

Figure (b): the top-left quarter has a tilted square joining the midpoints of its sides; the two shaded corner triangles on its right are each $\frac{1}{8}$ of that quarter. Shaded $=2\times \frac{1}{8}\times \frac{1}{4}=$ $\frac{1}{16}$ of the big square.

(a) 3/8 (b) 1/16

$\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \frac{1}{5}\times \frac{1}{16}\times \frac{1}{4}=\frac{6}{7680}=\frac{1}{1280}$ dramma $=$ 1 cowrie shell.

1 cowrie $=\frac{1}{30}$ pana, and 1 pana $=\frac{1}{48}$ dinar, so 1 cowrie $=\frac{1}{30}\times \frac{1}{48}=$ $\frac{1}{1440}$ dinar.

1 cowrie; 1 cowrie = 1/30 pana = 1/1440 dinar

27/7, 7/4, 1, 1, 16/9, 49/4, 15, 9/5, 2/11, 8/3

(a) (iii) $8÷\frac{1}{4}$ $=8\times 4=$ 32 bags

(b) (iv) $\frac{1}{2}÷8$ $=\frac{1}{2}\times \frac{1}{8}=$ $\frac{1}{16}$ m

(c) (iii) $5÷\frac{1}{6}$ $=5\times 6=$ 30 loaves (note (iv) $5\times 6$ gives the same value)

(a) 8 ÷ 1/4 = 32 (b) 1/2 ÷ 8 = 1/16 m (c) 5 ÷ 1/6 = 30

6 rotis are half of 12: $\frac{1}{2}\times \frac{1}{4}=$ $\frac{1}{8}$ kg (125 g).

1/8 kg

$6+10+13+9+2=$ 40

40

Read: $\frac{1}{5}+\frac{3}{10}=\frac{5}{10}=\frac{1}{2}$, i.e. 200 pages. Left: $400-200=$ 200 pages.

200 pages

$16\times \frac{11}{4}=4\times 11=$ 44 km

44 km

$5\frac{1}{6}-\frac{1}{2}=5\frac{1}{6}-\frac{3}{6}=4\frac{4}{6}=$ $4\frac{2}{3}$ hours (4 hours 40 minutes).

4 2/3 hours

Rest: $1-\frac{4}{5}=\frac{1}{5}$. Each friend: $\frac{1}{5}÷3=$ $\frac{1}{15}$ of the cake.

1/15 of the cake

Both fractions are greater than 1 (numerator > denominator). Multiplying by a number greater than 1 makes a number bigger, so the product is greater than each of them, and greater than 1.

Correct: (a), (c) and (e).

(a), (c), (e)

In the bottom-right quarter, the arms of the Y run from the top corners of the quarter to its centre, and the stem goes straight down. The shaded piece = the left half of the quarter minus the small triangle above the arm $=\frac{1}{2}-\frac{1}{8}=\frac{3}{8}$ of the quarter.

Of the whole square: $\frac{3}{8}\times \frac{1}{4}=$ $\frac{3}{32}$.

3/32

Follow the splits from the bottom:

• 1st point (2 ways): $\frac{1}{2}$ to the mango tree, $\frac{1}{2}$ on.
• 2nd point (2 ways): $\frac{1}{4}$ to the mango tree, $\frac{1}{4}$ on.
• 3rd point (4 ways): each $\frac{1}{16}$: two paths to the mango tree, one to the sugarcane field, one on.
• 4th point (2 ways): $\frac{1}{32}$ to the mango tree, $\frac{1}{32}$ to the sugarcane field.

Mango tree: $\frac{1}{2}+\frac{1}{4}+\frac{2}{16}+\frac{1}{32}=\frac{16+8+4+1}{32}=$ $\frac{29}{32}$

Sugarcane field: $\frac{1}{16}+\frac{1}{32}=$ $\frac{3}{32}$

Mango tree 29/32; sugarcane field 3/32

$1-\frac{1}{2}=\frac{1}{2}$

$\frac{1}{2}\times \frac{2}{3}=\frac{1}{3}$

$\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}=\frac{1}{5}$

$\frac{1}{2}\times \frac{2}{3}\times \cdots \times \frac{9}{10}=\frac{1}{10}$

General statement: $\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\cdots \left(1-\frac{1}{n}\right)=\frac{1}{n}$.

Why: $1-\frac{1}{k}=\frac{k-1}{k}$, so the product is $\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{n-1}{n}$. Every numerator cancels with the denominator just before it, leaving only $\frac{1}{n}$.

1/2, 1/3, 1/5, 1/10; in general the product up to (1 − 1/n) is 1/n, since all middle factors cancel.

A queen attacks along its row, column and both diagonals, so each row and each column must have exactly one queen and no two may share a diagonal.

4 × 4: queens in the squares (row, column) = (1, 2), (2, 4), (3, 1), (4, 3).

8 × 8: one solution places the queens in rows 1 to 8 in columns 1, 5, 8, 6, 3, 7, 2, 4.

(There are 92 solutions in all for 8 queens.)

4 queens: columns 2, 4, 1, 3 in rows 1–4. 8 queens: columns 1, 5, 8, 6, 3, 7, 2, 4 in rows 1–8.