Problems Based on Articles and their cost
Question 4: pens
pencils together cost
pens
pencils cost
. Find the cost of one pen and one pencil.
Answer:
Let the cost of a pen . and the cost of a pencil
.
Therefore given:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by and equation (ii) by
… … … … … (iii)
… … … … … (iv)
Subtracting (iv) from (iii) we get
in (i) we get
Question 5: A person has pens and pencils with together are in number.If she had
more pencils
less pens, then the number of pencils would become
times the number of pens. Find the number of pens and pencils that the person had.
Answer:
Let the number of pens . and the number of pencil
.
Therefore given:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by
… … … … … (iii)
Subtracting (i1) from (iii) we get
in (i) we get
Question 6: books
pens together cost
whereas
books
pens together cost
. Find the cost of one books and two pens.
Answer:
Let the cost of a book . and the cost of a pen
.
Therefore given:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by and equation (ii) by
… … … … … (iii)
… … … … … (iv)
Subtracting (iv) from (iii) we get
in (i) we get
Question 7: On selling a TV at gain and a fridge at
gain, a shopkeeper gains
. But if he sells the TV at
gain and the fridge at
loss, he gains
on the transaction. Find the actual prices of the TV and the fridge.
Answer:
Let the cost of TV . and the cost of Fridge
.
Therefore given:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by
… … … … … (iii)
Adding (iii) from (i) we get
in (i) we get
Question 8: A lending library has a fixed charge for the first three days and an additional charge for each additional day. Person A paid for a book kept for
while Person B paid
for the book kept for
days. Find the fixed charge and the charge for each additional day.
Answer:
Let the fixed cost . and the cost per day
.
Therefore given:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
in (i) we get
Problems Based on Numbers
Question 9: The sum of the digits of a two digit number is and the difference between the number and that formed by reversing the digits is
. Find the number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given: … … … … … (i)
… … … … … (ii)
Adding (i) and (ii)
and the number is
.
Question 10: The sum of two digit number and the number obtained by reversing the order of its digits is , and the two digits differ by
. Find the number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
Consider +ve sign
and the number is
Consider -ve sign
and the number is
Question 11: The sum of two digit number and the number formed by interchanging its digits is . If
is subtracted from the first number, the new number is
more than
times the sum of the digits in the first number. Find the first number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
and the number is
Question 12: The sum of two numbers is . If their sum is
times their difference, find the numbers.
Answer:
Let the two numbers be
… … … … … (i)
… … … … … (ii)
Now multiplying equation (i) by and adding (i) and (ii) we get
Question 13: The sum of the digits of a two digit number is . The number obtained by reversing the order of the digits of the given number exceeds the given number by
. Find the given number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
and the number is
Question 14: The sum of two numbers is and the difference between their squares is
. Find the numbers.
Answer:
Let the two numbers be
… … … … … (i)
… … … … … (ii)
Adding (i) and (ii) we get
Question 15: A digit number is four times the sum of its digits. If
is added to the number, the digits are reversed. Find the number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
and the number is
Question 16: A digit number is such that the product of its digits is
. If
is added to the number, the digits interchange their places. Find the number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
Substituting (ii) in (i) we get
(not possible)
For
and the number is
Question 17: times two digit number is equal to
times the number obtained by reversing the digits. If the difference between the digits is
. Find the number.
Answer:
Let the digit in unit place is and that in tenth place is
.
Therefore Number
Number formed by reversing the digits
Given:
… … … … … (i)
… … … … … (ii)
and the number is
Problems Based on Fractions
Question 18: A fraction become if
is added to both numerator and denominator. If, however,
is subtracted from both numerator and denominator, the fraction becomes
. What is the fraction?
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Multiplying (ii) by and subtracting it from (i) we get
.
in (i)
Question 19: A denominator of a fraction is more than twice the numerator. When both the numerator and denominator are decreased by
, the denominator becomes
times the numerator. Determine the fraction.
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
.
in (i)
Question 20: A fraction becomes is
is subtracted from both its numerator and denominator. If
is added to both numerator and denominator, it becomes
. Find the fraction.
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
.
in (i)
Question 21: If the numerator of the fraction is multiplied by and the denominator is reduced by
the fraction becomes
. And if the denominator is doubled and the numerator is increased by
, the fraction becomes
. Determine the fraction.
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
.
in (i)
Question 22: The sum of the numerator and denominator of a fraction is . If the denominator is increased by
, the fraction reduces to
. Find the fraction.
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Adding (i) and (i) we get
.
in (i)
Question 23: The sum of the numerator and denominator of a fraction is less than twice the denominator. If the numerator and denominator are decreased by
, the numerator becomes half the denominator. Determine the fraction.
Answer:
Therefore based on the given conditions:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
.
in (i)
Problems Based on Ages
Question 24: If twice the son’s age is added to father’s age, the sum is . But if twice the father’s age is added to the son’s age, the sum is
. Find the ages of father and son.
Answer:
Let Son’s age be and Father’s age by
years.
… … … … … (i)
… … … … … (ii)
Multiplying (i) by 2 and subtracting (ii) from (i) we get
years.
years.
Hence the age of son is and that of father is
.
Question 25: Ten year ago, father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be. Find their present ages.
Answer:
Present Age | Age |
Age |
|
Son’s Age | |||
Father’s Age |
Given:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i)
years.
years.
Hence the age of son is and that of father is
.
Question 26: Ten years later, will be twice as old as
and five year ago,
was three times as old as
. Find the present ages of
?
Answer:
Present Age | Age |
Age |
|
A | |||
B |
Given:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i)
years.
years.
Hence the age of A is and that of B is
.
Question 27: Ten years ago, a father was twelve times as old as his son and ten year hence, he will be twice as old as his son will be then. Find their present ages.
Answer:
Present Age | Age |
Age |
|
Father | |||
Son |
Given:
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i)
years.
years.
Hence the age of Father is and that of Son is
.
Question 28: The present age of father is three years more then three times that age of the son. Three years hence father’s age will be years more than twice the age of the son. Determine their present ages.
Answer:
Let Father’s age be and Son’s age by
years.
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
years.
years.
Hence the age of son is and that of father is
.
Question 29: Father’s age is three times the sum of ages of his two children. After years his age will be twice the sum of the two children. Find the age of father.
Answer:
Let Father’s age be and Son’s age by
years.
… … … … … (i)
… … … … … (ii)
Substituting (i) into (ii) we get
Hence Father’s age is .
Question 30: The ages of two friends differ by
years. A’s father
is twice as old as
is twice as old as here sister
. The ages of
differ by
years. Find the ages of
.
Answer:
Let the age of be
respectively.
Therefore
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
years.
years.
Hence the age of A is and that of B is
.
Problems based on Time, Distance and Speed
Question 31: Point are
km apart from each other on a road. A car starts from point
and another from point
at the same time. If they go in the same direction they meet in
hours and if they go in opposite direction, they meet in
hours. Find their speeds.
Answer:
Let the speed of the car be
and car
be
km/ hr.
Let the cars meet at point .
Case 1: The cars are travelling in the same direction
… … … … … (i)
Case 2: The cars are travelling in the same direction
… … … … … (ii)
Question 32: Point are
km apart from each other on a road. A car starts from point
and another from point
at the same time. If they go in the same direction they meet in
hours and if they go in opposite direction, they meet in
hour. Find their speeds.
Answer:
Let the speed of the car be
and car
be
km/ hr.
Let the cars meet at point .
Case 1: The cars are travelling in the same direction
… … … … … (i)
Case 2: The cars are travelling in the same direction
… … … … … (ii)
Question 33: A train covers a certain distance with uniform speed. If the train would have been faster, it would have taken
hours less than the scheduled time. And if the train was slower by
it would have taken
more hours than the scheduled time. Find the length of the journey.
Answer:
Let the actual speed of the train be and the actual time taken by the train be
hours.
Therefore the distance covered by the train
If the speed was more:
… … … … … (i)
If the speed was slower:
… … … … … (ii)
hours
Question 34: A man travels km partly by train and partly by car. If he covers
km by train and rest by car, it takes him
hours. But if he travels
km by train and rest by car, it takes him
minutes longer. Find the speed of the train and the car.
Answer:
Let the speed of the train be and the speed of the car by
When he covers km by train and rest by car, it takes him
hours.
Therefore:
… … … … … (i)
When he travels km by train and rest by car, it takes him
minutes longer
… … … … … (i)
… … … … … (iii)
… … … … … (iv)
Question 35: A man travels km partly by train and partly by car. If he covers
km by train and rest by car, it takes him
hours. But if he travels
km by train and rest by car, it takes him
minutes longer. Find the speed of the train and the car.
Answer:
Let the speed of the train be and the speed of the car by
When he covers km by train and rest by car, it takes him
hours.
Therefore:
… … … … … (i)
When he travels km by train and rest by car, it takes him
minutes longer
… … … … … (i)
… … … … … (iii)
… … … … … (iv)
Question 36: A girl travels km partly by train and partly by car. If she covers
km by train and rest by car, it takes her
hours. But if she travels
km by train and rest by car, it takes her
minutes longer. Find the speed of the train and the car.
Answer:
Let the speed of the train be and the speed of the car by
When she covers km by train and rest by car, it takes her
hours.
Therefore:
… … … … … (i)
When she travels km by train and rest by car, it takes her
minutes longer
… … … … … (ii)
… … … … … (iii)
… … … … … (iv)
Question 37: A boat covers km upstream
km downstream in
hours. Also, it covers
km upstream
km downstream in
hours. Find the speed of the boat in still water and that of the stream.
Answer:
Let the speed of the boat be and that of the stream is
.
Therefore speed of boat up stream
and Speed of boat downstream
… … … … … (i1)
… … … … … (ii)
Therefore we get
… … … … … (iii)
… … … … … (vi)
. Solving these two equations
Question 38: A sailor goes km downstream in
minutes and returns in
hours. Determine the speed of the sailor in still water and the speed of the current.
Answer:
Let the speed of the sailor be and that of the stream is y km/hr.
Therefore speed of sailor up stream
and Speed of boat downstream
… … … … … (i1)
… … … … … (ii)
Question 39: A boat covers km upstream
km downstream in
hours. Also, it covers 30 km upstream
km downstream in
hours. Find the speed of the boat in still water and that of the stream.
Answer:
Let the speed of the boat be and that of the stream is y km/hr.
Therefore speed of boat up stream
and Speed of boat downstream
… … … … … (i1)
… … … … … (ii)
Therefore we get
… … … … … (iii)
… … … … … (vi)
. Solving these two equations
Question 40: A boat covers km upstream
km downstream in
hours. Also, it covers
km upstream
km downstream in
hours. Find the speed of the boat in still water and that of the stream.
Answer:
Let the speed of the boat be and that of the stream is y km/hr.
Therefore speed of boat up stream
and Speed of boat downstream
… … … … … (i1)
… … … … … (ii)
Therefore we get
… … … … … (iii)
… … … … … (vi)
. Solving these two equations
Question 41: takes
hours more than
to walk
km. But if
doubles his pace, he is ahead of
by
hours. Find their speed of walking.
Answer:
Let the speed of the X be and that of Y is y km/hr.
… … … … … (iii)
… … … … … (iv)
Question 42: A man walks a certain distance with a certain speed. If he walks faster he would take
hours lesser. But if he walks
slower, he takes
more hours. Find the distance covered by the man and his speed of walking.
Answer:
Let the actual speed of the man be and the actual time taken by the train be
hours.
Therefore the distance covered by the man
If the speed was more:
… … … … … (i)
If the speed was slower:
… … … … … (ii)
hours
Distance covered
Question 43: A train covered distance at a uniform speed. If the train could have been faster, it would have taken two hours lesser then the scheduled time. But if it were slower by
, it would have taken
more hours than the scheduled time. Find the distance covered by the train.
Answer:
Let the actual speed of the train be and the actual time taken by the train be
hours.
Therefore the distance covered by the train
If the speed was more:
… … … … … (i)
If the speed was slower:
… … … … … (ii)
hours
Distance covered km
Miscellaneous Problems
Question 44: A taxi charge comprises of a fixed charge and a variable charge per km. For a journey of km the charge paid is
and for a journey of
km, the charge paid is
. What will the person pay for a journey of
km.
Answer:
Let the fixed charges and the Variable charges
Therefore: … … … … … (i)
… … … … … (ii)
Subtracting (i) from (ii)
Question 45: A part of the monthly hostel charges is fixed and the other part depends on the number of days that you have stayed. When a student stays for
, he pays
as the hostel charge where as the student
who stays for
pays
for the hostel charges. Find the fixed charge and the daily charge for the stay.
Answer:
Let the fixed charges and the Variable charges
Therefore: … … … … … (i)
… … … … … (ii)
Subtracting (i) from (ii)
Question 46: A man starts his job with a certain monthly salary and earns a fixed increment every year. If his slary was after
years,
after
years, what was his starting salary and his annual increment.
Answer:
Let the starting salary and the increment
Therefore: … … … … … (i)
… … … … … (ii)
Subtracting (i) from (ii)
Therefore starting salary
Question 47: Students of a class are made to stand in rows. If there is one student extra in a row, there would be rows less. If one student is less in a row, there would be
rows more. Find the number of students in the class.
Answer:
Let the number of students in a row be and the number of rows be
.
Therefore the total number of students
If there was one student more in each row:
… … … … … (i)
If there was one student less in a row:
… … … … … (ii)
rows
Total number of students
Question 48: men
boys can finish the work in
while
men
boys can finish the work in
. Find the time taken by one men alone and that by one boy alone to finish the work.
Answer:
Let one man can complete the work in and one boy can finish the work in
alone.
Since men
boys can finish the work in
, we get
… … … … … (i)
Similarly men
boys can finish the work in
, we get
… … … … … (ii)
… … … … … (iii)
… … … … … (iv)
Therefore we can say that a man alone will take while a boy working alone will finish the work in
.
Question 49: men
boys can finish the work in
while
men
boys can finish the work in
. Find the time taken by one men and one boy to finish the work.
Answer:
Let one man can complete the work in and one boy can finish the work in
alone.
Since men
boys can finish the work in
, we get
… … … … … (i)
Similarly men
boys can finish the work in
, we get
… … … … … (ii)
… … … … … (iii)
… … … … … (iv)
Therefore we can say that a man alone will take while a boy working alone will finish the work in
.
Question 50: women
men can finish the work in
while
women
men can finish the work in
. Find the time taken by one woman alone and that by one man alone to finish the work.
Answer:
Let one woman can complete the work in and one men can finish the work in
alone.
Since women
men can finish the work in
, we get
… … … … … (i)
Similarly women
men can finish the work in
, we get
… … … … … (ii)
… … … … … (iii)
… … … … … (iv)
Therefore we can say that a woman alone will take while a boy working alone will finish the work in
.
Question 51: The ratio of income of two people is and the ratio of their expenditure is
. If each one of them saves
per month, find the monthly incomes.
Answer:
Let the income of the people be respectively. Similarly their expenses will be
respectively.
Therefore given based on savings
… … … … … (i)
… … … … … (ii)
Therefore the incomes are respectively.
Question 52: The income of are in the ratio of
and their expenditures are in the ratio of
. If each save
, find their income.
Answer:
Let the income of the people be respectively. Similarly their expenses will be
respectively.
Therefore given based on savings
… … … … … (i)
… … … … … (ii)
Therefore the incomes are respectively.
Question 53: Find the four angles of a cyclic quadrilateral in which
,
,
.
Answer:
In a cyclic quadrilateral
Therefore
… … … … … (i)
… … … … … (ii)
Hence the angles are ,
,
.
Question 54: In a ,
,
. If
, prove that the triangle is a right-angled triangle.
Answer:
In a triangle, the sum of all the three angles is .
Therefore
… … … … … (i)
and given … … … … … (i)
Solving (i) and (ii) we get ,
. Therefore the triangle is a right angled triangle.
Question 55: If in a rectangle, the length is increased and the breadth is decreases each by units, the area is reduced by
sq. units. However, if the length is reduced by
units and breadth is increased by
units, the area increases by
sq. units. Find the area of the rectangle.
Answer:
Let the length be and the breadth be
.
Therefore the area
If the length is increased and the breadth is decreases each by units, the area is reduced by
sq. units:
… … … … … (i)
If the length is reduced by units and breadth is increased by
units, the area increases by
sq. units
… … … … … (ii)
units
Total area sq. units
Question 56: Half the perimeter of the garden, whose length is more than its width is
. Find the dimensions of the garden.
Answer:
Let the width units
Therefore the length units
units
Therefore length is units.
Question 57: If gives
to
, then
will have twice the amount of money left with
. But if
gives
to
, then
will have thrise as much as as is left with
. How much money does each have.
Answer:
have
have
… … … … … (i)
… … … … … (ii)
Question 58: A scored marks in a test, getting
marks for each right answer
mark for wrong answer. Had
marks been awarded for each right answer
marks deducted for each wrong answer then A would have scored
marks. How many questions were in the test.
Answer:
Let the number of questions answered right are x and that answered wrong are y
… … … … … (i)
… … … … … (ii)
Therefore the total number of questions in the test are .
Question 59: Students of a class are made to stand in rows. If there are students extra in a row, there would be
row less. If
students are less in a row, there would be
rows more. Find the number of students in the class.
Answer:
Let the number of students in a row be and the number of rows be
.
Therefore the total number of students
If there was one student more in each row:
… … … … … (i)
If there was one student less in a row:
… … … … … (ii)
rows
Total number of students