$\displaystyle \text{ Question 1: Find the sum of } \frac{4}{7} \text{ of Rs.} 588 \text{ and } \frac{5}{9} \text{ of Rs. } 432$

$\displaystyle \text{ Sum } = \frac{4}{7} \times 588+ \frac{5}{9} \times 432=4 \times 84+5 \times 48=336+240=576$

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$\displaystyle \text{ Question 2: } \frac{16}{25} \text{ of a number is 816 . Find the number.}$

Let the number be $\displaystyle x$

$\displaystyle \Rightarrow \frac{16}{25} \times x=816$

$\displaystyle \Rightarrow x= \frac{25}{16} \times 816=25 \times 51=1275$

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Question 3: One-fifth of a number exceeds its one-ninth by $\displaystyle 16$. Find the number.

Let the number be $\displaystyle x$

$\displaystyle \Rightarrow \frac{1}{5} x- \frac{1}{9} x=16$

$\displaystyle \Rightarrow \frac{4}{45} x=16$

$\displaystyle \Rightarrow x= \frac{45}{4} \times 16=180$

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Question 4: If $\displaystyle 3/8$ of an estate is worth of Rs. $\displaystyle 3546000$, what is the value of the whole estate?

Let the value of the whole estate $\displaystyle = x$

$\displaystyle \Rightarrow \frac{3}{8} x=3546000$

$\displaystyle \Rightarrow x= \frac{8}{3} \times 3546000=9456000$

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$\displaystyle \text{ Question 5: The product of two fractions is } \frac{3}{20} . \text{ If one of the fractions is } \frac{13}{25} \\ \text{ , find the other.}$

Let one fraction be $\displaystyle x$

$\displaystyle \Rightarrow x \times \frac{13}{15} = \frac{3}{20}$

$\displaystyle \Rightarrow x= \frac{25}{13} \times \frac{3}{20} = \frac{15}{52}$

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$\displaystyle \text{ Question 6: } \frac{4}{15} \text{ of } \frac{5}{7} \text{ of a number is greater than } \frac{4}{9} \text{ of } \frac{2}{5} \text{ of the same number by 8.} \\ \\ \text{ Find the number.}$

Let the number be $\displaystyle x$

$\displaystyle \Rightarrow \frac{4}{15} \times \frac{5}{7} x- \frac{4}{9} \times \frac{2}{5} x=8$

$\displaystyle \Rightarrow \frac{4}{21} x- \frac{8}{45} x=8$

$\displaystyle \Rightarrow \frac{(60-56)}{315} x=8$

$\displaystyle \Rightarrow x=630$

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$\displaystyle \text{ Question 7: Find the fraction which is as much greater than} \frac{5}{8} \text{ as is less than} \frac{6}{7}$

Let the fraction be $\displaystyle x$

$\displaystyle \Rightarrow ( x- \frac{5}{8} )=( \frac{6}{7} -x )$

$\displaystyle \Rightarrow 2x= \frac{5}{8} + \frac{6}{7}$

$\displaystyle \Rightarrow 2x= \frac{83}{56}$

$\displaystyle \Rightarrow x= \frac{83}{112}$

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Question 8: On a certain day, $\displaystyle 810$ people visited the school exhibition. Out of this, $\displaystyle \frac{7}{15}$ were men and $\displaystyle \frac{11}{30}$ were ladies and the rest children. How many children visited the exhibition?

Let the number of children $\displaystyle = x$

$\displaystyle \Rightarrow \frac{7}{15} \times 810+ \frac{11}{30} \times 810+x=810$

$\displaystyle \Rightarrow \frac{25}{30} \times 810+x=810$

$\displaystyle \Rightarrow x=810- \frac{25}{30} \times 810$

$\displaystyle \Rightarrow x= \frac{(5 \times 810)}{30} =135$

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Question 9: A cake weighs $\displaystyle 2$ kg. If $\displaystyle \frac{2}{7}$ of its weight is flour, $\displaystyle \frac{1}{8}$ of its weight is sugar, $\displaystyle \frac{3}{14}$ of its weight is milk and the rest is nuts and plums, find the weight of nuts and plums in the cake.

Let the weight of nuts and plums be $\displaystyle x$ kg.

$\displaystyle \Rightarrow \frac{2}{7} \times 2+ \frac{1}{8} \times 2+ \frac{3}{14} \times 2+x=2$

$\displaystyle \Rightarrow x=2- \frac{5}{4} = \frac{3}{4} \ kg \ or \ 750$ gms

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Question 10: $\displaystyle \frac{1}{10}$ of a pole is colored white, $\displaystyle \frac{1}{20}$ is colored red, $\displaystyle \frac{1}{30}$ is colored green and the rest is colored black. If the length of the black portion is $\displaystyle 14$ m $\displaystyle 70$ cm, then find the length of the pole.

Let the length of the pole is $\displaystyle x$ meters

$\displaystyle \Rightarrow \frac{1}{10} \times x+ \frac{1}{20} \times x+ \frac{1}{30} \times x+14.70=x$

$\displaystyle \Rightarrow (\frac{1}{10} + \frac{1}{20} + \frac{1}{30} )x+14.70=x$

$\displaystyle \Rightarrow x- \frac{11}{60} x=14.70$

$\displaystyle \Rightarrow x= \frac{60}{49} \times 14.70=18$

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Question 11: A man had Rs. $\displaystyle 121200$. He gave $\displaystyle \frac{1}{3}$ of it to his wife, $\displaystyle \frac{3}{10}$ of the remainder to his daughter and $\displaystyle \frac{4}{7}$ of the remainder to his elder son. The rest of the money he gave to his younger son. How much money did the younger son get?

$\displaystyle \text{ Wife's Share in Rs.} = \frac{1}{3} \times 121200=40400$

$\displaystyle \text{ Daughter's Share in Rs.} = \frac{3}{10} \times (121200-40400)=24240$

$\displaystyle \text{ Elder Son's Share in Rs.} = \frac{4}{7} \times (121200-40400-24240)=32320$

$\displaystyle \text{ Younger Son's Share in Rs. }= (121200-40400-24240-32320)=24240$

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Question 12: If $\displaystyle \frac{1}{8}$ of a pencil is black, $\displaystyle \frac{1}{2}$ of the remaining is white and the remaining $\displaystyle 3 \frac{1}{2}$ cm is blue, find the total length of the pencil.

Let the length of the pencil be $\displaystyle x$ cm

$\displaystyle \frac{1}{8} \times x+ \frac{1}{2} \Big(x- \frac{1}{8} x \Big)+3 \frac{1}{2} =x$

$\displaystyle \Big( \frac{1}{8} + \frac{1}{2} - \frac{1}{16} \Big )x+ \frac{7}{2} =x$

$\displaystyle \frac{16}{2} = \frac{1}{2} x \ or \ x=8$

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Question 13: A man spends $\displaystyle \frac{3}{7}$ of his monthly income on food and clothes and $\displaystyle \frac{7}{10}$ of the remainder on house rent. What fraction of income is left with him? If money left is Rs. $\displaystyle 2400$, what is his monthly income?

Let the monthly income is $\displaystyle x$

$\displaystyle \frac{3}{7} x+ \frac{7}{10} \Big(x- \frac{3}{7} x \Big)+2400=x$

$\displaystyle \Big( \frac{3}{7} + \frac{7}{10} - \frac{3}{10} \Big) x+2400=x$

$\displaystyle \frac{58}{70} x+2400=x$

$\displaystyle x= \frac{70}{12} \times 2400 \ or \ x=14000$

Salary (Rs.) $\displaystyle = 14000$

$\displaystyle \text{ Fraction of Income left} = \frac{2400}{14000} = \frac{24}{140} = \frac{6}{35}$

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Question 14: A student was asked to multiply a given number by $\displaystyle \frac{8}{17}$ . Instead, he divided the given number by $\displaystyle \frac{8}{17}$ . His answer was $\displaystyle 225$ more than the correct answer. What was the given number?

Let the number be $\displaystyle x$

$\displaystyle \frac{17}{8} x- \frac{8}{17} x=225$

$\displaystyle \frac{17 \times 17-8 \times 8}{8 \times 17} x=225$

$\displaystyle x= \frac{8 \times 17225}{17 \times 17-8 \times 8} \ or \ x=136$

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Question 15: The highest score in an inning was $\displaystyle \frac{3}{11}$ of the total and the next highest was $\displaystyle \frac{3}{11}$ of the remainder. If the scores differed by $\displaystyle 63$, find the total score.

Let the score be $\displaystyle x$

$\displaystyle \frac{3}{11} x- \frac{3}{11} \Big(x- \frac{3}{11} x \Big)=63$

$\displaystyle x= \frac{121 \times 63}{9} \ or \ x=847$

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Question 16: In a school, $\displaystyle {\frac{1}{5}}^{th}$ of the girls and $\displaystyle {\frac{1}{8}}^{th}$ of the boys took part in a social camp. What fraction of the total strength took part in the camp?

$\displaystyle 1$ girl out of $\displaystyle 5$ girls and $\displaystyle 1$ boy out of $\displaystyle 8$ boys took part in the camp.

That means $\displaystyle 2$ pupils out of every $\displaystyle (5 + 8)$, took part in the camp.

Hence the fraction that took part in the camp is $\displaystyle \frac{2}{13}$

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Question 17: A train starts full of passengers. At the first station, it drops one-third of the passengers and takes $\displaystyle 280$ more. At the second station, it drops one-half of the new total and takes $\displaystyle 12$ more. On arriving at the third station, it is found to have $\displaystyle 248$ passengers. Find the number of passengers in the beginning.

Let the initial passengers be $\displaystyle x$

After the train leaves the first station the number of passengers in the train $\displaystyle = \frac{2}{3} x+280$

After the train leaves the Second Station, the number of passengers in the train $\displaystyle = \frac{1}{2} ( \frac{2}{3} x+280)+12$

Hence:

$\displaystyle \frac{1}{2} ( \frac{2}{3} x+280)+12=248$

Solving for $\displaystyle x$ we get $\displaystyle x=288$

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Question 18: In a fraction, the denominator exceeds the numerator by $\displaystyle 4$. If the numerator and the denominator are both increased by $\displaystyle 9$, the fraction becomes $\displaystyle v \frac{7}{8}$ . Find the original fraction.

Let the fraction be: $\displaystyle \frac{x}{(x+4)}$

If you add $\displaystyle 9$ to both the numerator and denominator then we have:

$\displaystyle \frac{(x+9)}{(x+4+9)} = \frac{7}{8}$

Solving for $\displaystyle x$ we get:

$\displaystyle x=19$ and the fraction $\displaystyle = \frac{19}{23}$

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Question 19: A water tank was $\displaystyle \frac{5}{8}$ filled with water. $\displaystyle \frac{7}{10}$ of this water was consumed and then $\displaystyle 125$ litres of water was added through a filling pipe. If the tank was now half-full, then find the capacity of the tank.

Let the capacity of the tank be $\displaystyle x$

$\displaystyle \frac{5}{8} x- \frac{7}{10} \times \frac{5}{8} x+125= \frac{1}{2} x$

Solving for $\displaystyle x$ we get $\displaystyle x= 400$ liter

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Question 20: A car covers $\displaystyle 313 \frac{31}{5}$ km in $\displaystyle 3 \frac{11}{15}$ hours. Find the speed of the car in km/hr.

$\displaystyle \text{ Speed } = \frac{(\text{Distance \ Covered})}{(\text{Time \ taken}) }$

$\displaystyle \text{ Speed } = \frac{ 313 \frac{3}{5}}{ 3 \frac{11}{15}}$

$\displaystyle \text{ Speed } = 84 \text{ km/hr}$

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Question 21: Find the least fraction that must be subtracted from the sum of $\displaystyle 2 \frac{1}{2} , 3 \frac{2}{3} , 4 \frac{3}{4}$ ,and $\displaystyle 5 \frac{4}{5}$ so that the result is a whole number.

$\displaystyle \text { Sum of } 2 \frac{1}{2} +3 \frac{2}{3} + 4 \frac{3}{4} +5 \frac{4}{5} = \frac{1003}{60} = 16 \frac{43}{60}$

Hence we should subtract $\displaystyle \frac{43}{60}$ so that the result is a whole number.

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Question 22: Rajan earns twice as much in the month of June as in each of the other months of the year. What part of his entire annual earnings was earned in June?

Let the monthly income in months other than June is $\displaystyle x$

Income in June $\displaystyle = 2x$

Therefore $\displaystyle \frac{2x}{13x} = \frac{2}{13}$ of the entire annual earnings was earned in June

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Question 23: A certain amount is distributed among $\displaystyle A, B \text{ and } C$. $\displaystyle A$ gets $\displaystyle \frac{3}{16}$ and $\displaystyle B$ gets $\displaystyle \frac{1}{4}$ of the whole amount. If $\displaystyle C$ gets $\displaystyle 8100$, then find the amount which $\displaystyle B$ gets.

Let the amount be $\displaystyle x$

$\displaystyle \frac{3}{16} x+ \frac{1}{4} x+8100=x$

$\displaystyle \text{ Solving for } x , \text{ we get } x=14400$

$\displaystyle \text{ Therefore B gets } \frac{1}{4} \times 14400=3600$

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Question 24: In a family, the father took $\displaystyle \frac{1}{4}$ of the cake and he had as much as each of the other members had. Find the total number of family members.

Let the amount of cake is $\displaystyle x$ and the number of family members be $\displaystyle n$

Each family member gets $\displaystyle \frac{x}{n}$ cake

Therefore

$\displaystyle \frac{x}{n} = \frac{1}{4} \text{ or } n=4$

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Question 25: A sum of Rs. $\displaystyle 20060$ has been divided among $\displaystyle A, B \text{ and } C$ such that $\displaystyle A$ gets $\displaystyle \frac{2}{3}$ of what $\displaystyle B$ gets and $\displaystyle B$ gets $\displaystyle \frac{1}{4}$ of what $\displaystyle C$ gets. Find B’s share.

Let $\displaystyle C$ get $\displaystyle x$
$\displaystyle x+ \frac{1}{4} x+ \frac{2}{3} \times \frac{1}{4} x=20060$
Solving for $\displaystyle x, x= 14160$
B’s Share $\displaystyle = \frac{1}{4} \times 14160=3540$