Problems On Numbers
Name: _____________________
Date: _____________________
Instructions: Answer all questions. Write your answers clearly in the space provided.
The sum of two numbers is 40 and their product is 375. What will be the sum of their reciprocals ?
A two-digit number is 7 times the sum of its two digits. The number that is formed by reversing its digits is 18 less than the original number. What is the number ?
If the difference between the reciprocal of a positive proper fraction and the fraction itself be $$frac{9}{20}$$, then the fraction is :
The sum of two numbers is 75 and their difference is 25. The product of the two numbers is :
Three-forth of a number is 60 more than its one-third. The number is :
The product of two natural numbers is 17. Then, the sum of the reciprocals of their squares is :
A number whose fifth part increase by 4 is equal to its fourth part diminished by 10, is :
If $$2frac{1}{2}$$ is added tp a number and the sum multiplied by $$4frac{1}{2}$$ and 3 is added to the product and the sum is divided by $$1frac{1}{5}$$, the quotient becomes 25. What is the number ?
The product of two numbers is 120 and the sum of their square is 289. The sum of the numbers is :
If the digit in the unit's place of a two-digit number is halved and the digit in the ten's place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true ?
If one-third of one-fourth of a number is 15, then three-tenth of that number is:
Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:
The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?
A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is:
The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?
The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is:
A number consists of two digits. If the digits interchange places and the new number is added to the original number, then the resulting number will be divisible by:
In a two-digit, if it is known that its unit's digit exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:
Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.
If a number is multiplied by two-third of itself the value so obtained is 864. What is the number ?
The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is :
If the product of three consecutive integers is 120, then the sum of the integers is :
A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is :
The sum of the squares of three numbers is 138, while the sum of their product taken two at a time is 131. Their sum is :
The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the numbers is ?
A number when multiplied by 13 is increased by 180. The number is :
A positive number when decreased by 4 is equal to 21 times the reciprocal of the number. The number is ?
The difference between two numbers is 1365. When the large number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is :
A number of two digits has 3 for its unit's digit, and the sum of digits is $$frac{1}{7}$$ of the itself. The number is :
The sum of the numerator and denominator of a fraction is 11. If 1 is added to the numerator and 2 is subtracted from the denominator, it becomes $$frac{2}{3}$$. The fraction is :
In a Mathematics examination the number scored by 5 candidates are 5 successive odd integers. If their total marks are 185, the highest score is :
Three numbers are in in the ratio of 3 : 4 : 6 and their product is 1944. The largest of these numbers is :
The sum of the squares of two numbers is 3341 and the difference of their squares is 891. The numbers are :
In a two-digit number, if it is known that its unit's digits exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is :
The difference between the numerator and the denominator of a fraction is 5. If 5 is added to its denominator, the fraction is decreased by $$1frac{1}{4}$$. Find the value of the fraction.
A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is :
The ratio between a two-digit number and the sum of the digits of that number is 4 : 1. If the digit in the unit's place is 3 more than the digit in the ten's place, then the number is ?
The difference between two positive integers is 3. If the sum of their squares is 369, then the sum of the numbers is :
A number consists of two digits. If the digits interchange place and the new number is added to the original number, then the resulting number will be divisible by :
A number consists of two digits such that the digit in the ten's place is less by 2 than the digit in the unit's place. Three times the number added to $$frac{6}{7}$$ times the number obtained by reversing the digits equals 108. The sum of the digits in the number is :
If (73) 2 is subtracted from the square of a number, the answer so obtained is 5075. What is the number ?
The sum of three consecutive odd numbers is 20 more than the first of these numbers. What is the middle number ?
The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller these two numbers ?
If the square of a two-digit number is reduced by the square of the number formed by reversing the digits of the number, the final result is :
Three times the first of three consecutive odd integers is 3 more than twice the third. The integer is :
In a three-digit number, the digit in the unit's place is 75% of the digit in the ten's place. The digit in the ten's place is greater than the digit in the hundred's place by 1. If the sum of the digits in the ten's place and the hundred's place is 15. What is the number ?
The sum of two numbers is 37 and the difference of their squares is 185, then the difference between the two numbers is :
The difference between $$frac{3}{5}$$th of $$frac{2}{3}$$rd of a number and $$frac{2}{5}$$th of $$frac{1}{4}$$th of the same number is 288. What is the number ?
The sum and product of two numbers are 12 and 35 respectively. The sum of their reciprocals will be :
The difference between two numbers is 16. If one-third of the smaller number is greater than one-seventh of the larger number by 4, then the two numbers are :
If a number of two digits is k times the sum of its digits, then the number formed by interchanging the digits is the sum of the digits multiplied by :
The product of two fractions is $$frac{14}{15}$$ and their quotient is $$frac{35}{24}$$. The greater fraction is :
A man bought some eggs of which 10% are rotten. He gives 80% of the remainder to his neighbours. Now he is left out with 36 eggs. How many eggs he bought ?
A number is double and 9 is added. If the resultant is trebled, it becomes 75. What is that number ?
The sum of a positive number and its reciprocal is thrice the difference of the number and its reciprocal. The number is :
If the numerator of a fraction is increased by $$frac{1}{4}$$ and the denominator is decreased byy $$frac{1}{3}$$, the new fraction obtained is $$frac{33}{64}$$. What was the original fraction ?
If doubling a number and adding 20 to the result gives the same answer as multiplying the number by 8 and taking away 4 from the product, the number is :
Out of six consecutive natural numbers if the sum of first three is 27, what is the sum of the other three ?
The difference between a two-digit number and the number obtained by interchanging the two digits is 63. Which is the smaller of the two numbers ?
If the numerator of a fraction is increased by 200% and the denominator is increased by 300%, the resultant fraction is $$frac{15}{26}$$. What was the original fraction ?
If 50 is subtracted from two-third of number, the result is equal to sum of 40 and one-fourth of that number. What is the number ?
The sum of two numbers is 25 and their difference is 13. Find their product :
The sum of seven consecutive numbers is 175. What is the difference between twice the largest number and thrice the smallest number ?
The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two digit of the number is 12, then what is the original number ?
A fraction is such that if the double of the numerator and the triple pf the denominator is changed by +10 percent and -30 percent respectively, then we get 11 percent of $$frac{16}{21}$$. Find the fraction :
A student was asked to divide the half of a certain number by 6 and the other half by 4 and then to add the two quantities so obtained. Instead of doing so the student divided the number by 5 and the result fell short by 4. The given number was ?
If the sum of numbers is 33 and their difference is 15, the smaller number is ?
In a two-digit positive number, the digit in the unit's place is equal to the square of the digit in ten's place, and the difference between the number and the number obtained by interchanging the digits is 54. What is 40% of the original number ?
243 has been divided into three parts such that half of the first part, one-third of the second part and one-fourth of the third part are equal. The largest part is :
If the sum of a number and its square is 182, what is the number ?
The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is :
The sum of three consecutive odd numbers and three consecutive even numbers together is 231. Also, the smallest odd number is 11 less than the smallest even number. What is the sum of the largest odd number and the largest even number ?
A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. The number is :
The sum of four numbers is 64. If you add 3 to the first number, 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results are equal. What is the difference between the largest and the smallest of the original numbers ?
The sum of five consecutive odd numbers is 575. What is the sum of the next set of five consecutive odd numbers ?
If the numerator of a fraction is increased by 2 and the denominator is increased by 3, the fraction becomes $$frac{7}{9}$$ and if both the numerator as well as he denominator are decreased by 1, the fraction becomes $$frac{4}{5}$$. What is the original fraction ?
The difference of two numbers is 20% of the larger number. If the smaller number is 12, the larger one is :
Two numbers are such that the ratio between them is 4 : 7. If each is increased by 4, the ratio becomes 3 : 5. The lager number is ?
the difference between two numbers is 3 and the difference between their squares is 63. Which is the larger number ?
twenty times a positive integer is less than its square by 96. What is the integer ?
There are two numbers such that the sum of twice the first number and thrice the second number is 100 and the sum of thrice the first number and twice the second number is 120. Which is the larger number ?
If one-seventh of a number exceeds its eleventh part by 100, then the number is :
The sum of three numbers is 264. If the first number be twice the second and third number be be one-third of the first, then the second number is :
A, B, C, D and E are five consecutive odd numbers. The sum of A and C is 146. What is the value of E ?
The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of the number ?
The digit in the unit's place of a numbers is equal to the digit in the ten's place of half of that number and the digit in the ten's place of that number is less than the digit in unit's place of half of the number by 1. If the sum of the digits of the number is 7, then what is the number ?
Of the three number, the sum of the first two is 73, the sum of the second and the third is 77 and the sum of the third and thrice the first is 104. The third number is ?
The difference between a number and its three-fifths is 50, What is the number ?
Two-third of a positive number and $$frac{25}{216}$$ of its reciprocal are equal, The number is :
The difference between two integers is 5. Their product is 500. Find the numbers.
What is the sum of two consecutive even numbers, the difference of whose squares is 84 ?
A certain number of two digits is three times the sum of its digits and if 45 be added to it, the digits are reversed. The number is ?
If the product of two numbers is 5 and one of the number is $$frac{3}{2}$$, then sum of two numbers is :
If a number is added to two-fifths of itself, the value so obtained is 455. What is the number ?
Find the whole number which when increased by 20 is equal to 69 times the reciprocal of the number.
The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is:
The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is:
A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed. The number is:
The sum of two number is 25 and their difference is 13. Find their product.
What is the sum of two consecutive even numbers, the difference of whose squares is 84?
The sum of twice a number and three times of 42 is 238. What is the sum of thrice the number and two times of 42 ?
By how much is $$frac{3}{4}$$th of 568 lesser than $$frac{7}{8}$$th of 1008 ?
Thrice the square of a natural number decreased by 4 times the number is equal to 50 more than the number. The number is :
What is the greater of the two numbers whose product is 1092 and the sum of the two numbers exceeds their difference by 42 ?
The product of three consecutive even numbers when divided by 8 is 720. The product of their square roots is :
Answer Key
& herefore frac{1}{x} + frac{1}{y} = frac{{x + y}}{{xy}} cr
& ,,,,,,,,,,,,,,,,,,,,,,, = frac{{40}}{{275}} cr
& ,,,,,,,,,,,,,,,,,,,,,,, = frac{8}{{75}} cr} $$
& herefore 10x + y = 7left( {x + y}
ight) cr
& Leftrightarrow 3x = 6y cr
& Leftrightarrow x = 2y cr} $$ Number formed by reversing the digits = 10y + x $$eqalign{
& herefore left( {10x + y}
ight) - left( {10y + x}
ight) = 18 cr
& Leftrightarrow 9x - 9y = 18 cr
& Leftrightarrow x - y = 2 cr
& Leftrightarrow 2y - y = 2 cr
& Leftrightarrow y = 2 cr
& { ext{So, }}x = 2y = 4 cr} $$ Hence, ∴ Required number = 10x + y = 40 + 2 = 42
& Leftrightarrow frac{1}{a} - a = frac{9}{{20}} cr
& Leftrightarrow frac{{1 - {a^2}}}{a} = frac{9}{{20}} cr
& Leftrightarrow 20 - 20{a^2} = 9a cr
& Leftrightarrow 20{a^2} + 9a - 20 = 0 cr
& Leftrightarrow 20{a^2} + 25a - 16a - 20 = 0 cr
& Leftrightarrow 5aleft( {4a + 5}
ight) - 4left( {4a + 5}
ight) = 0 cr
& Leftrightarrow left( {4a + 5}
ight)left( {5a - 4}
ight) = 0 cr
& Leftrightarrow a = frac{4}{5},,,,,,,,left[ {x08ecause a
e - frac{5}{4}}
ight] cr} $$
& a + b = 75 cr
& a - b = 25 cr
& x08ecause {left( {a + b}
ight)^2} - {left( {a - b}
ight)^2} = 4ab cr
& Rightarrow {75^2} - {25^2} = 4ab cr
& Rightarrow 4ab = left( {75 + 25}
ight)left( {75 - 25}
ight) cr
& left[ {x08ecause {a^2} - {b^2} = left( {a + b}
ight)left( {a - b}
ight)}
ight] cr
& Rightarrow 4ab = 100 imes 50 cr
& Rightarrow ab = frac{{100 imes 50}}{4} cr
& Rightarrow ab = 1250 cr} $$
& Leftrightarrow frac{3}{4}x - frac{1}{3}x = 60 cr
& Leftrightarrow frac{{5x}}{{12}} = 60 cr
& Leftrightarrow x = left( {frac{{60 imes 12}}{5}}
ight) cr
& Leftrightarrow x = 144 cr} $$
& = frac{1}{{{a^2}}} + frac{1}{{{b^2}}} cr
& = frac{{{a^2} + {b^2}}}{{{a^2}{b^2}}} cr
& = frac{{{1^2} + {{left( {17}
ight)}^2}}}{{{{left( {1 imes 17}
ight)}^2}}} cr
& = frac{{290}}{{289}} cr} $$
& Leftrightarrow frac{x}{5} + 4 = frac{x}{4} - 10 cr
& Leftrightarrow frac{x}{4} - frac{x}{5} = 14 cr
& Leftrightarrow frac{x}{{20}} = 14 cr
& Leftrightarrow x = 14 imes 20 cr
& Leftrightarrow x = 280 cr} $$
& Leftrightarrow frac{{4frac{1}{2}left( {x + 2frac{1}{2}}
ight) + 3}}{{1frac{1}{5}}} = 25 cr
& Leftrightarrow frac{{frac{9}{2}left( {x + frac{5}{2}}
ight) + 3}}{{frac{6}{5}}} = 25 cr
& Leftrightarrow frac{{9x}}{2} + frac{{45}}{4} + 3 = 25 imes frac{6}{5} cr
& Leftrightarrow frac{{9x}}{2} + frac{{45}}{4} + 3 = 30 cr
& Leftrightarrow frac{{9x}}{2} = 30 - frac{{57}}{4} cr
& Leftrightarrow frac{{9x}}{2} = frac{{63}}{4} cr
& Leftrightarrow x = left( {frac{{63}}{4} imes frac{2}{9}}
ight) cr
& Leftrightarrow x = frac{7}{2} cr
& Leftrightarrow x = 3frac{1}{2} cr} $$
& herefore {left( {x + y}
ight)^2} cr
& = {x^2} + {y^2} + 2xy cr
& = 289 + 240 cr
& = 529 cr
& herefore x + y cr
& = sqrt {529} cr
& = 23 cr} $$
& = 10 imes 2x + frac{y}{2} cr
& = 20x + frac{y}{2} cr} $$ $$eqalign{
& herefore 20x + frac{y}{2} = 10y + x cr
& Leftrightarrow 40x + y = 20y + 2x cr
& Leftrightarrow 38x = 19y cr
& Leftrightarrow y = 2x cr} $$ So, the unit's digit is twice the ten's digit.
& { ext{Let}},{ ext{the}},{ ext{number}},{ ext{be}},x cr
& { ext{Then}},,frac{1}{3},{ ext{of}},frac{1}{{4,}},{ ext{of}},x = 15cr
& Rightarrow , x = 15 imes 12 = 180 cr
& { ext{So,}},{ ext{required}},{ ext{number}} cr
& = {frac{3}{{10}} imes 180} cr
& = 54 cr} $$
& { ext{Then,}} cr
& left( {10x + frac{8}{x}}
ight) + 18 = 10 imes frac{8}{x} + x cr
& Rightarrow 10{x^2} + 8 + 18x = 80 + {x^2} cr
& Rightarrow 9{x^2} + 18x - 72 = 0 cr
& Rightarrow {x^2} + 2x - 8 = 0 cr
& Rightarrow left( {x + 4}
ight)left( {x - 2}
ight) = 0 cr
& Rightarrow x = 2 cr} $$ ∴ first digit will be 2 and second digit will be 4. i.e digit is 24.
& { ext{Let}},{ ext{the}},{ ext{number}},{ ext{be}},x cr
& { ext{Then}},,x + 17 = frac{{60}}{x} cr
& Rightarrow {x^2} + 17x - 60 = 0 cr
& Rightarrow (x + 20)(x - 3) = 0 cr
& Rightarrow x = 3 cr} $$
& Leftrightarrow x imes frac{2}{3}x = 864 cr
& Leftrightarrow frac{2}{3}{x^2} = 864 cr
& Leftrightarrow {x^2} = left( {frac{{864 imes 3}}{2}}
ight) cr
& Leftrightarrow {x^2} = 1296 cr
& Leftrightarrow x = sqrt {1296} cr
& Leftrightarrow x = 36 cr} $$
& Leftrightarrow frac{{xy}}{{left( {frac{x}{y}}
ight)}} = frac{{9375}}{{15}} cr
& Leftrightarrow {y^2} = 625 cr
& Leftrightarrow y = 25 cr
& herefore x = 15y cr
& Rightarrow x = 15 imes 25 cr
& Rightarrow x = 375 cr} $$ ∴ Sum of the numbers : = 375 + 25 = 400
& 120: cr
& = 2 imes 2 imes 2 imes 3 imes 5 cr
& = left( {2 imes 2}
ight) imes 5 imes left( {2 imes 3}
ight) cr
& = 4 imes 5 imes 6 cr} $$ Clearly, the three consecutive integers whose product is 120 are 4, 5 and 6 Required sum : = 4 + 5 + 6 = 15
& Leftrightarrow left( {10x + frac{8}{x}}
ight) + 18 = 10 imes frac{8}{x} + x cr
& Leftrightarrow 10{x^2} + 8 + 18x = 80 + {x^2} cr
& Leftrightarrow 9{x^2} + 18x - 72 = 0 cr
& Leftrightarrow {x^2} + 2x - 8 = 0 cr
& Leftrightarrow left( {x + 4}
ight)left( {x - 2}
ight) = 0 cr
& Leftrightarrow x = 2 cr} $$ So, ten's digit = 2 and unit's digit = 4 Hence, required number = 24
& {a^2} + {b^2} = 100.....( ext{i}) cr
& {a^2} - {b^2} = 28.....( ext{ii}) cr} $$ By adding (i) and (ii), we get : $$eqalign{
& herefore {a^2} + {b^2} + {a^2} - {b^2} = 100 + 28 cr
& Rightarrow 2{a^2} = 128 cr
& Rightarrow {a^2} = frac{{128}}{2} cr
& Rightarrow a = sqrt {64} cr
& Rightarrow a = 8 cr} $$ From equation (i) $$eqalign{
& Rightarrow {8^2} + {b^2} = 100 cr
& Rightarrow {b^2} = 100 - 64 cr
& Rightarrow b = sqrt {36} cr
& Rightarrow b = 6 cr
& herefore a + b = 8 + 6 cr
& ,,,,,,,,,,,,,,,,,,, = 14 cr} $$
& Rightarrow 13x = x + 180 cr
& Rightarrow 12x = 180 cr
& Rightarrow x = frac{{180}}{{12}} cr
& Rightarrow x = 15 cr} $$
& Leftrightarrow x - 4 = frac{{21}}{x} cr
& Leftrightarrow {x^2} - 4x - 21 = 0 cr
& Leftrightarrow left( {x - 7}
ight)left( {x + 3}
ight) = 0 cr
& Leftrightarrow x = 7 cr} $$
& Leftrightarrow x + y = 11.....( ext{i}) cr
& Leftrightarrow frac{{x + 1}}{{y - 2}} = frac{2}{3} cr
& Leftrightarrow 3left( {x + 1}
ight) = 2left( {y - 2}
ight) cr
& Leftrightarrow 3x - 2y = - 7.....( ext{ii}) cr} $$ Solving (i) and (ii), we get: x = 3 and y = 8 So, the fraction is $$frac{3}{8}$$
& Leftrightarrow 185 = 5x + 20 cr
& Leftrightarrow 5x = 185 - 20 cr
& Leftrightarrow 5x = 165 cr
& Leftrightarrow x = 33 cr} $$ Highest number = 33 + 8 = 41
& Leftrightarrow 3x imes 4x imes 6x = 1944 cr
& Leftrightarrow 72{x^3} = 1944 cr
& Leftrightarrow {x^3} = 27 cr
& Leftrightarrow x = 3 cr} $$ ∴ Largest number = 6x = 18
& herefore left( {11x + 2}
ight)left( {2x + 2}
ight) = 144 cr
& Leftrightarrow 22{x^2} + 26x - 140 = 0 cr
& Leftrightarrow 11{x^2} + 13x - 70 = 0 cr
& Leftrightarrow left( {x - 2}
ight)left( {11x + 35}
ight) = 0 cr
& Leftrightarrow x = 2 cr} $$ Hence, required number: = 11x + 2 = 11× 2 + 2 = 24
& Leftrightarrow frac{{x + 5}}{x} - frac{{x + 5}}{{x + 5}} = frac{5}{4} cr
& Leftrightarrow frac{{x + 5}}{x} = frac{5}{4} + 1 cr
& Leftrightarrow frac{{x + 5}}{x} = frac{9}{4} cr
& Leftrightarrow frac{{x + 5}}{x} = 2frac{1}{4} cr} $$ So, the fraction is $$2frac{1}{4}$$
& Leftrightarrow left( {frac{1}{5}x + 4}
ight) = left( {frac{1}{4}x - 10}
ight) cr
& Leftrightarrow frac{x}{{20}} = 14 cr
& Leftrightarrow x = 14 imes 20 cr
& Leftrightarrow x = 280 cr} $$
& herefore frac{{11x + 3}}{{2x + 3}} = frac{4}{1} cr
& Leftrightarrow 11x + 3 = 8x + 12 cr
& Leftrightarrow 3x = 9 cr
& Leftrightarrow x = 3 cr} $$ Hence, Required number = 11x + 3 = 11 × 3 + 3 = 36
& Leftrightarrow {x^2} + {left( {x + 3}
ight)^2} = 369 cr
& Leftrightarrow {x^2} + {x^2} + 9 + 6x = 369 cr
& Leftrightarrow 2{x^2} + 6x - 360 = 0 cr
& Leftrightarrow {x^2} + 3x - 180 = 0 cr
& Leftrightarrow left( {x + 15}
ight)left( {x - 12}
ight) = 0 cr
& Leftrightarrow x = 12 cr} $$ So, the numbers are 12 and 15 ∴ Required sum = (12 + 15) = 27
ight) + x}
ight] + frac{6}{7}$$ xa0 xa0 $$left[ {10x + left( {x - 2}
ight)}
ight]$$ xa0 $$ = 108$$ ⇔ 231x - 420 + 66x - 12 = 756 ⇔ 297x = 1188 ⇔ x = 4 Hence, sum of the digits : = x + (x - 2) = 2x - 2 = 6
& Leftrightarrow {x^2} - {left( {73}
ight)^2} = 5075 cr
& Leftrightarrow {x^2} - 5329 = 5075 cr
& Leftrightarrow {x^2} = 5075 + 5329 cr
& Leftrightarrow {x^2} = 10404 cr
& Leftrightarrow x = sqrt {10404} cr
& Leftrightarrow x = 102 cr} $$
& Leftrightarrow x + left( {x + 2}
ight) + left( {x + 4}
ight) = x + 20 cr
& Leftrightarrow 2x = 14 cr
& Leftrightarrow x = 7 cr} $$ ∴ Middle number : = x + 2 = 9
& Leftrightarrow xleft( {28 - x}
ight) = 192 cr
& Leftrightarrow {x^2} - 28x + 192 = 0 cr
& Leftrightarrow left( {x - 16}
ight)left( {x - 12}
ight) = 0 cr
& Leftrightarrow x = 16{ ext{ or }}x = 12 cr} $$ So, the numbers are 16 and 12
ight)^2} - {left( {10y + x}
ight)^2}$$ $$ = left( {100{x^2} + {y^2} + 20xy}
ight) - $$ xa0 xa0 $$left( {100{y^2} + {x^2} + 20xy}
ight)$$ $$ = 99left( {{x^2} - {y^2}}
ight)$$ xa0 xa0 which is divisible by both 9 and 11
& = 75\% { ext{ of }}left( {x + 1}
ight) cr
& = frac{3}{4}left( {x + 1}
ight) cr} $$ $$eqalign{
& herefore left( {x + 1}
ight) + x = 15 cr
& Leftrightarrow 2x = 14 cr
& Leftrightarrow x = 7 cr} $$ So, hundred's digit = 7 Ten's digit = 8 Unit's digit : $$eqalign{
& = frac{3}{4}left( {x + 1}
ight) cr
& = frac{3}{4}left( {7 + 1}
ight) cr
& = frac{3}{4}left( 8
ight) cr
& = 6 cr} $$ Hence, required number = 786
& a + b = 37& {a^2} - {b^2} = 185 cr
& Rightarrow left( {a + b}
ight)left( {a - b}
ight) = 185 cr
& Rightarrow 37left( {a - b}
ight) = 185 cr
& Rightarrow a - b = frac{{185}}{{37}} cr
& Rightarrow a - b = 5 cr} $$
& frac{3}{5}{ ext{of }}frac{2}{3}{ ext{of }}x - frac{2}{5}{ ext{of }}frac{1}{4}{ ext{of }}x = 288 cr
& Leftrightarrow left( {x imes frac{3}{5} imes frac{2}{3}}
ight) - left( {x imes frac{2}{5} imes frac{1}{4}}
ight) = 288 cr
& Leftrightarrow frac{2}{5}x - frac{1}{{10}}x = 288 cr
& Leftrightarrow frac{{3x}}{{10}} = 288 cr
& Leftrightarrow x = left( {frac{{288 imes 10}}{3}}
ight) cr
& Leftrightarrow x = 960 cr} $$
& x + y = 12,, & ,, xy = 35 cr
& herefore frac{1}{x} + frac{1}{y} = frac{{x + y}}{{xy}} = frac{{12}}{{35}} cr} $$
& Leftrightarrow frac{x}{3} - frac{{left( {x + 16}
ight)}}{7} = 4 cr
& Leftrightarrow 7x - 3left( {x + 16}
ight) = 84 cr
& Leftrightarrow 4x = 84 + 48 cr
& Leftrightarrow 4x = 132 cr
& Leftrightarrow x = 33 cr} $$ Hence, the numbers are 33 and 49
& herefore 10x + y = kleft( {x + y}
ight) cr
& Rightarrow k = frac{{10x + y}}{{x + y}} cr} $$ Number formed by interchanging the digits = 10y + x Let, 10y + x = h(x + y) Then, $$eqalign{
& h = frac{{10y + x}}{{x + y}} cr
& ,,,,, = frac{{11left( {x + y}
ight) - left( {10x + y}
ight)}}{{x + y}} cr
& ,,,,, = 11 - frac{{10x + y}}{{x + y}} cr
& ,,,,, = 11 - k cr} $$
& ab = frac{{14}}{{15}}, & , frac{a}{b} = frac{{35}}{{24}} cr
& Leftrightarrow frac{{ab}}{{left( {frac{a}{b}}
ight)}} = left( {frac{{14}}{{15}} imes frac{{24}}{{35}}}
ight) cr
& Leftrightarrow {b^2} = frac{{16}}{{25}} cr
& Leftrightarrow b = frac{4}{5} cr
& { ext{So, }} cr
& Leftrightarrow ab = frac{{14}}{{15}} cr
& Leftrightarrow a = left( {frac{{14}}{{15}} imes frac{5}{4}}
ight) cr
& Leftrightarrow a = frac{7}{6} cr} $$ Since a > b, So, greater fraction is $$frac{7}{6}$$
& = a - 10\% { ext{ of }}a cr
& = a - frac{{10a}}{{100}} cr
& = frac{{100a - 10a}}{{100}} cr
& = frac{{90a}}{{100}} cr
& = frac{{9a}}{{10}} cr} $$ Man gives 80% of $$frac{{9a}}{{100}}$$ eggs to his neighbour $$eqalign{
& = frac{{80}}{{100}} imes frac{{9a}}{{10}} cr
& = frac{{72a}}{{100}} cr} $$ Remaining eggs : $$eqalign{
& = frac{{9a}}{{10}} - frac{{72a}}{{100}} cr
& = frac{{90a - 72a}}{{100}} cr
& = frac{{18a}}{{100}} cr
& = frac{{9a}}{{50}} cr} $$ According the question, $$eqalign{
& Rightarrow frac{{9a}}{{50}} = 36 cr
& Rightarrow 9a = 36 imes 50 cr
& Rightarrow a = frac{{36 imes 50}}{9} cr
& Rightarrow a = 200 cr} $$ Hence, the total number of eggs bought be = 200
& Leftrightarrow x + frac{1}{x} = 3left( {x - frac{1}{x}}
ight) cr
& Leftrightarrow frac{{{x^2} + 1}}{x} = 3left( {frac{{{x^2} - 1}}{x}}
ight) cr
& Leftrightarrow {x^2} + 1 = 3{x^2} - 3 cr
& Leftrightarrow 2{x^2} = 4 cr
& Leftrightarrow {x^2} = 2 cr
& Leftrightarrow x = sqrt 2 cr} $$
& Leftrightarrow frac{{x + frac{1}{4}}}{{y - frac{1}{3}}} = frac{{33}}{{64}} cr
& Leftrightarrow frac{{3left( {4x + 1}
ight)}}{{4left( {3y - 1}
ight)}} = frac{{33}}{{64}} cr
& Leftrightarrow frac{{4x + 1}}{{3y - 1}} = frac{{33}}{{64}} imes frac{4}{3} cr
& Leftrightarrow frac{{4x + 1}}{{3y - 1}} = frac{{11}}{{16}} cr
& Leftrightarrow 16left( {4x + 1}
ight) = 11left( {3y - 1}
ight) cr
& Leftrightarrow 64x + 16 = 33y - 11 cr
& Leftrightarrow 64x - 33y = - 27 cr} $$ Which cannot be solved to find $$frac{x}{y}$$ Hence, the original fraction cannot be determined from the given data.
& Leftrightarrow left( {10x + y}
ight) - left( {10y + x}
ight) = 63 cr
& Leftrightarrow 9left( {x - y}
ight) = 63 cr
& Leftrightarrow x - y = 7 cr} $$
& Leftrightarrow frac{{x + 200\% { ext{ of }}x}}{{y + 300\% { ext{ of }}y}} = frac{{15}}{{26}} cr
& Leftrightarrow frac{{3x}}{{4y}} = frac{{15}}{{26}} cr
& Leftrightarrow frac{x}{y} = frac{{15}}{{26}} imes frac{4}{3} cr
& Leftrightarrow frac{x}{y} = frac{{10}}{{13}} cr} $$
& Leftrightarrow frac{2}{3}x - 50 = frac{1}{4}x + 40 cr
& Leftrightarrow frac{2}{3}x - frac{1}{4}x = 90 cr
& Leftrightarrow frac{{5x}}{{12}} = 90 cr
& Leftrightarrow x = left( {frac{{90 imes 12}}{5}}
ight) cr
& Leftrightarrow x = 216 cr} $$
& Leftrightarrow 4xy = {left( {x + y}
ight)^2} - {left( {x - y}
ight)^2} cr
& Leftrightarrow 4xy = {left( {25}
ight)^2} - {left( {13}
ight)^2} cr
& Leftrightarrow 4xy = 625 - 169 cr
& Leftrightarrow 4xy = 456 cr
& Leftrightarrow xy = 114 cr} $$
ight)}
ight] - $$ xa0 xa0$$left[ {10left( {12 - x}
ight) + x}
ight]$$ xa0 $$ = 54$$ $$eqalign{
& Leftrightarrow 18x - 108 = 54 cr
& Leftrightarrow 18x = 162 cr
& Leftrightarrow x = 9 cr} $$ So, ten's digit = 9 and unit's digit = 3 Hence, original number = 93
& Leftrightarrow frac{{110\% { ext{ of 2}}x}}{{70\% { ext{ of 3}}y}} = 11\% { ext{ of }}frac{{16}}{{21}} cr
& Leftrightarrow frac{{22x}}{{21y}} = frac{{11}}{{100}} imes frac{{16}}{{21}} cr
& Leftrightarrow frac{x}{y} = left( {frac{{11}}{{100}} imes frac{{16}}{{21}} imes frac{{21}}{{22}}}
ight) cr
& Leftrightarrow frac{x}{y} = frac{2}{{25}} cr} $$
& Leftrightarrow left[ {frac{{left( {frac{x}{2}}
ight)}}{6} + frac{{left( {frac{x}{2}}
ight)}}{4}}
ight] - frac{x}{5} = 4 cr
& Leftrightarrow frac{x}{{12}} + frac{x}{8} - frac{x}{5} = 4 cr
& Leftrightarrow frac{{10x + 15x - 24x}}{{120}} = 4 cr
& Leftrightarrow x = 4 imes 120 cr
& Leftrightarrow x = 480 cr} $$
& herefore left( {10{x^2} + x}
ight) - left( {10x + {x^2}}
ight) = 54 cr
& Leftrightarrow 9{x^2} - 9x = 54 cr
& Leftrightarrow {x^2} - x = 6 cr
& Leftrightarrow {x^2} - x - 6 = 0 cr
& Leftrightarrow {x^2} - 3x + 2x - 6 = 0 cr
& Leftrightarrow left( {x - 3}
ight)left( {x + 2}
ight) = 0 cr
& Leftrightarrow x = 3 cr} $$ So. ten's digit = 3, unit's digit = 3 2 = 9 ∴ Original number = 39 Required result : = 40% of 39 = 15.6
& { ext{Let }}frac{A}{2} = frac{B}{3} = frac{C}{4} = x cr
& { ext{Then, }}A = 2x,B = 3x{ ext{ and }}C = 4x cr
& { ext{So, }}A:B:C = 2:3:4 cr
& herefore { ext{Largest part :}} cr
& = left( {243 imes frac{4}{9}}
ight) cr
& = 108 cr} $$
& Leftrightarrow x + {x^2} = 182 cr
& Leftrightarrow {x^2} + x - 182 = 0 cr
& Leftrightarrow left( {x + 14}
ight)left( {x - 13}
ight) = 0 cr
& Leftrightarrow x = 13 cr} $$
& Leftrightarrow frac{{x + y}}{{x - y}} = frac{{40}}{4} cr
& Leftrightarrow frac{{x + y}}{{x - y}} = 10 cr
& Leftrightarrow left( {x + y}
ight) = 10left( {x - y}
ight) cr
& Leftrightarrow 9x = 11y cr
& Leftrightarrow frac{x}{y} = frac{{11}}{9} cr
& Leftrightarrow x:y = 11:9 cr} $$
& Leftrightarrow A + B + C + D = 64 cr
& Leftrightarrow left( {x - 3}
ight) + left( {x + 3}
ight) + frac{x}{3} + 3x = 64 cr
& Leftrightarrow 5x + frac{x}{3} = 64 cr
& Leftrightarrow 16x = 192 cr
& Leftrightarrow x = 12 cr} $$ Thus, the numbers are 9, 15, 4 and 36 ∴ Required difference : = (36 - 4) = 32
& Leftrightarrow frac{{x + 2}}{{y + 3}} = frac{7}{9} cr
& Leftrightarrow 9x - 7y = 3.....(i) cr
& { ext{And,}} cr
& Leftrightarrow frac{{x - 1}}{{y - 1}} = frac{4}{5} cr
& Leftrightarrow 5x - 4y = 1.....(ii) cr} $$ Solving (i) and (ii), we get : x = 5 and y = 6 Hence, the original fraction is $$frac{5}{6}$$
& Leftrightarrow x - 12 = 20\% { ext{ of }}x cr
& Leftrightarrow x - frac{x}{5} = 12 cr
& Leftrightarrow frac{{4x}}{5} = 12 cr
& Leftrightarrow x = frac{{12 imes 5}}{4} cr
& Leftrightarrow x = 15 cr} $$
& Leftrightarrow frac{{4x + 4}}{{7x + 4}} = frac{3}{5} cr
& Leftrightarrow 5left( {4x + 4}
ight) = 3left( {7x + 4}
ight) cr
& Leftrightarrow x = 8 cr} $$ ∴ Larger number : = 7x = 7 × 8 = 56
& Leftrightarrow {x^2} - 20x = 96 cr
& Leftrightarrow {x^2} - 20x - 96 = 0 cr
& Leftrightarrow left( {x + 4}
ight)left( {x - 24}
ight) = 0 cr
& Leftrightarrow x = 24 cr} $$
& Rightarrow frac{1}{7}x - frac{1}{{11}}x = 100 cr
& Rightarrow frac{{4x}}{{77}} = 100 cr
& Rightarrow x = frac{{7700}}{4} cr
& Rightarrow x = 1925 cr} $$
& herefore 2x + x + frac{{2x}}{3} = 264 cr
& Leftrightarrow frac{{11x}}{3} = 264 cr
& Leftrightarrow x = left( {frac{{264 imes 3}}{{11}}}
ight) cr
& Leftrightarrow x = 72 cr} $$
& Rightarrow A + C = 146 cr
& Rightarrow x + left( {x + 4}
ight) = 146 cr
& Rightarrow 2x = 142 cr
& Rightarrow x = 71 cr
& herefore E = x + 8 cr
& ,,,,,,,,,,, = 71 + 8 cr
& ,,,,,,,,,,, = 79 cr} $$
& Leftrightarrow left( {10x + y}
ight) - left( {10y + x}
ight) = 36 cr
& Leftrightarrow 9left( {x - y}
ight) = 36 cr
& Leftrightarrow x - y = 4 cr} $$
& Leftrightarrow frac{{10x + y}}{2} = 10y + left( {x + 1}
ight) cr
& Leftrightarrow 10x + y = 20y + 2x + 2 cr
& Leftrightarrow 8x - 19y = 2.....(i) cr
& { ext{And}} Leftrightarrow x + y = 7.....(ii) cr} $$ Solving (i) and (ii), we get : x = 5, y = 2 Hence, required number = 52
& {x08f{x + y = 73}} cr
& Leftrightarrow y = 73 - x cr
& {x08f{y + z = 77 }} cr
& Leftrightarrow z = 77 - y cr
& Leftrightarrow z = 77 - left( {73 - x}
ight) cr
& Leftrightarrow z = 4 + x cr
& {x08f{3x + z = 104}} cr
& Leftrightarrow 3x + 4 + x = 104 cr
& Leftrightarrow 4x = 100 cr
& Leftrightarrow x = 25 cr
& x08ecause y = left( {73 - 25}
ight) cr
& Leftrightarrow y = 48 cr
& x08ecause z = left( {4 + 25}
ight) cr
& Leftrightarrow z = 29 cr} $$ ∴ Third number = 29
& Leftrightarrow x - frac{3}{5}x = 50 cr
& Leftrightarrow frac{2}{5}x = 50 cr
& Leftrightarrow x = left( {frac{{50 imes 5}}{2}}
ight) cr
& Leftrightarrow x = 125 cr} $$
& frac{2}{3}x = frac{25}{216} imes frac{1}{x} cr
& Leftrightarrow {x^2} = frac{{25}}{{216}} imes frac{3}{2} cr
& Leftrightarrow {x^2} = frac{{25}}{{144}} cr
& Leftrightarrow x = sqrt {frac{{25}}{{144}}} cr
& Leftrightarrow x = frac{5}{{12}} cr} $$
& Leftrightarrow xleft( {x + 5}
ight) = 500 cr
& Leftrightarrow {x^2} + 5x - 500 = 0 cr
& Leftrightarrow left( {x + 25}
ight)left( {x - 20}
ight) = 0 cr
& Leftrightarrow x = 20 cr} $$ So, the numbers are 20 and 25
& Rightarrow 10x + y = 3left( {x + y}
ight) cr
& Rightarrow 7x - 2y = 0.....(i) cr
& 10x + y + 45 = 10y + x cr
& Rightarrow y - x = 5.....(ii) cr} $$ Solving (i) and (ii), we get : x = 2 and y = 7 ∴ Required number = 27
& Rightarrow b = frac{5}{a} cr
& Rightarrow b = frac{5}{{frac{3}{2}}} cr
& Rightarrow b = frac{{5 imes 2}}{3} cr
& Rightarrow b = frac{{10}}{3} cr} $$ Required sum of : ⇒ a + b = $$frac{{3}}{2}$$ + $$frac{{10}}{3}$$ L.C.M. of 2 and 3 = 6 ⇒ a + b = $$frac{{9 + 20}}{6}$$ ⇒ a + b = $$frac{{29}}{6}$$ ⇒ a + b = $$4frac{{5}}{6}$$
& Leftrightarrow x + frac{2}{5}x = 455 cr
& Leftrightarrow frac{7}{5}x = 455 cr
& Leftrightarrow x = left( {frac{{455 imes 5}}{7}}
ight) cr
& Leftrightarrow x = 325 cr} $$
& Leftrightarrow x + 20 = frac{{69}}{x} cr
& Leftrightarrow {x^2} + 20x - 69 = 0 cr
& Leftrightarrow {x^2} + 23x - 3x - 69 = 0 cr
& Leftrightarrow xleft( {x + 23}
ight) - 3left( {x + 23}
ight) = 0 cr
& Leftrightarrow left( {x + 23}
ight)left( {x - 3}
ight) = 0 cr
& Leftrightarrow x = 3,,,,,,,,,left[ {x08ecause x
e - 23}
ight] cr} $$
& { ext{Let}},{ ext{the}},{ ext{number}},{ ext{be}},x,{ ext{and}},y cr
& { ext{Then}},,xy = 9375,{ ext{and}},frac{x}{y} = 15 cr
& frac{{xy}}{{left( {x/y}
ight)}} = frac{{9375}}{{15}} cr
& Rightarrow {y^2} = 625 cr
& Rightarrow y = 25 cr
& Rightarrow x = 15y = left( {15 imes 25}
ight) = 375 cr
& herefore { ext{Sum}},{ ext{of}},{ ext{the}},{ ext{number}} cr
& = x + y = 375 + 25 = 400 cr} $$
& Leftrightarrow 2x + 3 imes 42 = 238 cr
& Leftrightarrow 2x + 126 = 238 cr
& Leftrightarrow 2x = 112 cr
& Leftrightarrow x = 56 cr} $$ ∴ Required sum : $$eqalign{
& = 3x + 2 imes 42 cr
& = 3 imes 56 + 2 imes 42 cr
& = 168 + 84 cr
& = 252 cr} $$
& = frac{7}{8}{ ext{ of }}1008 - frac{3}{4}{ ext{ of }}568 cr
& = left( {1008 imes frac{7}{8}}
ight) - left( {568 imes frac{3}{4}}
ight) cr
& = 882 - 426 cr
& = 456 cr} $$
& Leftrightarrow 3{x^2} - 4x = x + 50 cr
& Leftrightarrow 3{x^2} - 5x - 50 = 0 cr
& Leftrightarrow left( {3x + 10}
ight)left( {x - 5}
ight) = 0 cr
& Leftrightarrow x = 5 cr} $$
& Leftrightarrow left( {x + y}
ight) - left( {x - y}
ight) = 42 cr
& Leftrightarrow 2y = 42 cr
& Leftrightarrow y = 21 cr} $$ Putting y = 21 in (i), we get : $$x = frac{{1092}}{{21}} = 52$$ Hence, greater number = 52
& Leftrightarrow frac{{xleft( {x + 2}
ight)left( {x + 4}
ight)}}{8} = 720 cr
& Leftrightarrow xleft( {x + 2}
ight)left( {x + 4}
ight) = 5760 cr
& herefore sqrt x imes sqrt {left( {x + 2}
ight)} imes sqrt {left( {x + 4}
ight)} cr
& = sqrt {xleft( {x + 2}
ight)left( {x + 4}
ight)} cr
& = sqrt {5760} cr
& = 24sqrt {10} cr} $$