Key Concept: Linear Equations in One Variable

b) $\frac{y}{2} + 5 = 10$

[Solution Description] A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. The highest power of the variable should be 1.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The equation $2x - 3 = 7$ is a linear equation in one variable because it contains only one variable, $x$, and the highest power of $x$ is 1. This satisfies the definition of a linear equation in one variable. Therefore, both the Assertion and the Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Linear Equations in One Variable, Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the equation $3x + 5 = 20$ is a linear equation in one variable because the highest power of the variable $x$ is 1. This is true as the equation satisfies the definition of a linear equation in one variable: it has only one variable ($x$) and the highest power of $x$ is 1.

The reason states that an algebraic equation is considered linear if the highest power of its variable is 1 and it has only one variable. This is also true and correctly explains why the equation $3x + 5 = 20$ is a linear equation in one variable.

Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

A linear equation in one variable is an equation where the highest power of the variable is 1. The given equation $3x + 5 = 11$ is linear because the highest power of $x$ is 1. This satisfies the definition of a linear equation in one variable. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Equations vs. Expressions

b) $2x + 3 = 7$

[Solution Description]

An equation is a statement that asserts the equality of two expressions, and it must contain an equality sign ($=$). Among the options, $2x + 3 = 7$ is an equation because it contains the equality sign and sets two expressions equal to each other.

Key Concept: Algebraic Expressions

b) $2x + 3$

[Solution Description]

A linear expression in one variable has only one variable with the highest power of 1. Let's analyze each option:

Option 1: $x^2 + 1$ is not linear because the power of $x$ is 2.

Option 2: $2x + 3$ is linear because the power of $x$ is 1.

Option 3: $y + y^2$ is not linear because the power of $y$ is 2.

Option 4: $xyz + x + y + z$ is not linear because it has multiple variables and powers greater than 1.

Therefore, the correct answer is $2x + 3$.

Key Concept: Linear Equations in One Variable, Algebraic Expressions

b) $10$

[Solution Description]

To solve for $z$, first multiply both sides by $2$:

$\frac{z - 4}{2} \times 2 = 3 \times 2$

Simplifying gives:

$z - 4 = 6$

Next, add $4$ to both sides:

$z - 4 + 4 = 6 + 4$

This results in:

$z = 10$

Therefore, the solution is $z = 10$.

Key Concept: Linear Equations in One Variable

b) 5

[Solution Description]

To solve the equation $3x + 5 = 20$, follow these steps:

Step 1: Subtract 5 from both sides of the equation.

$3x + 5 - 5 = 20 - 5$

This simplifies to:

$3x = 15$

Step 2: Divide both sides by 3.

$x = \frac{15}{3}$

Therefore, $x = 5$.

Key Concept: Equations vs Expressions

b) $2x - 3 = 9$

[Solution Description]

An equation always contains an equality sign ($=$). Let's analyze each option:

Option 1: $5x$ is an expression because it does not have an equality sign.

Option 2: $2x - 3 = 9$ is an equation because it has an equality sign.

Option 3: $\frac{y}{2} + \frac{z}{2}$ is an expression because it does not have an equality sign.

Option 4: $x^2 + 1$ is an expression because it does not have an equality sign.

Therefore, the correct answer is $2x - 3 = 9$.

Key Concept: Definition of algebraic expressions and equations, Linear equations in one variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3x + 5 = 2x - 7$ is a linear equation in one variable. This is true because the equation involves only one variable $x$ and the highest power of $x$ is 1, which satisfies the definition of a linear equation in one variable. The reason given is that all equations involving only one variable with the highest power of 1 are considered linear equations in one variable. This is also true and correctly explains why the assertion is valid.

Key Concept: Linear expressions

a) $5x + 3$

[Solution Description] A linear expression in one variable has the highest power of the variable equal to 1. Option (a) is a linear expression because the highest power of \$x\$ is 1. Options (b), (c), and (d) are not linear because they either have more than one variable or the highest power of the variable is greater than 1.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that the expression $3x + 2y$ is a linear equation in one variable. However, this is incorrect because $3x + 2y$ contains two variables, $x$ and $y$. A linear equation in one variable must have only one variable with its highest power as 1. The reason states that a linear equation in one variable has the highest power of the variable as 1, which is correct. Therefore, the assertion is false, but the reason is true.

Key Concept: Algebraic Expressions vs. Equations

c) Presence of an equality sign

[Solution Description]

An algebraic equation contains an equality sign (\$) showing that two expressions are equal, while an algebraic expression does not have an equality sign.

Key Concept: Linear Expressions, Equations

c) Only $4y - 7$ can be used.

[Solution Description]

A linear equation in one variable is formed by setting a linear expression (where the highest power of the variable is 1) equal to another expression or number. $4y - 7$ is a linear expression because the highest power of $y$ is 1. However, $y^2 + 3$ is not linear because the highest power of $y$ is 2. Therefore, only $4y - 7$ can be used to form a linear equation in one variable.

Key Concept: Forming equations from expressions

b) $3x - 7 = 10$

[Solution Description] To form an equation using the expression \$3x - 7\$, we need to set it equal to a value or another expression. Option (b) correctly forms an equation by setting \$3x - 7\$ equal to \$10\$. The other options either do not form an equation or use different expressions.

Key Concept: Equations vs Expressions

b) $2x - 3 = 9$

[Solution Description]

An equation must have an equality sign ($=$). Among the given options, $2x - 3 = 9$ is an equation because it contains an equality sign.

Key Concept: Equations vs. Expressions

b) $5x = 25$

[Solution Description]

An equation must contain an equality sign ($=$). Let's check each option:

Option 1: $2x + 3$ is an expression, not an equation, because it does not have an equality sign.

Option 2: $5x = 25$ is an equation because it contains an equality sign.

Option 3: $3y - 7$ is an expression, not an equation, because it does not have an equality sign.

Option 4: $xyz + x + y + z$ is an expression, not an equation, because it does not have an equality sign.

Key Concept: Linear expressions, constants

b) $3x + 2$

[Solution Description]

A linear expression in one variable has the highest power of the variable equal to 1. Among the given options:

$3x + 2$

is linear because the highest power of $x$ is 1.

Key Concept: Linear Equations, Variables and Constants

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the equation $3x + 5 = 2x + 10$ is a linear equation in one variable. This is true because it satisfies the definition of a linear equation in one variable, which requires the equation to have only one variable and the highest power of that variable to be 1.

The reason states that linear equations in one variable are those where the highest power of the variable is 1. This is also true as per the definition of linear equations in one variable.

Furthermore, the reason correctly explains why the equation in the assertion is a linear equation in one variable. Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear Expressions

b) $3y - 7$

[Solution Description]

A linear expression in one variable has the highest power of the variable as 1. Let's check each option:

Option 1: $x^2 + 2x + 1$ is not linear because the highest power of x is 2.

Option 2: $3y - 7$ is linear because the highest power of y is 1.

Option 3: $2xy + 5$ is not linear because it has two variables.

Option 4: $1 + z + z^2 + z^3$ is not linear because the highest power of z is 3.

Key Concept: Linear Equations, Algebraic Expressions

b) $2x - 3 = 7$

[Solution Description]

A linear equation in one variable has the highest power of the variable as 1 and contains only one variable. Let's analyze each option:

1. $x + y = 10$: This equation involves two variables, so it is not linear in one variable.

2. $2x - 3 = 7$: The highest power of $x$ is 1 and it contains only one variable, so it is linear.

3. $x^2 - 4 = 0$: The highest power of $x$ is 2, so it is not linear.

4. $y^2 + y = 6$: The highest power of $y$ is 2, so it is not linear.

Therefore, the correct answer is $2x - 3 = 7$.

Key Concept: Linear Equations, Algebraic Expressions

b) $2x + 5$

[Solution Description]

A linear expression has the highest power of the variable as 1. Let's analyze each option:

1. $x^2 + 3$: The highest power of $x$ is 2, so it is not linear.

2. $2x + 5$: The highest power of $x$ is 1, so it is linear.

3. $y^3 + y^2 + 1$: The highest power of $y$ is 3, so it is not linear.

4. $z^2 + z + 1$: The highest power of $z$ is 2, so it is not linear.

Therefore, the correct answer is $2x + 5$.

Key Concept: Equations vs. Expressions

c) An equation has an equality sign, whereas an expression does not.

[Solution Description]

An equation always contains an equality sign (\$=\$) and expresses a relationship between two expressions, known as the Left Hand Side (LHS) and Right Hand Side (RHS). An expression does not contain an equality sign and represents a mathematical phrase without equating to anything. Therefore, the statement "An equation has an equality sign, whereas an expression does not" is correct.

Key Concept: Linear Equations, Algebraic Expressions

b) $3y + 7 = 10$

[Solution Description]

A linear equation in one variable has the highest power of the variable as 1. Let's analyze each option:

1. $x^2 + 2x = 5$: The highest power of $x$ is 2, so it is not linear.

2. $3y + 7 = 10$: The highest power of $y$ is 1, so it is linear.

3. $z^3 - z = 0$: The highest power of $z$ is 3, so it is not linear.

4. $2xy = 8$: This equation involves two variables, so it is not linear in one variable.

Therefore, the correct answer is $3y + 7 = 10$.

Key Concept: Linear Equations

a) $3x + 5 = 11$

[Solution Description]

A linear equation in one variable is an equation where the highest power of the variable is 1 and there is only one variable. Let us check each option:

- $3x + 5 = 11$ is a linear equation because the highest power of $x$ is 1.

- $x^2 + 2x = 4$ is not a linear equation because the highest power of $x$ is 2.

- $y + 3 = 7$ is a linear equation but it has a different variable $y$.

- $2xy = 10$ is not a linear equation because it has two variables $x$ and $y$.

Therefore, the correct answer is $3x + 5 = 11$.

Key Concept: Solving Equations with Variables on Both Sides

c) $-9$

[Solution Description]

Start by subtracting $2x$ from both sides to collect like terms:

$3x - 2x + 4 = 2x - 2x - 5$

Simplify to get:

$x + 4 = -5$

Next, subtract $4$ from both sides to isolate $x$:

$x + 4 - 4 = -5 - 4$

This simplifies to:

$x = -9$

Key Concept: Variable on Both Sides, Fractional Equations

c) $x = \frac{29}{2}$

[Solution Description]

Start with the equation:

$\frac{4x - 7}{3} = \frac{2x + 5}{2}$

Multiply both sides by 6 (the LCM of 3 and 2) to eliminate denominators:

$6 \times \frac{4x - 7}{3} = 6 \times \frac{2x + 5}{2}$

Simplify to obtain:

$2(4x - 7) = 3(2x + 5)$

Expand both sides:

$8x - 14 = 6x + 15$

Subtract $6x$ from both sides:

$8x - 6x - 14 = 15$

Simplify to obtain:

$2x - 14 = 15$

Add $14$ to both sides:

$2x = 15 + 14$

Simplify to obtain:

$2x = 29$

Divide both sides by 2:

$x = \frac{29}{2}$

Key Concept: Solving Equations with Variables on Both Sides

c) 5

[Solution Description]

To solve the equation $3x + 5 = 2x + 10$, we first subtract $2x$ from both sides to get:

$3x - 2x + 5 = 10$

Simplifying, we have:

$x + 5 = 10$

Next, subtract $5$ from both sides:

$x = 10 - 5$

So, the solution is:

$x = 5$

Key Concept: Variable on Both Sides, Brackets

b) $x = 2$

[Solution Description]

Start with the equation:

$3(x - 4) + 2 = 5x - 2(3x + 1)$

Expand both sides:

$3x - 12 + 2 = 5x - 6x - 2$

Simplify both sides:

$3x - 10 = -x - 2$

Add $x$ to both sides:

$3x + x - 10 = -2$

Simplify to obtain:

$4x - 10 = -2$

Add $10$ to both sides:

$4x = -2 + 10$

Simplify to obtain:

$4x = 8$

Divide both sides by 4:

$x = 2$

Key Concept: Solving Equations with Variables on Both Sides

b) $4$

[Solution Description]

Multiply both sides by $2$ to eliminate the denominators:

$2 \times \left(\frac{4x - 3}{2}\right) = 2 \times \left(\frac{3x + 1}{2}\right)$

Simplify to get:

$4x - 3 = 3x + 1$

Subtract $3x$ from both sides:

$4x - 3x - 3 = 3x - 3x + 1$

Simplify to get:

$x - 3 = 1$

Add $3$ to both sides:

$x - 3 + 3 = 1 + 3$

This simplifies to:

$x = 4$

Key Concept: Balancing Equations

b) -5

[Solution Description]

To solve the equation $3x + 4 = 2x - 1$, we need to isolate the variable $x$.

Step 1: Subtract $2x$ from both sides to get:

$$3x - 2x + 4 = -1$$

$$x + 4 = -1$$

Step 2: Subtract $4$ from both sides to isolate $x$:

$$x = -1 - 4$$

$$x = -5$$

The solution to the equation is $x = -5$.

Key Concept: Balancing Equations, Transposing Terms

c) 17

[Solution Description]

To solve the equation $\frac{3x + 4}{5} = \frac{2x - 1}{3}$, we first eliminate the denominators by multiplying both sides by 15 (the least common multiple of 5 and 3). This gives:

$15 \times \left(\frac{3x + 4}{5}\right) = 15 \times \left(\frac{2x - 1}{3}\right)$

Simplifying both sides:

$3(3x + 4) = 5(2x - 1)$

Expanding both sides:

$9x + 12 = 10x - 5$

Next, we transpose the terms involving $x$ to one side and the constants to the other:

$9x - 10x = -5 - 12$

Simplifying:

$-x = -17$

Multiplying both sides by $-1$:

$x = 17$

Key Concept: Balancing Equations

a) $\frac{11}{8}$

[Solution Description]

To solve the equation $\frac{4x - 3}{2} = \frac{2x + 1}{3}$, we first eliminate the fractions by multiplying both sides by 6 (the least common multiple of 2 and 3):

Step 1: Multiply both sides by 6:

$$6 \times \left(\frac{4x - 3}{2}\right) = 6 \times \left(\frac{2x + 1}{3}\right)$$

$$3(4x - 3) = 2(2x + 1)$$

Step 2: Expand both sides:

$$12x - 9 = 4x + 2$$

Step 3: Subtract $4x$ from both sides:

$$12x - 4x - 9 = 2$$

$$8x - 9 = 2$$

Step 4: Add $9$ to both sides:

$$8x = 11$$

Step 5: Divide both sides by 8:

$$x = \frac{11}{8}$$

The solution to the equation is $x = \frac{11}{8}$.

Key Concept: Solving Equations with Variables on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $2x - 3 = x + 2$, we need to isolate the variable $x$. Subtracting $x$ from both sides is a crucial step because it helps in moving all terms containing $x$ to one side of the equation. Here is the step-by-step process:

1. Start with the equation: $2x - 3 = x + 2$.

2. Subtract $x$ from both sides: $2x - x - 3 = x - x + 2$.

3. Simplify: $x - 3 = 2$.

4. Add $3$ to both sides: $x - 3 + 3 = 2 + 3$.

5. Simplify: $x = 5$.

The assertion is true because subtracting $x$ from both sides is indeed necessary to solve for $x$. The reason is also true because this step helps in isolating the variable $x$ on one side of the equation. Moreover, the reason correctly explains the assertion.

Key Concept: Balancing Equations

c) $x = 5$

[Solution Description]

Start with the equation: $5x - 4 = 3x + 6$. Subtract $3x$ from both sides: $5x - 3x - 4 = 6$, which simplifies to $2x - 4 = 6$. Add $4$ to both sides: $2x = 6 + 4$, so $2x = 10$. Divide both sides by $2$: $x = 5$.

Key Concept: Solving Linear Equations

b) 8

[Solution Description]

To solve the equation $5x - 4 = 3x + 12$, we first subtract $3x$ from both sides:

$5x - 3x - 4 = 3x - 3x + 12$. This simplifies to:

$2x - 4 = 12$. Next, add 4 to both sides:

$2x = 12 + 4$. This gives:

$2x = 16$. Finally, divide both sides by 2 to find:

$x = \frac{16}{2}$. Therefore, the solution is:

$x = 8$.

Key Concept: Transposing terms

b) $-9$

[Solution Description]

To solve the equation $3x + 4 = 2x - 5$, we need to transpose the terms involving $x$ to one side and the constant terms to the other side.

Step 1: Subtract $2x$ from both sides of the equation:

$3x - 2x + 4 = -5$

This simplifies to:

$x + 4 = -5$

Step 2: Subtract $4$ from both sides:

$x = -5 - 4$

$x = -9$

Thus, the solution to the equation is $x = -9$.

Key Concept: Solving Equations with Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let us solve the given equation step by step:

1. The original equation is $3x + 4 = 2x - 1$.

2. To transpose $2x$ to the LHS, subtract $2x$ from both sides:

$3x + 4 - 2x = 2x - 1 - 2x$

Simplifying this gives $x + 4 = -1$.

3. Now, subtract $4$ from both sides to isolate $x$:

$x + 4 - 4 = -1 - 4$

This simplifies to $x = -5$.

4. Substituting $x = -5$ back into the original equation:

$3(-5) + 4 = 2(-5) - 1$

This results in $-15 + 4 = -10 - 1$, which simplifies to $-11 = -11$, verifying that $x = -5$ is indeed the correct solution.

Therefore, the assertion and reason are both true, and the reason correctly explains why the assertion holds.

Key Concept: Solving Equations with Variables on Both Sides, Transposing Terms

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = 2x - 5$, we need to isolate the variable $x$. Start by transposing $2x$ to the left-hand side (LHS) and $4$ to the right-hand side (RHS). This involves subtracting $2x$ from both sides and adding $5$ to both sides.

Step 1: Subtract $2x$ from both sides:

$3x - 2x + 4 = 2x - 2x - 5$

Simplifies to:

$x + 4 = -5$

Step 2: Add $5$ to both sides:

$x + 4 + 5 = -5 + 5$

Simplifies to:

$x + 9 = 0$

Step 3: Subtract $9$ from both sides:

$x + 9 - 9 = 0 - 9$

Simplifies to:

$x = -9$

The solution to the equation is $x = -9$. The assertion correctly states that the equation can be solved by transposing $2x$ to LHS and $4$ to RHS. The reason is also true, as transposing terms indeed involves moving terms from one side to the other while maintaining equality. However, the reason does not explain why transposing works in this specific case; it only defines what transposing is. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Solving Linear Equations

a) -7

[Solution Description]

To solve the equation $\frac{4x + 6}{2} = \frac{2x - 8}{2}$, we first multiply both sides by 2 to eliminate the denominators:

$2 \times \left(\frac{4x + 6}{2}\right) = 2 \times \left(\frac{2x - 8}{2}\right)$. This simplifies to:

$4x + 6 = 2x - 8$. Next, subtract $2x$ from both sides:

$4x - 2x + 6 = 2x - 2x - 8$. This simplifies to:

$2x + 6 = -8$. Then, subtract 6 from both sides:

$2x = -8 - 6$. This gives:

$2x = -14$. Finally, divide both sides by 2 to find:

$x = \frac{-14}{2}$. Therefore, the solution is:

$x = -7$.

Key Concept: Solving Equations Having Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x - 5 = x + 7$, we can follow these steps:

1. Transpose all terms involving the variable $x$ to one side and constants to the other side: $3x - x = 7 + 5$.

2. Simplify both sides: $2x = 12$.

3. Solve for $x$ by dividing both sides by 2: $x = 6$.

Thus, the equation has a unique solution, and transposing terms involving variables to one side simplifies the equation and helps in finding the solution.

Key Concept: Solving Equations Having Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = x + 10$, follow these steps:

Step 1: Subtract $x$ from both sides to consolidate the variable terms.

$3x + 4 - x = x + 10 - x$

Step 2: Simplify both sides.

$2x + 4 = 10$

Step 3: Subtract $4$ from both sides to isolate the term with $x$.

$2x = 10 - 4$

Step 4: Simplify the right-hand side.

$2x = 6$

Step 5: Divide both sides by $2$ to solve for $x$.

$x = \frac{6}{2} = 3$

Thus, the solution is $x = 3$. The Assertion is true because subtracting $x$ from both sides is the correct first step. The Reason is also true and correctly explains why this step is necessary.

Key Concept: Solving Equations Having Variable on Both Sides

b) -7

[Solution Description]

Start with the equation $\frac{4x + 6}{2} = \frac{2x - 8}{2}$. Multiply both sides by 2 to eliminate the denominators: $4x + 6 = 2x - 8$. Subtract $2x$ from both sides to get $2x + 6 = -8$. Subtract 6 from both sides to get $2x = -14$. Finally, divide both sides by 2 to find $x = -7$.

Key Concept: Solving Equations, Transposing terms

c) $-12$

[Solution Description]

To solve the equation $3x + 7 = 2x - 5$, follow these steps:

1. Subtract $2x$ from both sides to get all variable terms on one side:

$$3x - 2x + 7 = 2x - 2x - 5$$

$$x + 7 = -5$$

2. Now, subtract $7$ from both sides to isolate the variable term:

$$x + 7 - 7 = -5 - 7$$

$$x = -12$$

The solution to the equation is $x = -12$.

Key Concept: Solving Equations, Variable on Both Sides

d) Assertion is false, but Reason is true.

[Solution Description]

Let's solve the equation step by step:

Given the equation: $\frac{3x - 4}{2} = \frac{5x + 6}{3}$

Step 1: Eliminate the denominators by multiplying both sides by 6 (the least common multiple of 2 and 3):

$6 \times \left(\frac{3x - 4}{2}\right) = 6 \times \left(\frac{5x + 6}{3}\right)$

Step 2: Simplify both sides:

$3(3x - 4) = 2(5x + 6)$

Step 3: Distribute the constants:

$9x - 12 = 10x + 12$

Step 4: Move all terms involving $x$ to one side and constants to the other:

$9x - 10x = 12 + 12$

Step 5: Solve for $x$:

$-x = 24 \quad \Rightarrow \quad x = -24$

The solution to the equation is $x = -24$. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Solving linear equations, transposing terms, combining like terms

b) 4

[Solution Description]

To solve the equation $5x + 12 = 3x + 20$, we need to isolate the variable $x$.

Step 1: Subtract $3x$ from both sides of the equation to get all the $x$ terms on one side:

$5x - 3x + 12 = 3x - 3x + 20$

Simplifying this, we get:

$2x + 12 = 20$

Step 2: Subtract $12$ from both sides to solve for $x$:

$2x + 12 - 12 = 20 - 12$

Simplifying this, we get:

$2x = 8$

Step 3: Divide both sides by $2$ to find $x$:

$x = \frac{8}{2}$

Simplifying this, we get:

$x = 4$

Therefore, the solution is $x = 4$.

Key Concept: Solving Equations with Variables on Both Sides

c) -12

[Solution Description]

First, eliminate the denominators by multiplying both sides by the least common multiple of 3 and 2, which is 6:

$6 \times \left(\frac{4x + 6}{3}\right) = 6 \times \left(\frac{2x - 4}{2}\right)$

Simplify both sides:

$2(4x + 6) = 3(2x - 4)$

Expand the equations:

$8x + 12 = 6x - 12$

Subtract $6x$ from both sides:

$8x - 6x + 12 = -12$

Simplify to get:

$2x + 12 = -12$

Subtract $12$ from both sides:

$2x = -24$

Finally, divide both sides by 2:

$x = -12$

Key Concept: Solving Linear Equations with Variables on Both Sides

c) $x = 5$

[Solution Description]

To solve the equation $5x - 7 = 2x + 8$, follow these steps:

Step 1: Subtract $2x$ from both sides to get all $x$ terms on one side.

$5x - 2x - 7 = 2x - 2x + 8$

$3x - 7 = 8$

Step 2: Add 7 to both sides to isolate the $x$ term.

$3x - 7 + 7 = 8 + 7$

$3x = 15$

Step 3: Divide both sides by 3 to solve for $x$.

$\frac{3x}{3} = \frac{15}{3}$

$x = 5$

Therefore, the solution is $x = 5$.

Key Concept: Solving Equations Having Variable on Both Sides

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 5 = 2x - 7$, we first transpose terms involving $x$ to the left-hand side (LHS) and constants to the right-hand side (RHS). Subtracting $2x$ from both sides gives $3x - 2x + 5 = -7$, which simplifies to $x + 5 = -7$. Next, subtracting $5$ from both sides yields $x = -12$. This shows that the equation has a unique solution, confirming the Assertion. The Reason explains the method used to solve such equations by transposing variables and constants, which is correct but does not directly explain why the equation has a unique solution. Therefore, both are true, but the Reason is not the correct explanation of the Assertion.

Key Concept: Solving Equations with Variables on Both Sides

c) $\frac{23}{3}$

[Solution Description]

First, expand both sides of the equation:

$5x - 15 = 2x + 8$

Next, subtract $2x$ from both sides to gather like terms:

$5x - 2x - 15 = 8$

Simplify to get:

$3x - 15 = 8$

Add $15$ to both sides:

$3x = 23$

Finally, divide both sides by 3:

$x = \frac{23}{3}$

Key Concept: Reducing Equations to Simpler Form

b) $x = 2$

[Solution Description]

To solve the equation $4x - 3(2x - 5) = 2x + 7$, first expand the brackets:

$4x - 6x + 15 = 2x + 7$

Combine like terms:

$-2x + 15 = 2x + 7$

Move all terms involving $x$ to one side and constants to the other:

$-2x - 2x = 7 - 15$

Simplify:

$-4x = -8$

Solve for $x$:

$x = \frac{-8}{-4} = 2$

So, the solution is $x = 2$.

Key Concept: Reducing Equations to Simpler Form

b) $x = \frac{24}{7}$

[Solution Description]

To solve the equation, we first find the LCM of the denominators, which is 4. Multiply both sides by 4:

$4 \left( \frac{5x - 3}{2} \right) - 4 \left( \frac{3x + 4}{4} \right) = 4 \times \frac{7}{2}$

Simplifying, we get: $2(5x - 3) - (3x + 4) = 14$

Open the brackets: $10x - 6 - 3x - 4 = 14$

Combine like terms: $7x - 10 = 14$

Add 10 to both sides: $7x = 24$

Divide both sides by 7: $x = \frac{24}{7}$.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the given equation $\frac{3x + 4}{2} - \frac{x - 1}{4} = 5$, we first find the LCM of the denominators, which is 4. Multiplying both sides of the equation by 4:

$4 \left( \frac{3x + 4}{2} \right) - 4 \left( \frac{x - 1}{4} \right) = 4 \times 5$

Simplifying each term:

$2(3x + 4) - (x - 1) = 20$

Expanding the brackets:

$6x + 8 - x + 1 = 20$

Combining like terms:

$5x + 9 = 20$

Solving for $x$:

$5x = 11 \quad \text{and} \quad x = \frac{11}{5}$

Both Assertion and Reason are true, and Reason correctly explains how multiplying by the LCM simplifies the equation.

Key Concept: Reducing Equations to Simpler Form

a) $x = \frac{25}{31}$

[Solution Description]

First, expand the brackets on the left side:

$6x - 8 + \frac{5x + 1}{2} = \frac{3x - 5}{4}$

To eliminate the fractions, multiply both sides by 4 (the LCM of 2 and 4):

$4(6x - 8) + 4 \left( \frac{5x + 1}{2} \right) = 4 \left( \frac{3x - 5}{4} \right)$

Simplify each term:

$24x - 32 + 2(5x + 1) = 3x - 5$

Expand the brackets:

$24x - 32 + 10x + 2 = 3x - 5$

Combine like terms:

$34x - 30 = 3x - 5$

Move all $x$ terms to one side and constants to the other:

$34x - 3x = -5 + 30$

Simplify:

$31x = 25$

Divide both sides by 31:

$x = \frac{25}{31}$

The solution is $x = \frac{25}{31}$.

Key Concept: Reducing Equations to Simpler Form

a) $x = -\frac{8}{15}$

[Solution Description]

To solve the equation, first find the LCM of the denominators (4, 3, and 6), which is 12. Multiply both sides by 12 to eliminate the fractions:

$12 \left( \frac{3x - 2}{4} \right) + 12 \left( \frac{2x + 1}{3} \right) = 12 \left( \frac{x - 5}{6} \right)$

Simplifying each term:

$3(3x - 2) + 4(2x + 1) = 2(x - 5)$

Expand the brackets:

$9x - 6 + 8x + 4 = 2x - 10$

Combine like terms:

$17x - 2 = 2x - 10$

Move all $x$ terms to one side and constants to the other:

$17x - 2x = -10 + 2$

Simplify:

$15x = -8$

Divide both sides by 15:

$x = -\frac{8}{15}$

The solution is $x = -\frac{8}{15}$.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The given equation is $\frac{6x + 1}{3} + 1 = \frac{x - 3}{6}$. The denominators in the equation are 3 and 6, and their LCM is 6. Multiplying both sides of the equation by 6 eliminates the fractions, leading to a simpler form of the equation. This step is correct because it simplifies the equation without changing its solution. Therefore, the Assertion is true. The Reason is also true as multiplying both sides by the LCM of the denominators is a standard method to simplify equations containing fractions. Moreover, the Reason correctly explains why this step is valid.

Key Concept: Linear Equations

a) $\frac{9}{10}$

[Solution Description]

Find the LCM of the denominators 5, 10, and 2, which is 10. Multiply both sides by 10:

$10 \times \frac{2x + 3}{5} + 10 \times \frac{x - 2}{10} = 10 \times \frac{3x - 1}{2}$

Simplify each term:

$2(2x + 3) + (x - 2) = 5(3x - 1)$

Open the brackets:

$4x + 6 + x - 2 = 15x - 5$

Combine like terms:

$5x + 4 = 15x - 5$

Subtract $5x$ from both sides:

$4 = 10x - 5$

Add 5 to both sides:

$9 = 10x$

Divide both sides by 10:

$x = \frac{9}{10}$

Key Concept: Linear Equations

a) $\frac{5}{3}$

[Solution Description]

Open the brackets on both sides:

$6x - 10 + 4 = 3x + 6 - 7$

Simplify each side:

$6x - 6 = 3x - 1$

Subtract $3x$ from both sides:

$3x - 6 = -1$

Add 6 to both sides:

$3x = 5$

Divide both sides by 3:

$x = \frac{5}{3}$

Key Concept: Linear Equations, Simplification

a) $\frac{13}{9}$

[Solution Description]

To solve the equation $\frac{5x - 3}{4} + \frac{x}{2} = \frac{3x + 1}{3}$, first find the LCM of the denominators 4, 2, and 3, which is 12. Multiply both sides by 12 to eliminate the denominators:

$12 \times \left(\frac{5x - 3}{4}\right) + 12 \times \left(\frac{x}{2}\right) = 12 \times \left(\frac{3x + 1}{3}\right)$

Simplify each term:

$3(5x - 3) + 6x = 4(3x + 1)$

Expand the terms:

$15x - 9 + 6x = 12x + 4$

Combine like terms:

$21x - 9 = 12x + 4$

Transpose $12x$ to the left side and $-9$ to the right side:

$21x - 12x = 4 + 9$

Simplify:

$9x = 13$

Divide both sides by 9:

$x = \frac{13}{9}$

The solution is $x = \frac{13}{9}$.

Key Concept: Reducing Equations to Simpler Form

b) $x = -8$

[Solution Description]

To solve the equation, first eliminate the fractions by multiplying each term by 4:

$4 \left( \frac{3x + 2}{4} \right) + 4 \times 1 = 4 \left( \frac{x - 1}{2} \right)$

Simplifying, we get:

$3x + 2 + 4 = 2(x - 1)$

Combine like terms:

$3x + 6 = 2x - 2$

Subtract $2x$ from both sides:

$x + 6 = -2$

Subtract 6 from both sides:

$x = -8$

So, the solution is $x = -8$.

Key Concept: Use of LCM to eliminate fractions

c) $\frac{3}{5}$

[Solution Description]

To solve the equation, we first find the LCM of the denominators 4, 2, and 8, which is 8. We then multiply both sides of the equation by 8 to eliminate the fractions:

$8 \left( \frac{3x - 2}{4} \right) + 8 \left( \frac{x + 1}{2} \right) = 8 \left( \frac{5x + 3}{8} \right)$

Simplifying each term:

$2(3x - 2) + 4(x + 1) = 5x + 3$

Expanding the brackets:

$6x - 4 + 4x + 4 = 5x + 3$

Combining like terms:

$10x = 5x + 3$

Subtracting $5x$ from both sides:

$5x = 3$

Dividing both sides by 5:

$x = \frac{3}{5}$

Key Concept: Reducing Equations to Simpler Form, Use of LCM to eliminate fractions

b) $\frac{2}{3}$

[Solution Description]

To solve the equation $\frac{4x + 5}{3} - \frac{3x + 2}{2} = \frac{5x}{6}$, first find the LCM of the denominators 3, 2, and 6. The LCM is 6. Multiply both sides of the equation by 6 to eliminate the fractions:

$6 \left( \frac{4x + 5}{3} \right) - 6 \left( \frac{3x + 2}{2} \right) = 6 \left( \frac{5x}{6} \right)$

Simplify each term:

$2(4x + 5) - 3(3x + 2) = 5x$

Expand the terms:

$8x + 10 - 9x - 6 = 5x$

Combine like terms:

$-x + 4 = 5x$

Add x to both sides:

$4 = 6x$

Divide both sides by 6:

$x = \frac{2}{3}$

Therefore, the solution is $x = \frac{2}{3}$.

Key Concept: Reducing Equations to Simpler Form, Use of LCM to eliminate fractions

b) 10

[Solution Description]

To solve the equation $\frac{3x + 2}{4} - \frac{2x - 1}{3} = \frac{x}{6}$, first find the LCM of the denominators 4, 3, and 6. The LCM is 12. Multiply both sides of the equation by 12 to eliminate the fractions:

$12 \left( \frac{3x + 2}{4} \right) - 12 \left( \frac{2x - 1}{3} \right) = 12 \left( \frac{x}{6} \right)$

Simplify each term:

$3(3x + 2) - 4(2x - 1) = 2x$

Expand the terms:

$9x + 6 - 8x + 4 = 2x$

Combine like terms:

$x + 10 = 2x$

Subtract x from both sides:

$10 = x$

Therefore, the solution is $x = 10$.

Key Concept: Use of LCM to eliminate fractions

c) $1$

[Solution Description]

To solve the equation, we first find the LCM of the denominators 5, 10, and 2, which is 10. We then multiply both sides of the equation by 10 to eliminate the fractions:

$10 \left( \frac{4x - 3}{5} \right) + 10 \left( \frac{2x + 1}{10} \right) = 10 \left( \frac{3x - 2}{2} \right)$

Simplifying each term:

$2(4x - 3) + (2x + 1) = 5(3x - 2)$

Expanding the brackets:

$8x - 6 + 2x + 1 = 15x - 10$

Combining like terms:

$10x - 5 = 15x - 10$

Subtracting $15x$ from both sides:

$-5x - 5 = -10$

Adding 5 to both sides:

$-5x = -5$

Dividing both sides by $-5$:

$x = 1$

Key Concept: Reducing Equations to Simpler Form, Use of LCM to eliminate fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The given equation is $\frac{3x + 2}{4} + \frac{x - 1}{2} = \frac{5x - 3}{6}$. The denominators are 4, 2, and 6. The LCM of 4, 2, and 6 is 12. Multiplying both sides of the equation by 12, we get:

$12 \left( \frac{3x + 2}{4} \right) + 12 \left( \frac{x - 1}{2} \right) = 12 \left( \frac{5x - 3}{6} \right)$

Simplifying each term:

$3(3x + 2) + 6(x - 1) = 2(5x - 3)$

Expanding the terms:

$9x + 6 + 6x - 6 = 10x - 6$

Combining like terms:

$15x = 10x - 6$

Solving for $x$:

$15x - 10x = -6$

$5x = -6$

$x = -\frac{6}{5}$

The assertion is true as multiplying by the LCM simplifies the equation. The reason is also true since multiplying by the LCM eliminates the fractions, simplifying the equation. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Handling brackets and distributed terms

b) $y = \frac{9}{5}$

[Solution Description]

Step 1: Expand the left-hand side by opening the bracket:

$2(3y - 1) = 6y - 2$

Step 2: The equation now becomes:

$6y - 2 = y + 7$

Step 3: Transpose $y$ to LHS and $-2$ to RHS:

$6y - y = 7 + 2$

Step 4: Simplify:

$5y = 9$

Step 5: Divide both sides by 5:

$y = \frac{9}{5}$

Key Concept: Handling brackets and distributed terms

b) $a = 15$

[Solution Description]

Step 1: Expand the left-hand side by opening the bracket:

$4(a - 3) + 2 = 4a - 12 + 2 = 4a - 10$

Step 2: The equation now becomes:

$4a - 10 = 3a + 5$

Step 3: Transpose $3a$ to LHS and $-10$ to RHS:

$4a - 3a = 5 + 10$

Step 4: Simplify:

$a = 15$

Key Concept: Handling brackets, distributed terms

a) $\frac{1}{12}$

[Solution Description]

Eliminate the denominators by multiplying both sides by 20:

$20 \left( \frac{2(x - 3)}{5} \right) + 20 \left( \frac{3(2x + 1)}{4} \right) = 20 \left( \frac{5x - 1}{2} \right)$

Simplify each term:

$8(x - 3) + 15(2x + 1) = 10(5x - 1)$

Open the brackets:

$8x - 24 + 30x + 15 = 50x - 10$

Combine like terms:

$38x - 9 = 50x - 10$

Subtract $38x$ from both sides:

$-9 = 12x - 10$

Add 10 to both sides:

$1 = 12x$

Divide by 12:

$x = \frac{1}{12}$

The solution is $x = \frac{1}{12}$.

Key Concept: Reducing Equations to Simpler Form, Handling brackets and distributed terms

d) Assertion is false, but Reason is true.

[Solution Description]

Step 1: Multiply both sides by 12 to eliminate the denominators:

$12 \left( \frac{3x + 2}{4} - \frac{2x - 1}{3} \right) = 12 \left( \frac{x + 5}{6} \right)$

Step 2: Simplify each term:

$3(3x + 2) - 4(2x - 1) = 2(x + 5)$

Step 3: Expand the brackets:

$9x + 6 - 8x + 4 = 2x + 10$

Step 4: Combine like terms:

$x + 10 = 2x + 10$

Step 5: Transpose terms to isolate $x$:

$x - 2x = 10 - 10$

Step 6: Solve for $x$:

$-x = 0 \implies x = 0$

Conclusion: The Assertion is false because the correct solution is $x = 0$, not $x = 7$. However, the Reason is true as multiplying by the LCM of the denominators is a valid method to simplify the equation.

Key Concept: Handling brackets and distributed terms

b) $x = 4$

[Solution Description]

First, expand the brackets on the left side:

$5(x + 3) = 5x + 15$

$-3(2x - 4) = -6x + 12$

Combine these to get:

$5x + 15 - 6x + 12 = 2x + 15$

Simplify by combining like terms:

$-x + 27 = 2x + 15$

Add $x$ to both sides:

$27 = 3x + 15$

Subtract $15$ from both sides:

$12 = 3x$

Divide both sides by $3$:

$x = 4$

Key Concept: Reducing Equations to Simpler Form, Stepwise approach

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the given equation, multiply both sides by the LCM of the denominators, which is 4. This eliminates the fractions and simplifies the equation:

$4 \left( \frac{3x - 1}{2} \right) + 4 \left( \frac{x + 2}{4} \right) = 4 \left( \frac{5x - 3}{4} \right)$

Simplifying each term:

$2(3x - 1) + (x + 2) = 5x - 3$

Expanding the terms:

$6x - 2 + x + 2 = 5x - 3$

Combining like terms:

$7x = 5x - 3$

Transposing $5x$ to the left side:

$2x = -3$

Dividing by 2:

$x = -\frac{3}{2}$

Both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Reducing Equations to Simpler Form

b) $x = -3$

[Solution Description]

Find the LCM of the denominators, which is 12. Multiply both sides of the equation by 12:

$12 \left( \frac{2x - 1}{3} \right) + 12 \left( \frac{x + 2}{6} \right) = 12 \left( \frac{3x - 1}{4} \right)$

Simplify each term:

$4(2x - 1) + 2(x + 2) = 3(3x - 1)$

Expand the brackets:

$8x - 4 + 2x + 4 = 9x - 3$

Combine like terms:

$10x = 9x - 3$

Subtract $9x$ from both sides:

$x = -3$

The solution is $x = -3$.

Key Concept: Reducing Equations to Simpler Form

b) $-7$

[Solution Description]

Eliminate the denominators by multiplying both sides by 6 (the LCM of 3 and 2):

$6 \times \left( \frac{2x + 5}{3} \right) = 6 \times \left( \frac{x + 1}{2} \right)$

Simplifying both sides, we get:

$2(2x + 5) = 3(x + 1)$

Expanding both sides:

$4x + 10 = 3x + 3$

Transposing $3x$ to the left side and $10$ to the right side:

$4x - 3x = 3 - 10$

Simplifying further:

$x = -7$

Therefore, the solution is $x = -7$.

Key Concept: Reducing Equations to Simpler Form

c) $9$

[Solution Description]

Start by expanding both sides of the equation:

$2x + 6 = 3x - 3$

Transpose $2x$ to the right side and $-3$ to the left side:

$6 + 3 = 3x - 2x$

Simplifying both sides:

$9 = x$

Therefore, the solution is $x = 9$.

Key Concept: Reducing Equations, Combining Like Terms

a) $-\frac{3}{5}$

[Solution Description]

To solve the equation, first find the LCM of the denominators (4, 2, 8), which is 8. Multiply both sides by 8 to eliminate the fractions:

$8 \left( \frac{3x + 2}{4} \right) + 8 \left( \frac{x - 1}{2} \right) = 8 \left( \frac{5x - 3}{8} \right)$

Simplify each term:

$2(3x + 2) + 4(x - 1) = 5x - 3$

Expand and combine like terms:

$6x + 4 + 4x - 4 = 5x - 3$

$10x = 5x - 3$

Subtract $5x$ from both sides:

$5x = -3$

Finally, divide both sides by 5:

$x = -\frac{3}{5}$

Key Concept: Applications of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To find the width $w$, we use the perimeter formula for a rectangle: $P = 2(l + w)$. Given $P = 20$ units and $l = 6$ units, substituting these values into the formula gives $2(6 + w) = 20$. Simplifying this equation step by step:

$2(6 + w) = 20 \\ \\ 6 + w = \frac{20}{2} \\ \\ 6 + w = 10 \\ \\ w = 10 - 6 \\ \\ w = 4$

Therefore, the width $w$ is 4 units. Both the assertion and reason are true, and the reason correctly explains the assertion through the application of the perimeter formula.

Key Concept: Perimeters, Numbers

b) 6

[Solution Description]

Let the width of the rectangle be $w$. Then, the length is $2w + 5$. The perimeter of a rectangle is given by:

$2(\text{length} + \text{width}) = 50$

Substitute the expressions for length and width:

$2((2w + 5) + w) = 50$

Simplify inside the parentheses:

$2(3w + 5) = 50$

Distribute the 2:

$6w + 10 = 50$

Subtract 10 from both sides:

$6w = 40$

Divide by 6:

$w = \frac{40}{6} = \frac{20}{3} \approx 6.67$

Since the width must be a positive real number, the width is $\frac{20}{3}$ meters.

Key Concept: Numbers, Combination of Currency Notes

d) 8

[Solution Description]

Let the number of \$10 notes be $x$. Then, the number of \$20 notes is $x + 5$. The total amount can be expressed as:

$10x + 20(x + 5) = 200$

Simplify the equation:

$10x + 20x + 100 = 200$

Combine like terms:

$30x + 100 = 200$

Subtract 100 from both sides:

$30x = 100$

Divide by 30:

$x = \frac{100}{30} = \frac{10}{3} \approx 3.33$

Since the number of notes must be an integer, there is no valid solution with this condition.

Key Concept: Ages

c) 12

[Solution Description]

Let Alice's current age be $x$. Then John's current age is $4x$. In 6 years:

$4x + 6 = 2(x + 6)$

Solve the equation:

$4x + 6 = 2x + 12$

$2x = 6$

$x = 3$

John's current age is $4x = 12$.

Key Concept: Ages, Numbers

c) 24

[Solution Description]

Let the current age of the son be $x$. Then, the current age of the father is $65 - x$. Five years ago, the son's age was $x - 5$ and the father's age was $(65 - x) - 5 = 60 - x$. According to the problem:

$60 - x = 2(x - 5)$

Expand the right side:

$60 - x = 2x - 10$

Add $x$ to both sides:

$60 = 3x - 10$

Add 10 to both sides:

$70 = 3x$

Divide by 3:

$x = \frac{70}{3} \approx 23.33$

Since age must be an integer, there is no valid solution with this condition.

Key Concept: Word problems on numbers

d) 12

[Solution Description]

Let the width be $w$. Then, the length is $2w$. The perimeter of a rectangle is given by $P = 2l + 2w$, where $l$ is the length and $w$ is the width. Substituting the known values, we get $36 = 2(2w) + 2w = 4w + 2w = 6w$. Solving for $w$, we find $w = 6$. Therefore, the length is $2w = 2 \times 6 = 12$.

Key Concept: Linear Equations, Word Problems on Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the first number be $x$ and the second number be $y$. According to the problem, $x = 2y + 6$. Also, $x + y = 24$. We can substitute the expression for $x$ from the first equation into the second equation: $2y + 6 + y = 24$. Simplifying this, we get $3y + 6 = 24$. Subtracting 6 from both sides, $3y = 18$. Dividing both sides by 3, $y = 6$. Substituting $y = 6$ back into the first equation, $x = 2(6) + 6 = 12 + 6 = 18$. Therefore, the larger number is indeed 18. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Word problem on numbers

c) 30 and 15

[Solution Description]

Let the two numbers be $x$ and $y$. According to the problem, we have two equations:

$x + y = 45$

$x - y = 15$

Adding both equations gives us:

$2x = 60$

Solving for $x$:

$x = 30$

Substituting $x = 30$ in the first equation:

$30 + y = 45$

Solving for $y$:

$y = 15$

Therefore, the two numbers are 30 and 15.

Key Concept: Word problems on numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the two consecutive integers be $n$ and $n+1$. Their sum is $n + (n+1) = 2n + 1$. Since $2n$ is always even, adding 1 makes it odd. Therefore, the assertion is true. The reason states that if one integer is even, the other must be odd, which is also true for consecutive integers. Moreover, this is the correct explanation for why their sum is odd. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Age ratio

a) 12 years

[Solution Description]

Let the ages of the two brothers be in the ratio 3:5. Given that the elder brother is 20 years old, we can set up the equation:

$\frac{3}{5} = \frac{x}{20}$

Solving for $x$:

$3 \times 20 = 5 \times x$

$60 = 5x$

Dividing by 5:

$x = 12$

So, the younger brother is 12 years old.

Key Concept: Future age

c) 20 years

[Solution Description]

Let the number of years required be $x$. After $x$ years, the mother's age will be $40 + x$ and the daughter's age will be $10 + x$. According to the problem:

$40 + x = 2(10 + x)$

Expanding the right side:

$40 + x = 20 + 2x$

Subtracting $x$ from both sides:

$40 = 20 + x$

Subtracting 20 from both sides:

$x = 20$

So, the mother will be twice as old as her daughter in 20 years.

Key Concept: Age-related problems

a) 12

[Solution Description]

Let the current age of the daughter be $x$ and the man be $y$. According to the problem:

$y = 4x$ (Equation 1)

In six years, their ages will be $(x + 6)$ and $(y + 6)$, respectively. According to the second condition:

$y + 6 = 3(x + 6)$ (Equation 2)

Substituting Equation 1 into Equation 2:

$4x + 6 = 3x + 18$

Solving for $x$:

$x = 12$

Thus, the daughter's current age is 12 years.

Key Concept: Age-related problems

b) 8

[Solution Description]

Let the daughter's current age be $x$. Then the mother's current age is $x + 24$. In 4 years, the daughter's age will be $x + 4$ and the mother's age will be $(x + 24) + 4 = x + 28$. According to the problem, in 4 years, the mother will be three times as old as her daughter:

$x + 28 = 3(x + 4)$

Simplify the equation:

$x + 28 = 3x + 12$

Subtract $x$ from both sides:

$28 = 2x + 12$

Subtract 12 from both sides:

$16 = 2x$

Divide both sides by 2:

$x = 8$

Therefore, the daughter's current age is 8.

Key Concept: Perimeter, Linear Equations

c) Width: 12.5 units, Length: 37.5 units

[Solution Description]

Let the width of the rectangle be $x$ units. Then, the length is $3x$ units. The perimeter $P$ of a rectangle is given by:

$P = 2(\text{length} + \text{width})$

Substituting the given perimeter:

$100 = 2(3x + x)$

Simplifying:

$100 = 8x$

Solving for $x$:

$x = 12.5$

Therefore, the width is $12.5$ units and the length is $37.5$ units.

Key Concept: Perimeter-based problems, Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let's analyze both the Assertion and Reason:

1. **Assertion Analysis**: The perimeter of a rectangle is calculated using the formula $P = 2(l + w)$. If the perimeter is increased by 10 units, the new perimeter becomes $P_{\text{new}} = P + 10$. Since 10 is a positive number, $P_{\text{new}}$ will always be greater than $P$, regardless of the values of $l$ and $w$.

2. **Reason Analysis**: The Reason states that increasing any positive quantity by 10 units will result in a larger value. This is mathematically correct because adding a positive number to any positive quantity will always increase its value.

3. **Relationship Between Assertion and Reason**: The Reason correctly explains why the Assertion is true. The increase in perimeter is directly due to the addition of a positive quantity (10 units), which aligns with the Reason provided.

Key Concept: Perimeter of a square

b) 6 cm

[Solution Description]

The perimeter of a square is given by the formula: $P = 4 \times \text{side length}$. Let the side length be $x$ cm. Substituting the given perimeter: $24 = 4x$. Dividing both sides by 4 gives $x = 6$. Therefore, the length of one side of the square is 6 cm.

Key Concept: Perimeter, Linear Equations

b) 12 units

[Solution Description]

The perimeter of the square is $4 \times 9 = 36$ units. Let the side length of the equilateral triangle be $x$. Its perimeter is $3x$. Since both perimeters are equal:

$3x = 36$

Solving for $x$:

$x = 12$

Therefore, the side length of the equilateral triangle is 12 units.

Key Concept: Money-related problem

b) 6

[Solution Description]

Let the number of dimes be $x$. Then, the number of quarters is $3x$. The total value of the dimes and quarters can be expressed as:

$0.10x + 0.25(3x) = 5.25$

Simplifying the equation:

$0.10x + 0.75x = 5.25$

$0.85x = 5.25$

Solving for $x$:

$x = \frac{5.25}{0.85} = 6.176$

Since the number of coins must be an integer, this problem has no valid solution within the given constraints.

Key Concept: Currency Conversion

c) 127.5 EUR

[Solution Description]

To convert USD to EUR, multiply the amount in USD by the exchange rate.

Given the exchange rate is 1 USD = 0.85 EUR, the conversion for 150 USD is:

$150 \times 0.85 = 127.5$

Therefore, you will get 127.5 Euros for 150 USD.

Key Concept: Combination of currency notes, Linear equations

b) 3

[Solution Description]

Let the number of \$5 notes be $x$. Then, the number of \$10 notes is $x + 3$. The total amount of money can be expressed as:

$5x + 10(x + 3) = 50$

Simplify the equation:

$5x + 10x + 30 = 50$

Combine like terms:

$15x + 30 = 50$

Subtract 30 from both sides:

$15x = 20$

Divide both sides by 15:

$x = \frac{20}{15} = \frac{4}{3}$

Since the number of notes must be an integer, this problem has no valid solution. However, for the purpose of this question, we will proceed with the calculated value.

Key Concept: Applications of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the smaller part be $x$. According to the problem, the larger part is twice the smaller part, so it is $2x$. The total amount is \$20, so we can write the equation $x + 2x = 20$. Simplifying this, we get $3x = 20$. Dividing both sides by 3, we find $x = \frac{20}{3} \approx 6.67$. Therefore, the smaller part is indeed \$6.67. Both the assertion and the reason are true, and the reason correctly explains how the assertion is derived.

Key Concept: Application of Linear Equations - Currency Combination

c) 5

[Solution Description]

Let the number of 5-dollar bills be $x$. Then, the number of 10-dollar bills is $(7 - x)$. The total amount of money from 5-dollar bills is $5x$, and from 10-dollar bills is $10(7 - x)$. The total amount of money is given as \$45. So, we can write the equation: $5x + 10(7 - x) = 45$. Simplifying this, we get: $5x + 70 - 10x = 45$. Combining like terms, we get: $-5x + 70 = 45$. Subtracting 70 from both sides, we get: $-5x = -25$. Dividing both sides by -5, we get: $x = 5$. Therefore, there are 5 five-dollar bills.

Key Concept: Combination of Currency Notes

c) 10

[Solution Description]

Let the number of \$5 notes be $x$. Then, the number of \$10 notes is $(20 - x)$.

The total amount of money is given by:

$5x + 10(20 - x) = 150$

Expand the equation:

$5x + 200 - 10x = 150$

Combine like terms:

$-5x + 200 = 150$

Subtract 200 from both sides:

$-5x = -50$

Divide both sides by -5:

$x = 10$

Therefore, the person has 10 \$5 notes.

Key Concept: Real-life application of linear equations

c) \$35

[Solution Description]

The cost of one pen is \$5 and one notebook is \$10. The total cost for 3 pens and 2 notebooks can be calculated as follows:

$\text{Total Cost} = (3 \times 5) + (2 \times 10) = 15 + 20 = 35$

Therefore, the total cost is \$35.

Key Concept: Application of Linear Equations - Perimeter Problem

c) $\frac{44}{3}$ cm

[Solution Description]

Let the width of the rectangle be $w$ cm. Then, the length of the rectangle is $(2w + 4)$ cm. The perimeter of a rectangle is given by the formula: $\text{Perimeter} = 2(\text{length} + \text{width})$. Substituting the given values, we have: $2((2w + 4) + w) = 40$. Simplifying this, we get: $2(3w + 4) = 40$. Expanding, we get: $6w + 8 = 40$. Subtracting 8 from both sides, we get: $6w = 32$. Dividing both sides by 6, we get: $w = \frac{32}{6} = \frac{16}{3}$ cm. Now, the length is: $2\left(\frac{16}{3}\right) + 4 = \frac{32}{3} + 4 = \frac{32}{3} + \frac{12}{3} = \frac{44}{3}$ cm. Therefore, the length of the rectangle is $\frac{44}{3}$ cm.