Chapter 7 Algebra (Class 6 - Maths NCERT Exemplar Solutions)

Solution:

In algebra, when a number and a variable are written together such as $4a$, it indicates the operation of multiplication between the coefficient and the variable.

Correct Option: (B)

To Find:

Algebraic expression for the given verbal statement.

Solution:

Break the statement into mathematical components:

1. "three times the number x" is written as $3 \times x$ or $3x$.

2. "8 more than" indicates addition of 8 to the previous term.

Correct Option: (C)

Solution:

An equation is a mathematical condition on a variable which states that two expressions are equal. It must contain the equals sign ($=$).

Let us analyze the options:

(A) $x + 7$ : This is an expression, not an equation.

(B) $2y + 3 = 7$ : This contains an equality sign, hence it is an equation.

(C) $2p < 10$ : This is an inequality.

(D) $12x$ : This is a term/expression.

Correct Option: (B)

Solution:

Identify the terms in the statement:

1. "7 times of y" $= 7 \times y = 7y$.

2. "subtracted from 50" means we start with 50 and remove $7y$ from it.

Answer: $50 - 7y$

Given:

Equation: $3 - x = 8$

Value: $x = 5$

Solution:

To check if $x = 5$ is a solution, substitute the value in the Left Hand Side (LHS) of the equation:

Now, the Right Hand Side (RHS) is given as:

Since $-2 \neq 8$, the LHS is not equal to the RHS.

Answer: False

Alternate Solution:

Solve the equation for $x$:

Since the actual solution is $x = -5$, the statement that $x = 5$ is a solution is false.

To Find:

The algebraic expression for the given statement.

Solution:

Let the unknown number be $x$.

Step 1: "Thrice of a number" means we multiply the number by 3.

Step 2: "13 subtracted from" means we need to take away 13 from the product obtained above.

Therefore, the required expression is $3x - 13$.

To Find:

An algebraic expression representing Megha's age.

Solution:

Let the age of Megha's daughter be $x$ years.

Step 1: "5 times her daughter's age" is represented as:

Step 2: "2 more than" means we add 2 to the value calculated in the first step.

The expression for Megha's age is $5x + 2$.

Given:

Anagha's current position $= p$ (at step $p$)

To Find:

1. Position of Sushant

2. Position of Faizal

3. Total number of steps to the hill top

Solution:

Step 1: Find Sushant's position

It is given that Sushant is 10 steps ahead of Anagha.

Step 2: Find Faizal's position

It is given that Faizal is 6 steps behind Anagha.

Step 3: Find the Total number of steps

The total steps are 3 less than 8 times Anagha's position ($p$).

8 times Anagha's position $= 8p$

Summary:

Sushant is at step $(p + 10)$, Faizal is at step $(p - 6)$, and the total number of steps is $(8p - 3)$.

Given:

Cost of a pencil $= \textsf{₹} x$

Cost of a pen $= \textsf{₹} 6x$

Solution:

The expression $6x$ implies that the cost of a pen is obtained by multiplying the cost of a pencil by 6.

Statement in ordinary language:

The cost of a pen is six times the cost of a pencil.

Given:

Manisha's age $= z$ years

Uncle's age $= 5z$ years

Aunt's age $= (5z - 4)$ years

Solution:

We can interpret the relationships as follows:

1. The uncle's age ($5z$) is 5 times Manisha's age ($z$).

2. The aunt's age ($5z - 4$) is 4 years less than the uncle's age ($5z$).

Statement in ordinary language:

Manisha's uncle is five times as old as Manisha, and her aunt is 4 years younger than her uncle.

Given:

Number of matchsticks in one box = $50$

Number of boxes = $n$

To Find:

Total number of matchsticks required.

Solution:

To find the total number of matchsticks, we multiply the number of matchsticks in a single box by the total number of boxes.

Correct Option: (B)

Given:

Amulya's current age = $x$ years

To Find:

Amulya's age 5 years ago.

Solution:

To find the age in the past, we subtract the number of years from the current age.

Correct Option: (C)

Solution:

In algebra, the product of a constant (number) and a variable is typically written by placing them side by side without a multiplication sign.

Option (B) represents division, (C) represents addition, and (D) represents subtraction.

Correct Option: (A)

Solution:

An equation is a mathematical statement that asserts the equality of two expressions using the equals sign ($=$).

1. $x + 1$ and $x - 1$ are algebraic expressions.

2. $x + 1 > 0$ is an inequality.

3. $x - 1 = 0$ contains an "$=$" sign, relating two sides.

Correct Option: (C)

Given:

Expression: $x + 10$

Value of $x = 2$

Solution:

Substitute the value of $x$ into the given expression:

Correct Option: (B)

Given:

Perimeter of regular hexagon $= x$ metres

To Find:

Length of each side.

Solution:

A regular hexagon is a polygon with 6 equal sides.

To find the length of one side, we divide the perimeter by 6:

Correct Option: (B)

Solution:

We check each equation by substituting $x = 2$:

For Option (A): $x + 2 = 5$

For Option (B): $x - 2 = 0$

For Option (C): $2x + 1 = 0$

For Option (D): $x + 3 = 6$

Correct Option: (B)

Solution:

The commutative property of addition states that the order in which two numbers are added does not change the sum. For any two integers $x$ and $y$:

Option (D) shows the commutative property of multiplication, but the question specifically asks for the operation of addition.

Correct Option: (A)

Solution:

An integer is a whole number (positive, negative, or zero) that does not have a fractional part. Let's solve each equation:

(A) $x + 1 = 1$

(B) $x - 1 = 3$

(C) $2x + 1 = 6$

(D) $1 - x = 5$

Correct Option: (C)

Solution:

In algebra, we use symbols to represent multiplication by juxtaposition ($ab$ means $a \times b$). However, in arithmetic, we perform the actual calculation between numbers.

If we simply placed the digits together as in algebra (juxtaposition), it would result in the number 35, which is incorrect for a product in arithmetic.

Correct Option: (C)

Solution:

Algebra is a branch of mathematics where we use letters (like $x$, $y$, $z$) to represent unknown quantities or variables whose values are not yet determined or can vary.

Correct Option: (B)

Solution:

The word "variable" comes from the word "vary," which means to change. In mathematics, a variable is a symbol that can take different numerical values depending on the context or the equation.

Correct Option: (A)

Solution:

In the expression $10 - x$:

1. 10 is the starting quantity.

2. The minus sign ($-$) represents subtraction.

3. $x$ is the quantity being removed.

Thus, it translates to "$x$ is subtracted from 10".

Correct Option: (C)

Given:

Initial amount with Savitri = $\textsf{₹} x$

Expenditure on grocery = $\textsf{₹} 1000$

Expenditure on clothes = $\textsf{₹} 500$

Expenditure on education = $\textsf{₹} 400$

Amount received as gift = $\textsf{₹} 200$

To Find:

The money left with Savitri.

Solution:

Step 1: Calculate the total expenditure.

$\begin{array}{cc} & 1 & 0 & 0 & 0 \\ & & 5 & 0 & 0 \\ + & & 4 & 0 & 0 \\ \hline & 1 & 9 & 0 & 0 \\ \hline \end{array}$

Step 2: Calculate the total money after receiving the gift.

Step 3: Calculate the remaining balance.

Correct Option: (A)

Given:

The sides of the triangle are $x$, $x$, and $y$.

To Find:

The perimeter of the triangle.

Solution:

The perimeter of a triangle is the sum of the lengths of all its sides.

Correct Option: (A)

Given:

Side of the square $= x$

Solution:

The formula for the area of a square is the side multiplied by the side.

Correct Option: (A)

Solution:

We translate the word statement into an algebraic expression step-by-step:

Step 1: "x is multiplied by 2"

Step 2: "subtracted from 3" means we take the result ($2x$) and subtract it from 3.

Correct Option: (C)

Given:

Equation: $\frac{q}{2} = 3$

Solution:

To find the solution, we need to isolate the variable $q$. Since $q$ is divided by 2, we multiply both sides of the equation by 2.

Correct Option: (A)

Given:

Equation: $x - 4 = -2$

Solution:

To isolate the variable $x$, we add 4 to both sides of the equation:

Correct Option: (B)

Solution:

1. It is an equation because it contains an equals sign ($=$).

2. It is numerical because it consists only of numbers and no variables (letters like $x, y$).

3. It is a true statement since $\frac{4}{2}$ is indeed equal to 2.

Therefore, it is a numerical equation.

Correct Option: (A)

Given:

Initial number of pencils $= p$

Pencils added $= q$

Solution:

To find the total number of pencils, we add the number of pencils initially present to the number of pencils added later.

Correct Option: (A)

Given:

Equation: $4x = 16$

Solution:

To find the value of $x$ that satisfies the equation, we need to isolate $x$ by dividing both sides by 4.

Substituting $x = 4$ in LHS: $4 \times 4 = 16$, which is equal to RHS.

Correct Option: (A)

Solution:

Let the number thought of be $x$.

Step 1: "Adding 13 to it" is expressed as $x + 13$.

Step 2: "I get 27" means the result is equal to 27.

Correct Option: (D)

Given:

Constant speed $= 40$ km/hr

Time $= h$ hours

Solution:

The relationship between distance, speed, and time is given by the formula:

Answer: $40h$

Given:

Quantity of potatoes bought $= p$ kg

Total cost for $p$ kg $= \textsf{₹} 70$

Solution:

To find the cost of 1 kg of potatoes, we use the unitary method where the total cost is divided by the total quantity.

Answer: $\frac{70}{p}$

Given:

Charge for 1st km $= \textsf{₹} 10$

Charge for every subsequent km $= \textsf{₹} 8$

Total distance $= d$ km

Solution:

Step 1: The first kilometer costs $\textsf{₹} 10$.

Step 2: The remaining distance after the first kilometer is $(d - 1)$ km.

Step 3: The charge for the remaining $(d - 1)$ km is calculated at the rate of $\textsf{₹} 8$ per km.

Step 4: Total charge is the sum of both parts.

Answer: $8d + 2$

Given:

Equation: $7x + 4 = 25$

Solution:

Isolate the term containing $x$ by subtracting 4 from both sides:

Divide both sides by 7 to find $x$:

Answer: 3

Given:

Equation: $3x + 7 = -20$

Solution:

Subtract 7 from both sides of the equation:

Divide both sides by 3:

Answer: -9

Solution:

The word "exceeds" means that the first value is greater than the second value by a specific amount.

If $x$ exceeds $y$ by 7, it means the difference between $x$ and $y$ is 7.

Alternatively, this can be written as:

Answer: $x - y = 7$ (or $x = y + 7$)

Solution:

Step 1: "three times the number x" is $3x$.

Step 2: "8 more than" indicates adding 8 to that result.

Answer: $3x + 8$

Given:

Total amount available = $\textsf{₹} x$

Cost of one pencil = $\textsf{₹} 2$

To Find:

The number of pencils that can be bought.

Solution:

The number of items is calculated by dividing the total amount by the cost per item.

Answer: $\frac{x}{2}$

Given:

Number of weeks $= w$

Solution:

We know that there are 7 days in one week.

To find the total number of days in $w$ weeks, we multiply the number of weeks by 7.

Answer: $7w$

Given:

Monthly salary $= \textsf{₹} r$

Festival bonus $= \textsf{₹} 2000$

To Find:

Total annual salary.

Solution:

First, we calculate the salary for the entire year (12 months).

Now, we add the one-time festival bonus to this amount.

Answer: $12r + 2000$

Given:

Ten's digit $= t$

Unit's digit $= u$

Solution:

In a place value system, the value of a digit depends on its position. The digit in the ten's place is multiplied by 10, and the digit in the unit's place is multiplied by 1.

Answer: $10t + u$

Solution:

A variable is a symbol, usually a letter, that represents an unknown numerical value in an algebraic expression or equation.

In the equation $2p + 8 = 18$:

1. 2 and 8 are constants/coefficients.

2. 18 is a constant.

3. $p$ is the literal used to represent the unknown value.

Answer: $p$

Solution:

We know the standard conversion for length:

To convert $x$ metres into centimetres, we multiply $x$ by 100.

Answer: $100x$

Solution:

We know the standard conversion for capacity:

To convert $p$ litres into millilitres, we multiply $p$ by 1000.

Answer: $1000p$

Solution:

In the Indian currency system:

To convert $r$ rupees into paise, we multiply $r$ by 100.

Answer: $100r$

Given:

Present age of Ramandeep $= n$ years

Solution:

To find the age after a certain number of years (in the future), we add that number of years to the current age.

Answer: $n + 7$

Given:

Initial amount $= \textsf{₹} 100$

Amount spent $= \textsf{₹} f$

Solution:

The money left is the difference between the initial total and the amount spent.

Answer: $100 - f$

Given:

Equation: $x + 1 = 0$

Value to check: $x = 0$

Solution:

To check if $x = 0$ is a solution, we substitute this value into the Left Hand Side (LHS) of the equation.

Now, comparing LHS with the Right Hand Side (RHS):

Since $1 \neq 0$, the value $x = 0$ does not satisfy the equation.

Answer: False

Solution:

Let us solve both equations separately to find their solutions.

For Equation 1:

For Equation 2:

Since both equations result in the same value for $x$, they have the same solution.

Answer: True

Solution:

A multiple of 2 is any number that can be expressed in the form of $2 \times k$, where $k$ is an integer (including whole numbers).

Since $m$ is a whole number ($0, 1, 2, 3, \dots$), then $2m$ will always result in an even number (including 0), which are all multiples of 2.

Answer: True

Solution:

The additive inverse of a number is the value that, when added to the original number, results in zero.

Let the additive inverse of $x$ be $a$.

Therefore, the additive inverse of $x$ is $-x$, not $2x$.

Answer: False

Solution:

Let $x$ be a negative integer, for example, $x = -5$.

Then, $-x$ would be:

Since the product of two negative signs is positive, $-x$ will always be a positive integer when $x$ is negative.

Answer: True

Solution:

An equation must contain the "equal to" sign ($=$), which shows that the Left Hand Side is exactly equal to the Right Hand Side.

The expression $2x - 5 > 11$ uses a greater than sign ($>$), which makes it an inequality, not an equation.

Answer: False

Solution:

By definition, an equation is a mathematical statement that asserts the equality of two expressions. The expression to the left of the "$=$" sign is the LHS (Left Hand Side) and the expression to the right is the RHS (Right Hand Side).

Answer: True

Solution:

In the algebraic equation $7k - 7 = 7$:

1. 7 is a constant and also used as a coefficient.

2. $k$ is the letter that represents an unknown numerical value.

Therefore, the variable is $k$, not 7.

Answer: False

Given:

Equation: $2a - 1 = 5$

Value: $a = 3$

Solution:

Substitute $a = 3$ into the Left Hand Side (LHS):

Comparing with the Right Hand Side (RHS):

Since LHS $=$ RHS, $a = 3$ is indeed the solution.

Answer: True

Solution:

A variable is a quantity that can change or take different values. A constant is a quantity that remains fixed.

The distance between two fixed cities, New Delhi and Bhopal, is a fixed measurement and does not change. Therefore, it is a constant, not a variable.

Answer: True

Solution:

We know that in 1 minute, there are 60 seconds.

To convert $t$ minutes into seconds, we multiply by 60:

Answer: True

Given:

Equation: $3x + 2 = 20$

Value: $x = 5$

Solution:

Substitute the value $x = 5$ in the Left Hand Side (LHS):

Comparing with the Right Hand Side (RHS):

Since $17 \neq 20$, $x = 5$ is not the solution.

Answer: False

Solution:

Let the number be $x$.

1. "One third of a number" $= \frac{1}{3} \times x = \frac{x}{3}$.

2. "Added to itself" means we add $x$ to the previous term: $\frac{x}{3} + x$.

3. "Gives 8" means the sum is equal to 8.

So, the correct equation should be: $\frac{x}{3} + x = 8$.

The given statement says $\frac{x}{3} + 8 = x$, which is incorrect.

Answer: False

Solution:

Let Leela's age be $L$ and Yamini's age be $Y$. The difference is $L - Y$.

After $n$ years, Leela will be $L+n$ and Yamini will be $Y+n$.

The new difference $= (L+n) - (Y+n) = L + n - Y - n = L - Y$.

Since the difference between their ages remains the same throughout their lives, it is a constant, not a variable.

Answer: False

Solution:

According to the basic principles of geometry, an infinite number of lines can be drawn passing through a single point. This is a fixed geometrical fact and not a quantity that varies under different conditions in Euclidean geometry.

Answer: False

Solution:

Let the unknown number be $x$.

Step 1: "Twice the number" $= 2 \times x = 2x$.

Step 2: "One more than" indicates adding 1 to the result.

Answer: $2x + 1$

Solution:

Let the present temperature be $t^\circ \text{C}$.

"$20^\circ \text{C}$ less than" indicates subtracting 20 from the current temperature.

Answer: $t - 20$

Solution:

Let the integer be $z$.

The successor of any integer is the number that comes immediately after it, which is found by adding 1.

Answer: $z + 1$

Given:

Side of the equilateral triangle $= m$

Solution:

An equilateral triangle has 3 equal sides. The perimeter is the sum of all sides.

Answer: $3m$

Given:

Length $= k$ units

Breadth $= n$ units

Solution:

The formula for the area of a rectangle is:

Answer: $kn$

To Find:

The algebraic expression for the time Omar helps his mother.

Solution:

Let the time for which Omar's sister helps her mother be $x$ hours.

According to the statement, Omar helps for 1 hour more than his sister.

Answer: $x + 1$

To Find:

Algebraic expressions for two consecutive odd integers.

Solution:

An even number is always a multiple of 2, which can be represented as $2n$. An odd number is always 1 more than an even number.

Consecutive odd integers have a difference of 2. Therefore, the next consecutive odd integer is obtained by adding 2 to the first one.

Answer: $2n + 1$ and $2n + 3$

To Find:

Algebraic expressions for two consecutive even integers.

Solution:

An even number is defined as a multiple of 2. We can represent any even number using a variable $m$.

Consecutive even integers have a difference of 2. Therefore, the next consecutive even integer is obtained by adding 2 to the first one.

Answer: $2m$ and $2m + 2$

Solution:

A multiple of 5 is any number that is obtained by multiplying 5 by an integer.

Let the integer be $n$.

Answer: $5n$

Solution:

Let the numerator of the fraction be $n$.

As per the given statement, the denominator is 1 more than the numerator.

Therefore, the fraction can be expressed as:

Answer: $\frac{n}{n + 1}$

To Find:

Expression for the height of Mount Everest.

Solution:

Let the height of the Empire State building be $h$ metres.

The statement says Mount Everest is 20 times this height.

Answer: $20h$

Given:

Cost of 1 notebook $= \textsf{₹} p$

Cost of 1 pencil $= \textsf{₹} 3$

Solution:

1. Cost of 2 notebooks $= 2 \times p = 2p$

2. Cost of 1 pencil $= 3$

To find the total cost, we add these values together.

Answer: $2p + 3$

Solution:

Step 1: "$z$ is multiplied by $-3$"

Step 2: "the result is subtracted from 13"

Answer: $13 + 3z$

Solution:

Step 1: "$p$ is divided by 11"

Step 2: "added to 10"

Answer: $\frac{p}{11} + 10$

Solution:

We know that the smallest natural number is 1.

Step 1: "$x$ times of 3" means $3 \times x$ or $3x$.

Step 2: "Added to the smallest natural number"

Answer: $3x + 1$

To Find:

The algebraic expression for the given statement.

Solution:

Step 1: Identify the smallest two-digit number.

Step 2: Find the value of "6 times q".

Step 3: "Subtracted from" means we take $6q$ away from 10.

Answer: $10 - 6q$

To Find:

Two different algebraic equations where $x = 2$.

Solution:

We can construct equations by performing operations on the solution $x = 2$.

Equation 1:

Let's add 5 to both sides of the solution $x = 2$.

Equation 2:

Let's multiply the solution $x = 2$ by 4.

Answer: $x + 5 = 7$ and $4x = 8$

To Find:

An equation where the variable equals 0.

Solution:

Let the variable be $x$. If the solution is 0, then $x = 0$.

We can multiply both sides by any number, say 5:

Alternatively, we can add a number to $x$ and set the result to the same number:

Answer: $5x = 0$ (or $x + 12 = 12$)

Solution:

Whole numbers are $0, 1, 2, 3, \dots$. We need an equation where the solution is a fraction or a negative number.

Let's create an equation resulting in a fraction:

Solving this:

Since 2.5 is not a whole number, this equation satisfies the condition.

Alternate Solution:

Let's create an equation resulting in a negative number:

Since -6 is not a whole number, this is also a valid equation.

Answer: $2x = 5$

Solution:

The expression $5p$ indicates that the price of the pen is 5 times the price of the pencil ($p$).

Statement in ordinary language:

The cost of a pen is five times the cost of a pencil.

Solution:

The expression $(y + 10000)$ indicates that the remaining amount is obtained by adding $\textsf{₹} 10000$ to the contribution amount ($y$).

Statement in ordinary language:

The amount left with Leela is $\textsf{₹} 10,000$ more than the amount she contributed to the Prime Minister’s Relief Fund.

Solution:

The expression $7n$ indicates that the father's age is 7 times Kartik's age ($n$).

Statement in ordinary language:

Kartik's father is seven times as old as Kartik.

Solution:

The expression $(p - 10)$ indicates that the minimum temperature is $10^\circ\text{C}$ lower than the maximum temperature ($p$).

Statement in ordinary language:

The minimum temperature on a day in Delhi was $10^\circ\text{C}$ less than the maximum temperature.

Solution:

The expression $2t + 10$ consists of two parts: twice the plants ($2t$) and 10 more ($+ 10$).

Statement in ordinary language:

Jay planted 10 more than twice the number of plants planted by John.

Solution:

The expression $p - 5$ indicates that the number of cups has been reduced by 5 from the original amount ($p$).

Statement in ordinary language:

After having some health problem, Sharad takes 5 cups of tea less per day than he used to take.

Solution:

The expression $m - 30$ indicates a decrease of 30 students compared to last year's figure ($m$).

Statement in ordinary language:

The number of students dropping out of school this year is 30 less than the number of students who dropped out last year.

Solution:

The expression $p - 5$ indicates that the current price is $\textsf{₹} 5$ lower than last month's price ($p$).

Statement in ordinary language:

The price of petrol per litre now is $\textsf{₹} 5$ less than the price last month.

Solution:

The expression $P + 1000$ indicates that the salary in 2006 is higher than the salary in 2005 ($P$) by $\textsf{₹} 1000$.

Statement in ordinary language:

Khader’s monthly salary in the year 2006 was $\textsf{₹} 1,000$ more than his salary in the year 2005.

Solution:

The expression $3g - 10$ indicates that the enrollment this year is 10 less than three times the previous year's enrollment ($g$).

Statement in ordinary language:

The number of girls enrolled in the school this year is 10 less than three times the enrollment last year.

Solution:

Let the unknown number be $x$.

(a) 13 subtracted from twice a number gives 3.

Twice a number $= 2x$

(b) One fifth of a number is 5 less than that number.

One fifth of a number $= \frac{x}{5}$

5 less than the number $= x - 5$

(c) Two-third of number is 12.

(d) 9 added to twice a number gives 13.

Twice a number $= 2x$

(e) 1 subtracted from one-third of a number gives 1.

One-third of a number $= \frac{1}{3}x$ or $\frac{x}{3}$

Solution:

(a) Perimeter (p) is three times of side (a).

(b) Diameter (d) is twice its radius (r).

(c) Selling price (s) is the sum of cost price (c) and profit (p).

(d) Amount (a) is the sum of principal (p) and interest (i).

Given:

Kanika's present age $= x$ years

Solution:

Based on the given situations, we can derive the following expressions:

Given:

Equation: $2m - 5 = -1$

Variable $m$ is a whole number less than 5, i.e., $m \in \{0, 1, 2, 3, 4\}$.

Solution:

Let's calculate the value of the expression $2m - 5$ for each value of $m$:

For $m = 0$, $2(0) - 5 = -5$

For $m = 1$, $2(1) - 5 = -3$

For $m = 2$, $2(2) - 5 = -1$

For $m = 3$, $2(3) - 5 = 1$

For $m = 4$, $2(4) - 5 = 3$

Completed Table:

Final Answer:

By inspection of the table, we see that the value of $2m - 5$ is $-1$ when $m = 2$.

Therefore, the solution of the equation is $m = 2$.

Given:

To Find:

Amount of money left to spend on other items.

Solution:

First, we calculate the total amount of money collected from all students.

Now, we find the money left by subtracting the advance payment from the total collection.

Therefore, the amount of money left with them is $\textsf{₹} (50p - 1800)$.

Given:

To Find:

Total amount of water in all the tanks.

Solution:

Step 1: Calculate the total rain water collected in all 8 tanks.

Step 2: Calculate the total amount of water by adding the water already present in one tank.

Therefore, the total amount of water in the tanks is $(8x + 100)$ litres.

Given:

To Find:

Area of the square.

Solution:

The formula for the area of a square is given by the product of its side with itself.

The area of the square is $m^2$ $\text{cm}^2$.

Given:

Formula for perimeter: $P = a + b + c$

Where $a, b, c$ are the three sides of the triangle.

Solution:

The formula $P = a + b + c$ expresses the relationship between the perimeter and the side lengths of a triangle.

In ordinary language, this rule can be stated as:

"The perimeter of a triangle is equal to the sum of the lengths of its three sides."

Alternate Expression:

The perimeter is obtained by adding the lengths of all the boundary sides of the triangle.

The given formula for the perimeter ($P$) of a rectangle is $P = 2 (l + w)$, where $l$ is the length and $w$ is the breadth (or width) of the rectangle.

The formula involves two steps:

1. Find the sum of the length and the breadth ($l + w$).

2. Multiply the sum by 2 ($2 \times (l + w)$).

This corresponds to the fact that a rectangle has two sides of length $l$ and two sides of length $w$. The perimeter is $l + w + l + w = 2l + 2w = 2(l + w)$.

Expressing the rule in words:

The rule expressed by the formula $P = 2 (l + w)$ is: The perimeter of a rectangle is twice the sum of its length and breadth.

Alternate wording:

The perimeter of a rectangle is the sum of twice its length and twice its breadth.

We need to find the present weight by adding the weight gained to the weight from last year.

Given:

Weight on last birthday = 40 kg

Weight gained after a year = $m$ kg

The present weight is the sum of the weight on the last birthday and the weight gained over the year.

Present weight = (Weight on last birthday) + (Weight gained after a year)

Present weight = $40 \text{ kg} + m \text{ kg}$

Present weight = $(40 + m) \text{ kg}$

Therefore, the present weight is $(40 + m)$ kg.

We are given the dimensions of a bulletin board and asked to solve several problems related to it.

Given:

Length of the bulletin board = $r$ cm

Breadth of the bulletin board = $t$ cm

(i) Length of the aluminium strip required to frame the board, if 10cm extra strip is required to fix it properly.

The aluminium strip is required to frame the board, which means it goes around the perimeter of the board.

The perimeter of a rectangle is given by the formula $P = 2(l + w)$.

Here, length $l = r$ cm and breadth $w = t$ cm.

Perimeter of the board = $2(r + t)$ cm.

An extra 10 cm strip is required.

Total length of aluminium strip required = (Perimeter of the board) + (Extra strip)

Total length $= 2(r + t) \text{ cm} + 10 \text{ cm}$

Total length $= \mathbf{(2(r + t) + 10) \text{ cm}}$

(ii) If x nails are used to repair one board, how many nails will be required to repair 15 such boards?

Given:

Number of nails used to repair one board = $x$ nails.

We need to find the number of nails required for 15 such boards.

Total nails required = (Number of boards) $\times$ (Nails per board)

Total nails required $= 15 \times x$ nails

Total nails required $= \mathbf{15x \text{ nails}}$

(iii) If 500sqcm extra cloth per board is required to cover the edges, what will be the total area of the cloth required to cover 8 such boards?

To cover the board, we need cloth equal to the area of the board plus the extra area for edges.

The area of a rectangle is given by the formula $A = l \times w$.

Area of one board = $r \times t$ cm$^2$ = $rt$ cm$^2$.

Extra cloth required per board for edges = 500 cm$^2$.

Total cloth required per board = (Area of the board) + (Extra area for edges)

Total cloth per board $= rt \text{ cm}^2 + 500 \text{ cm}^2 = (rt + 500) \text{ cm}^2$.

We need to find the total area of cloth required for 8 such boards.

Total area of cloth for 8 boards = (Number of boards) $\times$ (Total cloth per board)

Total area of cloth $= 8 \times (rt + 500) \text{ cm}^2$

Total area of cloth $= \mathbf{8(rt + 500) \text{ cm}^2}$ or $\mathbf{(8rt + 4000) \text{ cm}^2}$

(iv) What will be the expenditure for making 23 boards, if the carpenter charges Rs x per board.

Given:

Cost charged by the carpenter per board = $\textsf{₹} x$.

We need to find the total expenditure for making 23 boards.

Total expenditure = (Number of boards) $\times$ (Cost per board)

Total expenditure $= 23 \times \textsf{₹} x$

Total expenditure $= \mathbf{\textsf{₹} 23x}$

Let Sunita's present age be $S$ years.

Let Geeta's present age be $G$ years.

The problem states that Sunita is half the age of her mother Geeta.

This means $S = \frac{1}{2}G$ or $G = 2S$.

Let's express both ages in terms of one variable, say Sunita's present age, $S$.

Sunita's present age = $S$ years.

Geeta's present age = $2S$ years.

(i) Ages after 4 years:

After 4 years, both Sunita and Geeta will be 4 years older.

Sunita's age after 4 years = (Sunita's present age) + 4

Sunita's age after 4 years $= S + 4$ years.

Geeta's age after 4 years = (Geeta's present age) + 4

Geeta's age after 4 years $= 2S + 4$ years.

So, their ages after 4 years will be $(S + 4)$ years and $(2S + 4)$ years respectively, where $S$ is Sunita's present age.

(ii) Ages before 3 years:

Before 3 years, both Sunita and Geeta were 3 years younger.

Sunita's age before 3 years = (Sunita's present age) - 3

Sunita's age before 3 years $= S - 3$ years.

Geeta's age before 3 years = (Geeta's present age) - 3

Geeta's age before 3 years $= 2S - 3$ years.

So, their ages before 3 years were $(S - 3)$ years and $(2S - 3)$ years respectively, where $S$ is Sunita's present age.

Given:

A list of mathematical statements and terms in Column I and their potential matches in Column II.

To Find:

The correct matching pair for each item in Column I from Column II.

Solution:

(i) The number of corners of a quadrilateral

A quadrilateral (like a square or rectangle) always has exactly 4 corners. Since this number is fixed and does not change, it is known as a constant.

Thus, (i) matches with (B).

(ii) The variable in the equation $2p + 3 = 5$

In algebra, a variable is a literal or letter that represents an unknown value. In the given equation, the letter used is $p$.

Thus, (ii) matches with (E).

(iii) The solution of the equation $x + 2 = 3$

To find the solution, we solve for $x$:

Thus, (iii) matches with (C).

(iv) Solution of the equation $2p + 3 = 5$

To find the solution, we solve for $p$:

Thus, (iv) also matches with (C).

(v) A sign used in an equation

The fundamental part of any equation is the equality sign ($=$), which relates the Left Hand Side (LHS) to the Right Hand Side (RHS).

Thus, (v) matches with (A).

Final Matching Table: