Question 1: Find two natural numbers which differ by 3 and the sum of whose squares is 117.
Let the natural numbers be and
Since is not a natural number
One number and other number
Therefore Numbers are and
Question 2: Five times a certain whole number is equal to three less than twice the square of the number. Find the number.
Let the number be
Given, five times the number less than twice the square of the number
Therefore required whole number is
Question 3: Divide 8 into two parts such that the sum of their reciprocals is
Let the two parts be and
Therefore Required parts are and
Question 4: For the same amount of work, A takes 6 hours less than B. If together they complete the work in 13 hours 20 minutes; find how much time will B alone take to, I complete the work.
If B alone takes hours then A alone takes hours for the same work.
B along will take 30 hours to complete the work.
Question 5: The hypotenuse of a right triangle is 13 cm and the difference between the other two sides is 7 cm. Taking x as the length of the shorter of the two sides, write an equation in x that represents the above statement and also solve the equation to find the two unknown sides of the triangle
Since, the shorter side cm.
Therefore Longer side cm.
Using Pythagoras Theorem, we get :
On solving, it gives:
Since the side of triangle cannot be negative, therefore,
Therefore one side of the triangle cm and the other side is cm
Question 6: The length of a verandah is 3 m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.
(i) Taking x as the breadth of the verandah, write an equation in x that represents the above statement.
(ii) Solve the equation obtained in (i) above and hence find the dimensions of the verandah.
Since breadth m , then Length m
(i) Given : Area of verandah = its perimeter
Since the breadth cannot be negative, therefore
Hence, the length of verandah and its breadth
Question 7: By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km is reduced by 36 minutes. Find the original speed of the car. [ ICSE Board 2005]
Let the original speed of the car km/hr
New speed of the car km/hr
Given : Time is reduced by 36 minutes :
Since speed cannot be negative hence the value of
The original speed of the car km/hr
Question 8: Car A travels x km for every liter of petrol, while car B travels (x + 5) km for every liter of petrol.
(i) Write down the number of liters of petrol used by car A and car B in covering a distance of 400 km.
(ii) If car A uses 4 liters of petrol more than car B in covering the 400 km, write down an equation in x and solve it to determine the number of liters of petrol used by car B for the journey. [ ICSE Board 1997]
Question 9: By selling an article for Rs. 24, a trader loses as much per cent as the cost price of the article. Calculate the cost price.
Let Cost Price of the article be Rs.
Therefore the cost price of the article is or
Question 10: The sum S of first n natural numbers is given by the relation: Find if the sum is
Since is a natural number, cannot be
Question 11: A two digit number is such that the product of its digits is 6. When 9 is added to this number; the digits interchange their places. Find the number. [ICSE Board 2014]
Let the required two digit number be
Since, is not a digit,
Therefore The required two digit number
Question 12: Five years ago, a woman’s age was the square of her son’s age. Ten years hence her age will be twice that of her son’s age. Find :
(i) the age of the son five years ago.
(ii) the present age of the woman [ICSE Board 20007]
Let the age of the son 5 years ago years
The woman’s age 5 years ago years
The present age of the woman years
The present age of her son years
The woman’s age will be years
Her son’s age will be years
According to the given statement :
10 years hence, woman’s age = twice her son’s age
Age cannot be negative so
(i) The age of the son 5 years ago years
(i0 The present age of the woman years
Question 13: A motor-boat, whose speed is 9 km/h in still water, goes 12 km downstream and comes back in a total time of 3 hours. Find the speed of the stream.
Let the speed of the stream km/hr
Speed of the boat down stream km/hr
Speed of the boat upstream km/hr
Since, speed can not be negative. Therefore
Hence, the speed of the stream km/hr
Question 14: A piece of cloth costs Rs. 200. If the piece was 5 m longer and each meter of cloth costs Rs. 2 less; the cost of the piece would have remained unchanged. How long is the piece and what is the original rate per meter ?
Let the length of the piece be meters.
Since the cost of x meters cloth
New length of cloth m
Since length cannot be negative, therefore
The length of the piece m
Question 15: A shopkeeper buys a certain number of boots for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be bought for Rs. 960 would be 4 more. Taking the original cost of each book to be Rs. x, write and equation in x and solve it. [ICSE Board 2013]
Original cost of each book
In 2nd Case:
The cost of each book
Since cost cannot be negative, therefore
Question 16: Some students planned a picnic. The budget for the food was Rs. 480. As eight of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic. [ICSE Board 2008]
Let the number of students who planned the picnic
Since, the budget for food for all
Given eight of them failed to join the party. Therefore students went for the picnic
Since, the cost of food for each member (student) is increased by
On solving, it gives
But the number of students cannot be negative, therefore
The number of student who went for picnic