Fractions — Exercise 4(E)

Find the product :

$\dfrac{5}{7} \times \dfrac{2}{3}$

Answer

$\Rightarrow \dfrac{5}{7} \times \dfrac{2}{3}\\[1em] \Rightarrow \dfrac{5 \times 2}{7 \times 3}\\[1em] \Rightarrow \dfrac{10}{21}$

Hence, $\dfrac{5}{7} \times \dfrac{2}{3} = \dfrac{10}{21}$.

Find the product :

$\dfrac{3}{5} \times \dfrac{4}{9}$

Answer

$\Rightarrow \dfrac{3}{5} \times \dfrac{4}{9}\\[1em] \Rightarrow \dfrac{3 \times 4}{5 \times 9}\\[1em] \Rightarrow \dfrac{12}{45}\\[1em] \Rightarrow \dfrac{4}{15}$

Hence, $\dfrac{3}{5} \times \dfrac{4}{9} = \dfrac{4}{15}$.

Find the product :

$\dfrac{12}{35} \times \dfrac{14}{27}$

Answer

$\Rightarrow \dfrac{12}{35} \times \dfrac{14}{27}\\[1em] \Rightarrow \dfrac{12}{27} \times \dfrac{14}{35} \\[1em] \Rightarrow \dfrac{4}{9} \times \dfrac{2}{5} \\[1em] \Rightarrow \dfrac{8}{45}.$

Hence, $\dfrac{12}{35} \times \dfrac{14}{27} = \dfrac{8}{45}$.

Find the product :

$\dfrac{30}{51} \times \dfrac{17}{25}$

Answer

$\Rightarrow \dfrac{30}{51} \times \dfrac{17}{25}\\[1em] \Rightarrow \dfrac{30}{25} \times \dfrac{17}{51} \\[1em] \Rightarrow \dfrac{6}{5} \times \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{2}{5}.$

Hence, $\dfrac{30}{51} \times \dfrac{17}{25} = \dfrac{2}{5}$.

Find the product :

$\dfrac{9}{16} \times \dfrac{14}{15}$

Answer

$\Rightarrow \dfrac{9}{16} \times \dfrac{14}{15}\\[1em] \Rightarrow \dfrac{9}{15} \times \dfrac{14}{16} \\[1em] \Rightarrow \dfrac{3}{5} \times \dfrac{7}{8} \\[1em] \Rightarrow \dfrac{21}{40}.$

Hence, $\dfrac{9}{16} \times \dfrac{14}{15} = \dfrac{21}{40}$.

Find the product :

$2\dfrac{11}{27} \times 18$

Answer

$\Rightarrow 2\dfrac{11}{27} \times 18\\[1em] \Rightarrow \dfrac{65}{27} \times 18\\[1em] \Rightarrow 65 \times \dfrac{18}{27} \\[1em] \Rightarrow 65 \times \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{130}{3} \\[1em] \Rightarrow 43\dfrac{1}{3}.$

Hence, $2\dfrac{11}{27} \times 18 = 43\dfrac{1}{3}$.

Find the product :

$7\dfrac{1}{12} \times \dfrac{8}{17}$

Answer

$\Rightarrow 7\dfrac{1}{12} \times \dfrac{8}{17}\\[1em] \Rightarrow \dfrac{85}{12} \times \dfrac{8}{17}\\[1em] \Rightarrow \dfrac{85}{17} \times \dfrac{8}{12} \\[1em] \Rightarrow 5 \times \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{10}{3}\\[1em] \Rightarrow 3\dfrac{1}{3}.$

Hence, $7\dfrac{1}{12} \times \dfrac{8}{17} = 3\dfrac{1}{3}$.

Find the product :

$8\dfrac{1}{21} \times 1\dfrac{1}{13}$

Answer

$\Rightarrow 8\dfrac{1}{21} \times 1\dfrac{1}{13}\\[1em] \Rightarrow \dfrac{169}{21} \times \dfrac{14}{13}\\[1em] \Rightarrow \dfrac{169}{13} \times \dfrac{14}{21} \\[1em] \Rightarrow 13 \times \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{26}{3} \\[1em] \Rightarrow 8\dfrac{2}{3}.$

Hence, $8\dfrac{1}{21} \times 1\dfrac{1}{13} = 8\dfrac{2}{3}$.

Find the product :

$7\dfrac{7}{11} \times 6\dfrac{3}{16}$

Answer

$\Rightarrow 7\dfrac{7}{11} \times 6\dfrac{3}{16}\\[1em] \Rightarrow \dfrac{84}{11} \times \dfrac{99}{16}\\[1em] \Rightarrow \dfrac{84}{16} \times \dfrac{99}{11} \\[1em] \Rightarrow \dfrac{21}{4} \times 9 \\[1em] \Rightarrow \dfrac{189}{4} \\[1em] \Rightarrow 47\dfrac{1}{4}.$

Hence, $7\dfrac{7}{11} \times 6\dfrac{3}{16} = 47\dfrac{1}{4}$.

Find the product :

$11\dfrac{1}{4} \times 6\dfrac{7}{9}$

Answer

$\Rightarrow 11\dfrac{1}{4} \times 6\dfrac{7}{9}\\[1em] \Rightarrow \dfrac{45}{4} \times \dfrac{61}{9}\\[1em] \Rightarrow \dfrac{45}{9} \times \dfrac{61}{4} \\[1em] \Rightarrow 5 \times \dfrac{61}{4} \\[1em] \Rightarrow \dfrac{305}{4} \\[1em] \Rightarrow 76\dfrac{1}{4}.$

Hence, $11\dfrac{1}{4} \times 6\dfrac{7}{9} = 76\dfrac{1}{4}$.

Find the product :

$1\dfrac{3}{4} \times 2\dfrac{1}{7} \times 4\dfrac{4}{5}$

Answer

$\Rightarrow 1\dfrac{3}{4} \times 2\dfrac{1}{7} \times 4\dfrac{4}{5}\\[1em] \Rightarrow \dfrac{7}{4} \times \dfrac{15}{7} \times \dfrac{24}{5}\\[1em] \Rightarrow \dfrac{7}{7} \times \dfrac{15}{5} \times \dfrac{24}{4} \\[1em] \Rightarrow 1 \times 3\times 6 \\[1em] \Rightarrow 18.$

Hence, $1\dfrac{3}{4} \times 2\dfrac{1}{7} \times 4\dfrac{4}{5} = 18$.

Find the product :

$3\dfrac{1}{6} \times 2\dfrac{3}{4} \times 2\dfrac{4}{11}$

Answer

$\Rightarrow 3\dfrac{1}{6} \times 2\dfrac{3}{4} \times 2\dfrac{4}{11}\\[1em] \Rightarrow \dfrac{19}{6} \times \dfrac{11}{4} \times \dfrac{26}{11}\\[1em] \Rightarrow \dfrac{19}{6} \times \dfrac{11}{11} \times \dfrac{26}{4} \\[1em] \Rightarrow \dfrac{19}{6} \times 1 \times \dfrac{13}{2} \\[1em] \Rightarrow \dfrac{247}{12} \\[1em] \Rightarrow 20\dfrac{7}{12}.$

Hence, $3\dfrac{1}{6} \times 2\dfrac{3}{4} \times 2\dfrac{4}{11} = 20\dfrac{7}{12}$.

Find the reciprocal of :

(i) $\dfrac{5}{9}$

(ii) $\dfrac{7}{13}$

(iii) 8

(iv) $\dfrac{1}{5}$

(v) $3\dfrac{2}{5}$

(vi) $8\dfrac{1}{9}$

(vii) $5\dfrac{3}{8}$

(viii) 23

Answer

The reciprocal of a non-zero fraction is a fraction obtained by swapping its numerator and denominator, so that their product is 1.

(i) $\dfrac{5}{9}$

Hence, the reciprocal of $\dfrac{5}{9} = \dfrac{9}{5}$.

(ii) $\dfrac{7}{13}$

Hence, the reciprocal of $\dfrac{7}{13} = \dfrac{13}{7}$.

(iii) $8$

Hence, the reciprocal of $8 = \dfrac{1}{8}$.

(iv) $\dfrac{1}{5}$

Hence, the reciprocal of $\dfrac{1}{5} = \dfrac{5}{1} = 5$.

(v) $3\dfrac{2}{5}$

$\Rightarrow \dfrac{17}{5}$

Hence, the reciprocal of $\dfrac{17}{5} = \dfrac{5}{17}$.

(vi) $8\dfrac{1}{9}$

$\Rightarrow \dfrac{73}{9}$

Hence, the reciprocal of $\dfrac{73}{9} = \dfrac{9}{73}$.

(vii) $5\dfrac{3}{8}$

$\Rightarrow \dfrac{43}{8}$

Hence, the reciprocal of $\dfrac{43}{8} = \dfrac{8}{43}$.

(viii) 23

Hence, the reciprocal of $23 = \dfrac{23}{1} = \dfrac{1}{23}$.

Divide:

$\dfrac{13}{21} ÷ \dfrac{2}{7}$

Answer

$\Rightarrow \dfrac{13}{21} ÷ \dfrac{2}{7}\\[1em] \Rightarrow \dfrac{13}{21} \times \dfrac{7}{2}\\[1em] \Rightarrow \dfrac{13}{2} \times \dfrac{7}{21} \\[1em] \Rightarrow \dfrac{13}{2} \times \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{13}{6} \\[1em] \Rightarrow 2\dfrac{1}{6}.$

Hence, $\dfrac{13}{21} ÷ \dfrac{2}{7} = 2\dfrac{1}{6}$.

Divide:

$3\dfrac{1}{8} ÷ 5$

Answer

$\Rightarrow 3\dfrac{1}{8} ÷ 5\\[1em] \Rightarrow \dfrac{25}{8} \times \dfrac{1}{5}\\[1em] \Rightarrow \dfrac{25 \times 1}{8 \times 5}\\[1em] \Rightarrow \dfrac{25}{40}\\[1em] \Rightarrow \dfrac{5}{8}.$

Hence, $3\dfrac{1}{8} ÷ 5 = \dfrac{5}{8}$.

Divide:

$1 ÷ 2\dfrac{3}{5}$

Answer

$\Rightarrow 1 ÷ 2\dfrac{3}{5}\\[1em] \Rightarrow 1 ÷ \dfrac{13}{5}\\[1em] \Rightarrow 1 \times \dfrac{5}{13}\\[1em] \Rightarrow \dfrac{5}{13}.$

Hence, $1 ÷ 2\dfrac{3}{5} = \dfrac{5}{13}$.

Divide:

$7\dfrac{5}{9} ÷ 34$

Answer

$\Rightarrow 7\dfrac{5}{9} ÷ 34\\[1em] \Rightarrow \dfrac{68}{9} ÷ 34\\[1em] \Rightarrow \dfrac{68}{9} \times \dfrac{1}{34}\\[1em] \Rightarrow \dfrac{68}{34} \times \dfrac{1}{9} \\[1em] \Rightarrow 2 \times \dfrac{1}{9} \\[1em] \Rightarrow \dfrac{2}{9}.$

Hence, $7\dfrac{5}{9} ÷ 34 = \dfrac{2}{9}$.

Divide:

$70 ÷ 8\dfrac{2}{5}$

Answer

Solving,

$\Rightarrow 70 ÷ 8\dfrac{2}{5}\\[1em] \Rightarrow \dfrac{70}{1} ÷ \dfrac{42}{5}\\[1em] \Rightarrow \dfrac{70}{1} \times \dfrac{5}{42}\\[1em] \Rightarrow \dfrac{70}{42} \times 5 \\[1em] \Rightarrow \dfrac{5}{3} \times 5 \\[1em] \Rightarrow \dfrac{25}{3}\\[1em] \Rightarrow 8\dfrac{1}{3}.$

Hence, $70 ÷ 8\dfrac{2}{5} = 8\dfrac{1}{3}$.

Divide:

$4\dfrac{1}{2} ÷ 6\dfrac{1}{2}$

Answer

Solving,

$\Rightarrow 4\dfrac{1}{2} ÷ 6\dfrac{1}{2}\\[1em] \Rightarrow \dfrac{9}{2} ÷ \dfrac{13}{2}\\[1em] \Rightarrow \dfrac{9}{2} \times \dfrac{2}{13}\\[1em] \Rightarrow \dfrac{9 \times 2}{2 \times 13}\\[1em] \Rightarrow \dfrac{9}{13}.$

Hence, $4\dfrac{1}{2} ÷ 6\dfrac{1}{2} = \dfrac{9}{13}$.

Divide:

$5\dfrac{7}{10} ÷ 3\dfrac{1}{6}$

Answer

Solving,

$\Rightarrow 5\dfrac{7}{10} ÷ 3\dfrac{1}{6}\\[1em] \Rightarrow \dfrac{57}{10} ÷ \dfrac{19}{6}\\[1em] \Rightarrow \dfrac{57}{10} \times \dfrac{6}{19}\\[1em] \Rightarrow \dfrac{57}{19} \times \dfrac{6}{10} \\[1em] \Rightarrow 3 \times \dfrac{3}{5}\\[1em] \Rightarrow \dfrac{9}{5}\\[1em] \Rightarrow 1\dfrac{4}{5}.$

Hence, $5\dfrac{7}{10} ÷ 3\dfrac{1}{6} = 1\dfrac{4}{5}$.

Divide:

$10\dfrac{5}{7} ÷ 1\dfrac{11}{14}$

Answer

Solving,

$\Rightarrow 10\dfrac{5}{7} ÷ 1\dfrac{11}{14}\\[1em] \Rightarrow \dfrac{75}{7} ÷ \dfrac{25}{14}\\[1em] \Rightarrow \dfrac{75}{7} \times \dfrac{14}{25}\\[1em] \Rightarrow \dfrac{75}{25} \times \dfrac{14}{7}\\[1em] \Rightarrow 3 \times 2 \\[1em] \Rightarrow 6.$

Hence, $10\dfrac{5}{7} ÷ 1\dfrac{11}{14} = 6$.

Divide:

$15\dfrac{8}{9} ÷ 3\dfrac{2}{3}$

Answer

Solving,

$\Rightarrow 15\dfrac{8}{9} ÷ 3\dfrac{2}{3}\\[1em] \Rightarrow \dfrac{143}{9} ÷ \dfrac{11}{3}\\[1em] \Rightarrow \dfrac{143}{9} \times \dfrac{3}{11}\\[1em] \Rightarrow \dfrac{143}{11} \times \dfrac{3}{9} \\[1em] \Rightarrow 13 \times \dfrac{1}{3} \\[1em] \Rightarrow \dfrac{13}{3} \\[1em] \Rightarrow 4\dfrac{1}{3}.$

Hence, $15\dfrac{8}{9} ÷ 3\dfrac{2}{3} = 4\dfrac{1}{3}$

Divide:

$9\dfrac{4}{5} ÷ 3\dfrac{23}{25}$

Answer

Solving,

$\Rightarrow 9\dfrac{4}{5} ÷ 3\dfrac{23}{25}\\[1em] \Rightarrow \dfrac{49}{5} ÷ \dfrac{98}{25}\\[1em] \Rightarrow \dfrac{49}{5} \times \dfrac{25}{98}\\[1em] \Rightarrow \dfrac{49}{98} \times \dfrac{25}{5} \\[1em] \Rightarrow \dfrac{1}{2} \times 5 \\[1em] \Rightarrow \dfrac{5}{2} \\[1em] \Rightarrow 2\dfrac{1}{2}.$

Hence, $9\dfrac{4}{5} ÷ 3\dfrac{23}{25} = 2\dfrac{1}{2}$.

Divide:

$2\dfrac{17}{38} ÷ 1\dfrac{12}{19}$

Answer

Solving,

$\Rightarrow 2\dfrac{17}{38} ÷ 1\dfrac{12}{19}\\[1em] \Rightarrow \dfrac{93}{38} ÷ \dfrac{31}{19}\\[1em] \Rightarrow \dfrac{93}{38} \times \dfrac{19}{31}\\[1em] \Rightarrow \dfrac{93}{31} \times \dfrac{19}{38}\\[1em] \Rightarrow 3 \times \dfrac{1}{2}\\[1em] \Rightarrow \dfrac{3}{2}\\[1em] \Rightarrow 1\dfrac{1}{2}.$

Hence, $2\dfrac{17}{38} ÷ 1\dfrac{12}{19} = 1\dfrac{1}{2}$.

Divide:

$8\dfrac{7}{25} ÷ 3\dfrac{1}{15}$

Answer

Solving,

$\Rightarrow 8\dfrac{7}{25} ÷ 3\dfrac{1}{15}\\[1em] \Rightarrow \dfrac{207}{25} ÷ \dfrac{46}{15}\\[1em] \Rightarrow \dfrac{207}{25} \times \dfrac{15}{46}\\[1em] \Rightarrow \dfrac{207}{46} \times \dfrac{15}{25} \\[1em] \Rightarrow \dfrac{9}{2} \times \dfrac{3}{5} \\[1em] \Rightarrow \dfrac{27}{10}\\[1em] \Rightarrow 2\dfrac{7}{10}.$

Hence, $8\dfrac{7}{25} ÷ 3\dfrac{1}{15} = 2\dfrac{7}{10}$.

The cost of 1 litre of milk is ₹ $42\dfrac{3}{5}$. Find the cost of $12\dfrac{1}{2}$ litres of milk.

Answer

Given, the cost of 1 litre of milk = ₹ $42\dfrac{3}{5}$

The cost of $12\dfrac{1}{2}$ litres of milk :

$= ₹42\dfrac{3}{5} \times 12\dfrac{1}{2}\\[1em] = ₹\dfrac{213}{5} \times \dfrac{25}{2}\\[1em] = ₹\dfrac{213}{2} \times \dfrac{25}{5}\\[1em] = ₹\dfrac{213}{2} \times 5 \\[1em] = ₹\dfrac{1065}{2}\\[1em] = ₹ 532\dfrac{1}{2}.$

Hence, the cost of $12\dfrac{1}{2}$ litres of milk = ₹ $532\dfrac{1}{2}$.

The cost of 1 litre of petrol is ₹ $65\dfrac{3}{4}$. Find the cost of 36 litres of petrol.

Answer

Given, the cost of 1 litre of petrol = ₹ $65\dfrac{3}{4}$

The cost of 36 litres of petrol

$= ₹ 65\dfrac{3}{4} \times 36\\[1em] = ₹ \dfrac{263}{4} \times 36\\[1em] = ₹ \dfrac{263 \times 36}{4} \\[1em] = ₹ \dfrac{9468}{4} \\[1em] = ₹ 2,367$

Hence, the cost of 36 litres of petrol = ₹ 2,367.

The cost of $3\dfrac{1}{2}$ metres of cloth is ₹ $547\dfrac{3}{4}$. Find the cost of 1 metre of cloth.

Answer

Given, the cost of $3\dfrac{1}{2}$ metres of cloth = ₹ $547\dfrac{3}{4}$

The cost of 1 m of cloth :

$= 547\dfrac{3}{4} ÷ 3\dfrac{1}{2}\\[1em] = \dfrac{2,191}{4} ÷ \dfrac{7}{2}\\[1em] = \dfrac{2,191}{4} \times \dfrac{2}{7}\\[1em] = \dfrac{2,191}{7} \times \dfrac{2}{4}\\[1em] = 313 \times \dfrac{1}{2}\\[1em] = \dfrac{313}{2}\\[1em] = 156\dfrac{1}{2}.$

Hence, the cost of 1 metre of cloth = ₹ $156\dfrac{1}{2}$.

Tanvy cuts 54 m of cloth into pieces, each of length $3\dfrac{3}{8}$ m. How many pieces does she get?

Answer

Given, total length of cloth = 54 m

Length of each piece = $3\dfrac{3}{8}$ m

Let n be the number of pieces.

By formula,

Total length = Number of pieces x length of each piece

$\Rightarrow n \times 3\dfrac{3}{8} = 54\\[1em] \Rightarrow n \times \dfrac{27}{8} = 54\\[1em] \Rightarrow n = 54 ÷ \dfrac{27}{8}\\[1em] \Rightarrow n = 54 \times \dfrac{8}{27}\\[1em] \Rightarrow n = \dfrac{54}{27} \times 8 \\[1em] \Rightarrow n = 2 \times 8 \\[1em] \Rightarrow n = 16.$

Hence, the number of pieces = 16.

A cord of length $126\dfrac{1}{2}$ m has been cut into 46 pieces of equal length. What is the length of each piece?

Answer

Given, total length of cord = $126\dfrac{1}{2}$ m

Number of pieces = 46

Let x be the length of each piece.

By formula,

Total length = Number of pieces x length of each piece

$\Rightarrow 46 \times x = 126\dfrac{1}{2}\\[1em] \Rightarrow 46 \times x = \dfrac{253}{2} \\[1em] \Rightarrow x = \dfrac{253}{2} ÷ 46\\[1em] \Rightarrow x = \dfrac{253}{2} \times \dfrac{1}{46} \\[1em] \Rightarrow x = \dfrac{253}{46} \times \dfrac{1}{2} \\[1em] \Rightarrow x = \dfrac{253}{92} \\[1em] \Rightarrow x = \dfrac{23 \times 11}{23 \times 4} \\[1em] \Rightarrow x = \dfrac{11}{4} \\[1em] \Rightarrow x = 2\dfrac{3}{4}.$

Hence, the length of each piece = $2\dfrac{3}{4}$ m.

A car travels $283\dfrac{1}{2}$ km in $4\dfrac{2}{3}$ hours. How far does it go in 1 hour?

Answer

Given, total distance = $283\dfrac{1}{2}$ km

Total time = $4\dfrac{2}{3}$ hours

Let x be the distance covered in 1 hour.

By formula,

Distance covered in 1 hour x Total time = Total distance

$\Rightarrow x \times 4\dfrac{2}{3} = 283\dfrac{1}{2}\\[1em] \Rightarrow x \times \dfrac{14}{3} = \dfrac{567}{2}\\[1em] \Rightarrow x = \dfrac{567}{2} ÷ \dfrac{14}{3}\\[1em] \Rightarrow x = \dfrac{567}{2} \times \dfrac{3}{14}\\[1em] \Rightarrow x = \dfrac{567}{14} \times \dfrac{3}{2} \\[1em] \Rightarrow x = \dfrac{81}{2} \times \dfrac{3}{2} \\[1em] \Rightarrow x = \dfrac{243}{4} \\[1em] \Rightarrow x = 60\dfrac{3}{4}.$

Hence, distance covered in 1 hour = $60\dfrac{3}{4}$ km.

The area of a rectangular plot of land is $46\dfrac{2}{5}$ sq.m. If its length is $7\dfrac{1}{4}$ m, find its breadth.

Answer

Given, the area of a rectangular plot of land = $46\dfrac{2}{5}$ sq. m

Length of rectangular plot = $7\dfrac{1}{4}$ m

Let b be the breath of the rectangular plot.

By formula, length x breadth = area

$\Rightarrow 7\dfrac{1}{4} \times b = 46\dfrac{2}{5}\\[1em] \Rightarrow \dfrac{29}{4} \times b = \dfrac{232}{5}\\[1em] \Rightarrow b = \dfrac{232}{5} ÷ \dfrac{29}{4}\\[1em] \Rightarrow b = \dfrac{232}{5} \times \dfrac{4}{29}\\[1em] \Rightarrow b = \dfrac{232 \times 4}{5 \times 29}\\[1em] \Rightarrow b = \dfrac{232}{29} \times \dfrac{4}{5} \\[1em] \Rightarrow b = 8 \times \dfrac{4}{5} \\[1em] \Rightarrow b = \dfrac{32}{5}\\[1em] \Rightarrow b = 6\dfrac{2}{5}.$

Hence, the breadth of the rectangular plot = $6\dfrac{2}{5}$ m.

The product of two fractions is $15\dfrac{3}{4}$. If one of them is $4\dfrac{1}{2}$, find the other.

Answer

Given, the product of two fractions = $15\dfrac{3}{4}$

One of the fraction = $4\dfrac{1}{2}$

Let x be the other fraction.

$\Rightarrow x \times 4\dfrac{1}{2} = 15\dfrac{3}{4} \\[1em] \Rightarrow x = 15\dfrac{3}{4} ÷ 4\dfrac{1}{2}\\[1em] = \dfrac{63}{4} ÷ \dfrac{9}{2}\\[1em] = \dfrac{63}{4} \times \dfrac{2}{9}\\[1em] = \dfrac{63 \times 2}{4 \times 9}\\[1em] = \dfrac{126}{36}\\[1em] = \dfrac{7}{2}\\[1em] = 3\dfrac{1}{2}.$

Hence, the other fraction = $3\dfrac{1}{2}$.