Fractions - Exercise 2(A)

Find the product :

$\dfrac{5}{6} \times \dfrac{3}{7}$

Answer

We have:

$\begin{array}{ll} \phantom{=} \dfrac{5}{6} \times \dfrac{3}{7} \\\\ = \dfrac{5 \times 3}{6 \times 7} \\\\ = \dfrac{5 \times 1}{2 \times 7} & \text{[Simplifying 3 and 6 ⇒ Divide by 3 ]} \\\\ = \dfrac{5}{14} \\\\ \end{array}$

∴ The answer is $\dfrac{5}{14}$

Find the product :

$\dfrac{7}{18} \times \dfrac{9}{14}$

Answer

We have:

$\begin{array}{ll} \phantom{=} \dfrac{7}{18} \times \dfrac{9}{14} \\\\ = \dfrac{7 \times 9}{18 \times 14} \\\\ = \dfrac{7 \times 1}{2 \times 14} & \text{[Simplifying 9 and 18 ⇒ Divide by 9]} \\\\ = \dfrac{1 \times 1}{2 \times 2} & \text{[Simplifying 7 and 14 ⇒ Divide by 7]} \\\\ = \dfrac{1}{4} & \end{array}$

∴ The answer is $\dfrac{1}{4}$

Find the product :

$28 \times \dfrac{7}{8}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 28 \times \dfrac{7}{8} \\\\ = \dfrac{28}{1} \times \dfrac{7}{8} \\\\ = \dfrac{7}{1} \times \dfrac{7}{2} & \text{[Simplifying 28 and 8 ⇒ Divide by 4]} \\\\ = \dfrac{7 \times 7}{1 \times 2} \\\\ = \dfrac{49}{2} & \\\\ = 24 \dfrac{1}{2} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The answer is 24 $\dfrac{1}{2}$

Find the product :

$7 \times \dfrac{1}{7}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 7 \times \dfrac{1}{7} \\\\ = \dfrac{7}{1} \times \dfrac{1}{7} \\\\ = \dfrac{7 \times 1}{1 \times 7} \\\\ = \dfrac{7}{7} \\\\ = 1 & \text{[Dividing both by 7]} \end{array}$

∴ The answer is 1

Find the product :

$2\dfrac{1}{25} \times \dfrac{5}{17}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 2\dfrac{1}{25} \times \dfrac{5}{17} \\\\ = \dfrac{51}{25} \times \dfrac{5}{17} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{51}{5} \times \dfrac{1}{17} & \text{[Simplifying 5 and 25 ⇒ Divide by 5]} \\\\ = \dfrac{3}{5} \times \dfrac{1}{1} & \text{[Simplifying 51 and 17 ⇒ Divide by 17]} \\\\ = \dfrac{3 \times 1}{5 \times 1} & \\\\ = \dfrac{3}{5} \end{array}$

∴ The answer is $\dfrac{3}{5}$

Find the product :

$1\dfrac{1}{13} \times 7\dfrac{3}{7}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 1\dfrac{1}{13} \times 7\dfrac{3}{7} \\\\ = \dfrac{14}{13} \times \dfrac{52}{7} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{2}{13} \times \dfrac{52}{1} & \text{[Simplifying 14 and 7 ⇒ Divide by 7]} \\\\ = \dfrac{2}{1} \times \dfrac{4}{1} & \text{[Simplifying 52 and 13 ⇒ Divide by 13]} \\\\ = \dfrac{2 \times 4}{1 \times 1} \\\\ = \dfrac{8}{1} \\\\ = 8 \end{array}$

∴ The answer is 8

Find the product :

$\dfrac{4}{17} \times 7\dfrac{1}{12}$

Answer

We have:

$\begin{array}{ll} \phantom{=} \dfrac{4}{17} \times 7\dfrac{1}{12} \\\\ = \dfrac{4}{17} \times \dfrac{85}{12} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{4}{1} \times \dfrac{5}{12} & \text{[Simplifying 85 and 17 ⇒ Divide by 17 ]} \\\\ = \dfrac{1}{1} \times \dfrac{5}{3} & \text{[Simplifying 4 and 12 ⇒ Divide by 4 ]} \\\\ = \dfrac{1 \times 5}{1 \times 3} \\\\ = \dfrac{5}{3} \end{array}$

∴ The answer is $\dfrac{5}{3}$

Find the product :

$7\dfrac{1}{4}\times\dfrac{7}{58}\times1\dfrac{11}{21}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 7\dfrac{1}{4}\times\dfrac{7}{58}\times1\dfrac{11}{21} \\\\ = \dfrac{29}{4} \times \dfrac{7}{58} \times \dfrac{32}{21} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{1}{4} \times \dfrac{7}{2} \times \dfrac{32}{21} & \text{[Simplifying 29 and 58 ⇒ Divide by 29 ]} \\\\ = \dfrac{1}{4} \times \dfrac{1}{2} \times \dfrac{32}{3} & \text{[Simplifying 7 and 21 ⇒ Divide by 7 ]} \\\\ = \dfrac{1}{1} \times \dfrac{1}{2} \times \dfrac{8}{3} & \text{[Simplifying 32 and 4 ⇒ Divide by 4 ]} \\\\ = \dfrac{1 \times 1 \times 8}{1 \times 2 \times 3} \\\\ = \dfrac{8}{6} \\\\ = \dfrac{4}{3} & \text{[Dividing both by 2]} \\\\ = 1\dfrac{1}{3} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The answer is $1\dfrac{1}{3}$

Find the product :

$\dfrac{1}{19}\times91\times5\dfrac{11}{13}$

Answer

We have:

$\begin{array}{ll} \phantom{=} \dfrac{1}{19}\times91\times5\dfrac{11}{13} \\\\ = \dfrac{1}{19} \times \dfrac{91}{1} \times \dfrac{76}{13} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{1}{19} \times \dfrac{7}{1} \times \dfrac{76}{1} & \text{[Simplifying 91 and 13 ⇒ Divide by 13 ]} \\\\ = \dfrac{1}{1} \times \dfrac{7}{1} \times \dfrac{4}{1} & \text{[Simplifying 76 and 19 ⇒ Divide by 19 ]} \\\\ = \dfrac{1 \times 7 \times 4}{1 \times 1 \times 1} \\\\ = \dfrac{28}{1} \\\\ = 28 \end{array}$

∴ The answer is 28

Find the product :

$7\dfrac{1}{4}\times2\dfrac{3}{16}\times2\dfrac{2}{7}$

Answer

We have:

$\begin{array}{ll} \phantom{=} 7\dfrac{1}{4}\times2\dfrac{3}{16}\times2\dfrac{2}{7} \\\\ = \dfrac{29}{4} \times \dfrac{35}{16} \times \dfrac{16}{7} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{29}{4} \times \dfrac{35}{1} \times \dfrac{1}{7} & \text{[Simplifying 16 and 16 ⇒ Divide by 16 ]} \\\\ = \dfrac{29}{4} \times \dfrac{5}{1} \times \dfrac{1}{1} & \text{[Simplifying 35 and 7 ⇒ Divide by 7 ]} \\\\ = \dfrac{29 \times 5 \times 1}{4 \times 1 \times 1} \\\\ = \dfrac{145}{4} \\\\ = 36\dfrac{1}{4} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The answer is $36\dfrac{1}{4}$

Find the value of :

$\dfrac{3}{4}$ of $\dfrac{8}{9}$

Answer

We have:

$\dfrac{3}{4}$ of $\dfrac{8}{9}$

$\begin{array}{ll} = \dfrac{8}{9} \times \dfrac{3}{4} \\\\ = \dfrac{2}{9} \times \dfrac{3}{1} & \text{[Simplifying 8 and 4 ⇒ Divide by 4 ]} \\\\ = \dfrac{2}{3} \times \dfrac{1}{1} & \text{[Simplifying 3 and 9 ⇒ Divide by 3 ]} \\\\ = \dfrac{2 \times 1}{3 \times 1} \\\\ = \dfrac{2}{3} \end{array}$

∴ The answer is $\dfrac{2}{3}$

Find the value of :

$\dfrac{1}{2}$ of $2\dfrac{2}{3}$

Answer

We have:

$\dfrac{1}{2}$ of $2\dfrac{2}{3}$

$\begin{array}{ll} = \dfrac{8}{3} \times \dfrac{1}{2} & \text{[Converting mixed fraction to improper fraction]} \\\\ = \dfrac{4}{3} \times \dfrac{1}{1} & \text{[Simplifying 8 and 2 ⇒ Divide by 2 ]} \\\\ = \dfrac{4}{3} \\\\ = 1\dfrac{1}{3} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The answer is 1 $\dfrac{1}{3}$

Find the value of :

$\dfrac{4}{5}$ of 1 hour

Answer

We have:

$\dfrac{4}{5}$ of 1 hour

$\begin{array}{ll} 1 \text{ hour} = 60 \text{ minutes} & \text{[Converting hour into minutes]} \\\\ = \Big(60 \times \dfrac{4}{5}\Big) \text{ minutes} \\\\ = \Big(\dfrac{60}{1} \times \dfrac{4}{5}\Big) \text{ minutes} \\\\ = \Big(\dfrac{12}{1} \times \dfrac{4}{1}\Big) \text{ minutes} & \text{[Simplifying 60 and 5 ⇒ Divide by 5 ]} \\\\ = \dfrac{48}{1} \text{ minutes} \\\\ = 48 \text{ minutes} \end{array}$

∴ The answer is 48 minutes

Find the value of :

$\dfrac{3}{5}$ of ₹1

Answer

We have:

$\dfrac{3}{5}$ of ₹1

₹1 = 100 paise $\hspace{1.5cm}$[Converting rupees into paise]

$\begin{array}{ll} = \Big(100 \times \dfrac{3}{5}\Big) \text{ paise} \\\\ = \Big(\dfrac{100}{1} \times \dfrac{3}{5}\Big) \text{ paise} \\\\ = \Big(\dfrac{20}{1} \times \dfrac{3}{1}\Big)\text{ paise} & \text{[Simplifying 100 and 5 ⇒ Divide by 5 ]} \\\\ = \Big(\dfrac{20 \times 3}{1 \times 1}\Big)\text{ paise} \\\\ = \dfrac{60}{1}\text{ paise} \\\\ = 60\text{ paise} \end{array}$

∴ The answer is 60 paise

Find the value of :

$\dfrac{8}{15}$ of $1\dfrac{1}{2}$ metres

Answer

We have:

$\dfrac{8}{15}$ of $1\dfrac{1}{2}$ metres

$1\dfrac{1}{2}$ metres = $\dfrac{3}{2}$ metres $\hspace{1.5cm}$[Converting mixed to improper fraction]

1 meter = 100 cm

∴$\dfrac{3}{2}$ metres = 150 cm $\hspace{1.5cm}$[Converting meter to centimeter]

Now we have:

$\dfrac{8}{15}$ of 150 centimetres

$\begin{array}{ll} = \Big(150 \times \dfrac{8}{15}\Big) \text{ cm} \\\\ = \Big(\dfrac{150}{1} \times \dfrac{8}{15}\Big) \text{ cm} \\\\ = \Big(\dfrac{10}{1} \times \dfrac{8}{1}\Big) \text{ cm} & \text{[Simplifying 150 and 15 ⇒ Divide by 15 ]} \\\\ = \Big(\dfrac{10 \times 8}{1 \times 1}\Big) \text{ cm} \\\\ = \dfrac{80}{1} \text{ cm} \\\\ = 80 \text{ cm} \end{array}$

∴ The answer is 80 cm

Find the value of :

$\dfrac{5}{7}$ of $2\dfrac{1}{3}$kg

Answer

We have:

$\dfrac{5}{7}$ of $2\dfrac{1}{3}$kg

$2\dfrac{1}{3}\text{ kg} = \dfrac{7}{3}\text{ kg} \hspace{1.5cm}$[Converting mixed to improper fraction]

Now we have:

$\dfrac{5}{7}$ of $\dfrac{7}{3}$ kg

$\begin{array}{ll} = \Big(\dfrac{7}{3} \times \dfrac{5}{7}\Big) \text{ kg} \\\\ = \Big(\dfrac{1}{3} \times \dfrac{5}{1}\Big) \text{ kg} & \text{[Simplifying 7 and 7 ⇒ Divide by 7 ]} \\\\ = \Big(\dfrac{1 \times 5}{3 \times 1}\Big) \text{ kg} \\\\ = \dfrac{5}{3} \text{ kg} \\\\ = 1\dfrac{2}{3} \text{ kg} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The answer is $1\dfrac{2}{3}$ kg

A car can travel $12\dfrac{1}{2}$km in 1 litre of petrol. How much distance can it travel in $42\dfrac{3}{5}$ litres of petrol?

Answer

Given:

Distance travelled in 1 litre of petrol = $12\dfrac{1}{2}$ km

= $\dfrac{25}{2}$ km $\hspace{1cm}$ [Converting mixed to improper fraction]

Distance travelled in $42\dfrac{3}{5}$ litres of petrol = ?

Number of litres = $42\dfrac{3}{5}$ = $\dfrac{213}{5}$ litres

Distance travelled in $\dfrac{213}{5}$ litres of petrol = (Distance travelled in 1 litre) x (Number of litres)

Substituting the values in above, we get:

$\begin{array}{ll} = \Big(\dfrac{25}{2} \times \dfrac{213}{5}\Big)\text{ km} \\\\ = \Big(\dfrac{5}{2} \times \dfrac{213}{1}\Big)\text{ km} & \text{[Simplifying 25 and 5 ⇒ Divide by 5 ]} \\\\ = \Big(\dfrac{5 \times 213}{2 \times 1}\Big) \text{ km} \\\\ = \dfrac{1065}{2}\text{ km} \\\\ = 532 \dfrac{1}{2}\text{ km} & \text{[Converting improper fraction to mixed fraction]} \end{array}$

∴ The distance travelled in $42\dfrac{3}{5}$ litres of petrol = $532\dfrac{1}{2}$ km.

A graphic designer charges ₹$27\dfrac{3}{5}$ for each diagram. Find the amount he will charge if he designs 186 diagrams for a book.

Answer

Given:

Charge of 1 diagram = ₹$27\dfrac{3}{5} = ₹\dfrac{138}{5}$

Charge for 186 diagrams = ?

Number of diagrams = 186

Charge for 186 diagrams = (Charge of 1 diagram) x (Number of diagrams)

Substituting the values in above, we get:

$\begin{array}{ll} = ₹\Big(\dfrac{138}{5} \times 186\Big) \\\\ = ₹\Big(\dfrac{138 \times 186}{5 \times 1}\Big) \\\\ = ₹\dfrac{25668}{5} \\\\ = ₹5133\dfrac{3}{5} & \text{[Converting improper fraction to mixed fraction]} \end{array}$

∴ The charge for 186 diagrams = ₹ $5133\dfrac{3}{5}$.

If a cloth costs ₹$715\dfrac{1}{4}$ per metre, find the cost of $3\dfrac{2}{5}$ metres of this cloth.

Answer

Cost of 1 meter of cloth = ₹$715\dfrac{1}{4} = ₹\dfrac{2861}{4}$

Cost of $3\dfrac{2}{5}$ meters of cloth = ?

Length of cloth = $3\dfrac{2}{5} \text{ meters} = \dfrac{17}{5}$ meters

Cost of $3\dfrac{2}{5}$ meters of cloth = (Cost of 1 meter of cloth) x (Length of cloth)

Substituting the values in above, we get:

$\begin{array}{ll} = ₹\Big(\dfrac{2861}{4} \times \dfrac{17}{5}\Big) \\\\ = ₹\Big(\dfrac{2861 \times 17}{4 \times 5}\Big) \\\\ = ₹\dfrac{48637}{20} \\\\ = ₹2431\dfrac{17}{20} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The cost of $3\dfrac{2}{5}$ meters of cloth = ₹ $2431\dfrac{17}{20}$.

Advertising in a magazine costs ₹$1472\dfrac{2}{5}$ per square inch. Find the cost of an advertisement of $17\dfrac{6}{7}$ square inch.

Answer

Cost of an advertisement of 1 square inch = ₹$1472\dfrac{2}{5} = ₹\dfrac{7362}{5}$

Cost of an advertisement of $17\dfrac{6}{7}$ square inch = ?

Area of an advertisement = $17\dfrac{6}{7}$ square inch

Cost of an advertisement of $17\dfrac{6}{7}$ square inch = (Cost of an advertisement of 1 square inch) x (Area of an advertisement)

Substituting the values in above, we get:

$\begin{array}{ll} = ₹\Big(\dfrac{7362}{5} \times 17\dfrac{6}{7}\Big) \\\\ = ₹\Big(\dfrac{7362}{5} \times \dfrac{125}{7}\Big) & \text{[Converting mixed to improper fraction]} \\\\ = ₹\Big(\dfrac{7362}{1} \times \dfrac{25}{7}\Big) \\\\ = ₹\Big(\dfrac{7362 \times 25}{1 \times 7}\Big) \\\\ = ₹\dfrac{184050}{7} \\\\ = ₹26292 \dfrac{6}{7} & \text{[Converting improper fraction to mixed fraction]} \end{array}$

∴ The cost of an advertisement of $17\dfrac{6}{7}$ square inch = ₹ $26292\dfrac{6}{7}$.

A car travelled from city A to city B with a uniform speed of $52\dfrac{2}{7}$ km per hour. Find the distance between the two cities, if it took $4\dfrac{3}{8}$ hours for the car to reach city B from city A.

Answer

Given:

Speed = $52\dfrac{2}{7}$ km/hour

Time = $4\dfrac{3}{8}$ hours

Distance = ?

We know the formula,

Distance = Speed x Time

By substituting the values we get,

$\begin{array}{ll} \text{Distance} = \Big(52\dfrac{2}{7} \times 4\dfrac{3}{8}\Big) \text{ km} & \space \\\\ = \Big(\dfrac{366}{7} \times \dfrac{35}{8}\Big) \text{ km} & \text{[Converting mixed to improper fraction]} \\\\ = \Big(\dfrac{366}{1} \times \dfrac{5}{8}\Big)\text{ km} & \text{[Simplifying 35 and 7 ⇒ Divide by 7 ]} \\\\ = \dfrac{366 \times 5}{1 \times 8}\text{ km} \\\\ = \dfrac{1830}{8} \text{ km} \\\\ = 228\dfrac{3}{4}\text{ km} & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The distance between two cities is $228\dfrac{3}{4}$ km.

The length of a rectangular plot of land is $29\dfrac{3}{7}$m. If its breadth is $12\dfrac{8}{11}$m, find its area.

Answer

Given:

Length of a rectangular plot of land = $29\dfrac{3}{7}$ m

Breadth of a rectangular plot of land = $12\dfrac{8}{11}$ m

Area of a rectangular plot of land = ?

We know the formula,

Area of rectangle = Length x Breadth

By substituting the values we get,

Area of rectangle = Length x Breadth

$\begin{array}{ll} = \Big(29\dfrac{3}{7} \times 12\dfrac{8}{11}\Big) \text{m}^2 \\\\ = \Big(\dfrac{206}{7} \times \dfrac{140}{11}\Big) \text{m}^2 & \text{ [Converting mixed to improper fraction]} \\\\ = \Big(\dfrac{206}{1} \times \dfrac{20}{11}\Big)\text{m}^2 & \text{[Simplifying 140 and 7 ⇒ Divide by 7 ]} \\\\ = \dfrac{206 \times 20}{1 \times 11} \text{m}^2 \\\\ = \dfrac{4120}{11}\text{m}^2 \\\\ = 374\dfrac{6}{11}\text{m}^2 & \text{[Converting improper to mixed fraction]} \end{array}$

∴ The area of a rectangular plot of land is $374\dfrac{6}{11}$ m²