Complete step-by-step solutions for Class 9 Mathematics Ganita Manjari Chapter 2 Exercise Set 2.2 (Questions 1–7). Covers evaluating linear and quadratic polynomials by substitution, solving linear age word problems, integer ratio problems, currency denomination problems, fence division, and rectangle dimensions.
To find the value of a polynomial for a given value of the variable, substitute that numerical value in place of the variable wherever it appears in the expression and simplify using standard arithmetic order of operations.
When evaluating terms with powers and negative coefficients, evaluate the exponent first before multiplying, and apply sign rules carefully: (−a)² = a² (positive) and −c(−a) = +ca.
Assign a variable (e.g. x) to the unknown base age. Express related ages in terms of x using the given relationships. After n years, add n to each person's age before setting up the equation according to the problem statement.
When two quantities are in the ratio a:b, represent them as ax and bx using a common multiplier x > 0. Then set up a linear equation using their given sum, difference, or other condition.
When dealing with different denominations of currency, let x represent the unknown number of one denomination and express the other count in terms of x. Multiply each count by its respective coin value to set up the total value equation.
For problems involving a rectangle's perimeter, express one dimension in terms of the other (e.g. width = x, length = 2x + 3), and substitute into the standard perimeter formula 2(length + width) = P to solve for x.
Find the value of the linear polynomial 5x − 3 if: (i) x = 0 (ii) x = −1 (iii) x = 2
Find the value of the quadratic polynomial 7s² − 4s + 6 if: (i) s = 0 (ii) s = −3 (iii) s = 4
The present age of Salil’s mother is three times Salil’s present age. After 5 years, their ages will add up to 70 years. Find their present ages.
Let Salil’s present age be x years.
Then his mother’s present age is:
3x years
Their ages will then add up to 70 years.
(x + 5) + (3x + 5) = 70
15 years
His mother’s present age is:
The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
The ratio of the two integers is:
Let the two integers be:
2x and 5x
Since the difference between them is 63:
and
42 and 105
Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total ₹88, how many coins does she have of each type?
Let the number of ₹5 coins be x.
Then the number of ₹2 coins is:
The total value of the coins is ₹88.
and the number of ₹2 coins is:
24 two-rupee coins and 8 five-rupee coins
24 two-rupee coins and 8 five-rupee coins
A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
Let the shorter piece be x feet.
Then the longer piece is:
4x feet
Since the total length is 300 feet:
60 feet
and the longer piece is:
60 feet and 240 feet
and
Both given conditions are satisfied.
If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?
Let the width of the rectangle be x cm.
Then the length is:
2x + 3 cm
The perimeter of a rectangle is:
Given perimeter = 24 cm:
3 cm
and the length is:
Both conditions are satisfied.
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