**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 years ago | Age years hence | |

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 years ago | Age years hence | |

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 years ago | Age years hence | |

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