Problems Based on Articles and their cost
Question 4: pens and
pencils together cost
and
pens and
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
we get
… … … … … (iii)
… … … … … (iv)
Subtracting (iv) from (iii) we get
Substituting in (i) we get
Question 5: A person has pens and pencils with together are in number.If she had
more pencils and
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 we get
… … … … … (iii)
Subtracting (i1) from (iii) we get
Substituting in (i) we get
Question 6: books and
pens together cost
whereas
books and
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
we get
… … … … … (iii)
… … … … … (iv)
Subtracting (iv) from (iii) we get
Substituting 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 we get
… … … … … (iii)
Adding (iii) from (i) we get
Substituting 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
days 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
Substituting 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) we get
Hence 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)
Also … … … … … (ii)
Consider +ve sign
Solving and
and
and the number is
Consider -ve sign
Solving and
and
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)
Also … … … … … (ii)
Solving and
and
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 and
Therefore … … … … … (i)
… … … … … (ii)
Now multiplying equation (i) by and adding (i) and (ii) we get
Therefore we get and
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)
Also … … … … … (ii)
Solving and
we get
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 and
Therefore … … … … … (i)
… … … … … (ii)
Adding (i) and (ii) we get
Therefore we get and
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)
Also … … … … … (ii)
Solving and
we get
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)
Also
… … … … … (ii)
Substituting (ii) in (i) we get
(not possible)
For we get
We get 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)
Also … … … … … (ii)
Solving and
we get
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and
… … … … … (ii)
Multiplying (ii) by and subtracting it from (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and … … … … … (ii)
Subtracting (ii) from (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and
… … … … … (ii)
Subtracting (ii) from (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and
… … … … … (ii)
Subtracting (ii) from (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and
… … … … … (ii)
Adding (i) and (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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:
Let the fraction
Therefore based on the given conditions:
… … … … … (i)
and … … … … … (ii)
Subtracting (ii) from (i) we get
Therefore .
Substituting in (i) we get
Therefore the fraction is
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.
Therefore … … … … … (i)
and … … … … … (ii)
Multiplying (i) by 2 and subtracting (ii) from (i) we get
years.
Therefore 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)
and … … … … … (ii)
Subtracting (ii) from (i)
years.
Therefore 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
and
?
Answer:
Present Age | Age |
Age |
|
A | |||
B |
Given:
… … … … … (i)
and … … … … … (ii)
Subtracting (ii) from (i)
years.
Therefore 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)
and … … … … … (ii)
Subtracting (ii) from (i)
years.
Therefore 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.
Therefore … … … … … (i)
and … … … … … (ii)
Subtracting (ii) from (i) we get
years.
Therefore 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.
Therefore
… … … … … (i)
and
… … … … … (ii)
Substituting (i) into (ii) we get
Hence Father’s age is .
Question 30: The ages of two friends and
differ by
years. A’s father
is twice as old as
and
is twice as old as here sister
. The ages of
and
differ by
years. Find the ages of
and
.
Answer:
Let the age of be
respectively.
Therefore
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (i) we get
years.
Therefore years.
Hence the age of A is and that of B is
.
Problems based on Time, Distance and Speed
Question 31: Point and
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
km/hr and car
be
km/ hr.
Let the cars meet at point .
Case 1: The cars are travelling in the same direction
Distance covered by car
Distance covered by car
Hence … … … … … (i)
Case 2: The cars are travelling in the same direction
Distance covered by car
Distance covered by car
Hence
… … … … … (ii)
Solving (i) and (ii) we get km/hr and
km/hr
Question 32: Point and
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
km/hr and car
be
km/ hr.
Let the cars meet at point .
Case 1: The cars are travelling in the same direction
Distance covered by car
Distance covered by car
Hence … … … … … (i)
Case 2: The cars are travelling in the same direction
Distance covered by car
Distance covered by car
Hence … … … … … (ii)
Solving (i) and (ii) we get km/hr and
km/hr
Question 33: A train covers a certain distance with uniform speed. If the train would have been km/hr faster, it would have taken
hours less than the scheduled time. And if the train was slower by
km/hr 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 km/hr and the actual time taken by the train be
hours.
Therefore the distance covered by the train
If the speed was km/hr more:
… … … … … (i)
If the speed was km/hr slower:
… … … … … (ii)
Solving (i) and (ii) we get km/hr and
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 km/hr and the speed of the car by
km/hr
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)
Now substituting
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr
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 km/hr and the speed of the car by
km/hr
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)
Now substituting
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr
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 km/hr and the speed of the car by
km/hr
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)
Now substituting
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr
Question 37: A boat covers km upstream and
km downstream in
hours. Also, it covers
km upstream and
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 km/hr and that of the stream is
km/hr.
Therefore speed of boat up stream km/hr
and Speed of boat downstream km/hr
Therefore we have
… … … … … (i1)
Similarly,
… … … … … (ii)
Let
and
Therefore we get
… … … … … (iii)
and … … … … … (vi)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr. Solving these two equations we get
km/hr and
km/hr
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 km/hr and that of the stream is y km/hr.
Therefore speed of sailor up stream km/hr
and Speed of boat downstream km/hr
Therefore we have
… … … … … (i1)
Similarly,
… … … … … (ii)
solving (i) and (ii) we get km/hr and
km/hr
Question 39: A boat covers km upstream and
km downstream in
hours. Also, it covers 30 km upstream and
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 km/hr and that of the stream is y km/hr.
Therefore speed of boat up stream km/hr
and Speed of boat downstream km/hr
Therefore we have
… … … … … (i1)
Similarly,
… … … … … (ii)
Let
and
Therefore we get
… … … … … (iii)
and … … … … … (vi)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr. Solving these two equations we get
km/hr and
km/hr
Question 40: A boat covers km upstream and
km downstream in
hours. Also, it covers
km upstream and
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 km/hr and that of the stream is y km/hr.
Therefore speed of boat up stream km/hr
and Speed of boat downstream km/hr
Therefore we have
… … … … … (i1)
Similarly,
… … … … … (ii)
Let
and
Therefore we get
… … … … … (iii)
and … … … … … (vi)
Solving (iii) and (iv) we get
and
Therefore km/hr and
km/hr. Solving these two equations we get
km/hr and
km/hr
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 km/hr and that of Y is y km/hr.
Therefore
and
Now substituting
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
km/hr
and
km/hr
Question 42: A man walks a certain distance with a certain speed. If he walks km/hr faster he would take
hours lesser. But if he walks
km/hr 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 km/hr and the actual time taken by the train be
hours.
Therefore the distance covered by the man
If the speed was km/hr more:
… … … … … (i)
If the speed was km/hr slower:
… … … … … (ii)
Solving (i) and (ii) we get km/hr and
hours
Distance covered km/hr
Question 43: A train covered distance at a uniform speed. If the train could have been km/hr faster, it would have taken two hours lesser then the scheduled time. But if it were slower by
km/hr, 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 km/hr and the actual time taken by the train be
hours.
Therefore the distance covered by the train
If the speed was km/hr more:
… … … … … (i)
If the speed was km/hr slower:
… … … … … (ii)
Solving (i) and (ii) we get km/hr and
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 Rs. and the Variable charges
Rs. / km
Therefore: … … … … … (i)
and … … … … … (ii)
Subtracting (i) from (ii) we get Rs. / km
Therefore Rs.
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
days, he pays
as the hostel charge where as the student
who stays for
days pays
for the hostel charges. Find the fixed charge and the daily charge for the stay.
Answer:
Let the fixed charges Rs. and the Variable charges
Rs. / day
Therefore: … … … … … (i)
and … … … … … (ii)
Subtracting (i) from (ii) we get Rs. / day
Therefore Rs.
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, and
after
years, what was his starting salary and his annual increment.
Answer:
Let the starting salary Rs. and the increment
Rs. / year
Therefore: … … … … … (i)
and … … … … … (ii)
Subtracting (i) from (ii) we get Rs. / year
Therefore starting salary Rs.
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)
Solving (i) and (ii) we get and
rows
Total number of students
Question 48: men and
boys can finish the work in
days while
men and
boys can finish the work in
days. 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 days and one boy can finish the work in
days alone.
Therefore one man’s work
and one boys work
Since men and
boys can finish the work in
days, we get
… … … … … (i)
Similarly men and
boys can finish the work in
days , we get
… … … … … (ii)
Let
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Hence days and
days
Therefore we can say that a man alone will take days while a boy working alone will finish the work in
days.
Question 49: men and
boys can finish the work in
days while
men and
boys can finish the work in
days. Find the time taken by one men and one boy to finish the work.
Answer:
Let one man can complete the work in days and one boy can finish the work in
days alone.
Therefore one man’s work
and one boys work
Since men and
boys can finish the work in
days, we get
… … … … … (i)
Similarly men and
boys can finish the work in
days , we get
… … … … … (ii)
Let
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Hence days and
days
Therefore we can say that a man alone will take days while a boy working alone will finish the work in
days.
Question 50: women and
men can finish the work in
days while
women and
men can finish the work in
days. 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 days and one men can finish the work in
days alone.
Therefore one man’s work
and one boys work
Since women and
men can finish the work in
days, we get
… … … … … (i)
Similarly women and
men can finish the work in
days , we get
… … … … … (ii)
Let
and
we get
… … … … … (iii)
… … … … … (iv)
Solving (iii) and (iv) we get
and
Hence days and
days
Therefore we can say that a woman alone will take days while a boy working alone will finish the work in
days.
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 and
respectively. Similarly their expenses will be
and
respectively.
Therefore given based on savings
… … … … … (i)
… … … … … (ii)
Solving (i) and (ii) we get and
Therefore the incomes are and
respectively.
Question 52: The income of and
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 and
respectively. Similarly their expenses will be
and
respectively.
Therefore given based on savings
… … … … … (i)
… … … … … (ii)
Solving (i) and (ii) we get and
Therefore the incomes are and
respectively.
Question 53: Find the four angles of a cyclic quadrilateral in which
,
,
and
.
Answer:
In a cyclic quadrilateral and
Therefore
… … … … … (i)
and … … … … … (ii)
Solving (i) and (ii) we get and
Hence the angles are ,
,
and
.
Question 54: In a ,
,
and
. 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 , and
Therefore and
. 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)
Solving (i) and (ii) we get and
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
Hence
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:
Let have
and
have
Therefore … … … … … (i)
and … … … … … (ii)
Solving (i) and (ii) we get and
Question 58: A scored marks in a test, getting
marks for each right answer and
mark for wrong answer. Had
marks been awarded for each right answer and
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
Therefore we have … … … … … (i)
and … … … … … (ii)
Solving (i) and (ii) we get and
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)
Solving (i) and (ii) we get and
rows
Total number of students