Maths-
General
Easy

Question

Is the given conjecture correct? Provide arguments to support your answer.
The product of any two consecutive odd numbers is 1 less than a perfect square.

Hint:

Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.
Deductive reasoning is the process by which a person makes conclusions based on previously known facts.

The correct answer is: odd numbers.


    Let the odd number be 2n+1. Its consecutive even number will be 2n+3
    Finding the product of (2n+1) and (2n+3)

    (2n+1)(2n+3) = 2n(2n+3) + 1(2n+3)

    (2n+1)(2n+3) = 2n(2n) + 2n(3) + 1(2n) + 1(3)

    (2n+1)(2n+3) = 4n2 + 6n + 2n + 3

    (2n+1)(2n+3) = 4n2 + 8n + 3
    The given conjecture is “The product of any two consecutive odd numbers is 1 less than a perfect square”
    So, add 1 to 4n2 + 8n + 3

    4n2 + 8n + 4 = (2n)2 + 2(2n)(2) + 22

    = (2n + 2)2
    So, we can see that the perfect square of 2n+2 is 4n2 + 8n + 4 and 1 less than 4n2 + 8n + 4 is a product of any two consecutive odd numbers.
    Final Answer:
    Hence, the given conjecture is right and we have proved it by Deductive reasoning.

    Related Questions to study

    General
    Maths-

    Name the polynomial based on its degree and number of terms.
    5 x y squared minus 3

    • We have been given a function in the question.
    • We will have to name the polynomial based on its degree and number of terms.
    Step 1 of 1:
    We have given a polynomial 5 x y squared minus 3.
    The degree of 5 x y squared is 3.
    And degree of 3 is zero.
    So, The degree of the polynomial is 3, So the given polynomial is cubic.
    And the number of terms is 2.

    Name the polynomial based on its degree and number of terms.
    5 x y squared minus 3

    Maths-General
    • We have been given a function in the question.
    • We will have to name the polynomial based on its degree and number of terms.
    Step 1 of 1:
    We have given a polynomial 5 x y squared minus 3.
    The degree of 5 x y squared is 3.
    And degree of 3 is zero.
    So, The degree of the polynomial is 3, So the given polynomial is cubic.
    And the number of terms is 2.
    General
    Maths-

    In an academic contest correct answers earn 12 points and incorrect answers lose 5
    points. In the final round, school A starts with 165 points and gives the same number
    of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.
    ii)How many answers did each school get correct in the final round?

    Answer:
    • Step by step explanation:
    ○    Step 1:
    ○    Solve equation: 165 + 12x - 5x = 65 + 12x
    165 + 12x - 5x = 65 + 12x
    rightwards double arrow 165 + 7x = 65 + 12x
    rightwards double arrow 165 - 65 = 12x - 7x
    rightwards double arrow 100 = 5x
    rightwards double arrow  100 over 5  = x
    rightwards double arrow20 = x
    • Final Answer:
    Hence, the school A gives 20 correct answers.

    In an academic contest correct answers earn 12 points and incorrect answers lose 5
    points. In the final round, school A starts with 165 points and gives the same number
    of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.
    ii)How many answers did each school get correct in the final round?

    Maths-General
    Answer:
    • Step by step explanation:
    ○    Step 1:
    ○    Solve equation: 165 + 12x - 5x = 65 + 12x
    165 + 12x - 5x = 65 + 12x
    rightwards double arrow 165 + 7x = 65 + 12x
    rightwards double arrow 165 - 65 = 12x - 7x
    rightwards double arrow 100 = 5x
    rightwards double arrow  100 over 5  = x
    rightwards double arrow20 = x
    • Final Answer:
    Hence, the school A gives 20 correct answers.
    General
    Maths-

    Determine the equation of the line that passes through left parenthesis negative 3 comma negative 6 right parenthesis text  and  end text left parenthesis 2 comma 34 right parenthesis

    Hint:
    We are given two points and we need to find the equation of the line passing through them. The equation of a line passing through two points (a, b) and (c, d) is
    fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction
    Step by step solution:
    Let the given points be denoted by
    (a, b) = (-3, -6)
    (c, d) = (2, 34)
    The equation of a line passing through two points  and  is
    fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction
    Using the above points, we have
    fraction numerator y minus 34 over denominator 34 minus left parenthesis negative 6 right parenthesis end fraction equals fraction numerator x minus 2 over denominator 2 minus left parenthesis negative 3 right parenthesis end fraction
    Simplifying the above equation, we have
    fraction numerator y minus 34 over denominator 34 plus 6 end fraction equals fraction numerator x minus 2 over denominator 2 plus 3 end fraction
    not stretchy rightwards double arrow fraction numerator y minus 34 over denominator 40 end fraction equals fraction numerator x minus 2 over denominator 5 end fraction
    Cross multiplying, we get
    5(y - 34) = 40(x - 2)
    Expanding the factors, we have
    5y - 170 = 40 x -80
    Taking all the terms in the left hand side, we have
    -40x + 5y - 170 + 80 = 0
    Finally, the equation of the line is
    -40x + 5y - 90=0
    Dividing the equation throughout by(- 5), we get
    This is the required equation.
    Note:
    We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by a + by + c=0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.

    Determine the equation of the line that passes through left parenthesis negative 3 comma negative 6 right parenthesis text  and  end text left parenthesis 2 comma 34 right parenthesis

    Maths-General
    Hint:
    We are given two points and we need to find the equation of the line passing through them. The equation of a line passing through two points (a, b) and (c, d) is
    fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction
    Step by step solution:
    Let the given points be denoted by
    (a, b) = (-3, -6)
    (c, d) = (2, 34)
    The equation of a line passing through two points  and  is
    fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction
    Using the above points, we have
    fraction numerator y minus 34 over denominator 34 minus left parenthesis negative 6 right parenthesis end fraction equals fraction numerator x minus 2 over denominator 2 minus left parenthesis negative 3 right parenthesis end fraction
    Simplifying the above equation, we have
    fraction numerator y minus 34 over denominator 34 plus 6 end fraction equals fraction numerator x minus 2 over denominator 2 plus 3 end fraction
    not stretchy rightwards double arrow fraction numerator y minus 34 over denominator 40 end fraction equals fraction numerator x minus 2 over denominator 5 end fraction
    Cross multiplying, we get
    5(y - 34) = 40(x - 2)
    Expanding the factors, we have
    5y - 170 = 40 x -80
    Taking all the terms in the left hand side, we have
    -40x + 5y - 170 + 80 = 0
    Finally, the equation of the line is
    -40x + 5y - 90=0
    Dividing the equation throughout by(- 5), we get
    This is the required equation.
    Note:
    We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by a + by + c=0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.
    parallel
    General
    Maths-

    Is the given conjecture correct? Provide arguments to support your answer.
    The product of any two consecutive even numbers is 1 less than a perfect square

    Let the even number be 2n. Its consecutive even number will be 2n+2
    Finding the product of 2n and (2n+2)

    2n(2n+2) = 2n(2n) + 2n(2)

    2n(2n+2) = 4n2 + 4n
    The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

    So, add 1 to 4n2 + 4n

    4n2 + 4n +1 = (2n)2 + 2(2n)(1) + 12

    = (2n + 1)2
    So, we can see that the perfect square of 2n+1 is 4n2 + 4n +1 and 1 less than 4n2 + 4n +1 is a product of any two consecutive even numbers.
    Final Answer:
    Hence, the given conjecture is right and we have proved it by Deductive reasoning.

    Is the given conjecture correct? Provide arguments to support your answer.
    The product of any two consecutive even numbers is 1 less than a perfect square

    Maths-General
    Let the even number be 2n. Its consecutive even number will be 2n+2
    Finding the product of 2n and (2n+2)

    2n(2n+2) = 2n(2n) + 2n(2)

    2n(2n+2) = 4n2 + 4n
    The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

    So, add 1 to 4n2 + 4n

    4n2 + 4n +1 = (2n)2 + 2(2n)(1) + 12

    = (2n + 1)2
    So, we can see that the perfect square of 2n+1 is 4n2 + 4n +1 and 1 less than 4n2 + 4n +1 is a product of any two consecutive even numbers.
    Final Answer:
    Hence, the given conjecture is right and we have proved it by Deductive reasoning.

    General
    Maths-

    Identify a linear polynomial.

    Explanation:
    We have to identify the linear polynomial from the given four options in the question
    Step 1 of 1:
    A linear polynomial is defined as any polynomial expressed in the form of an equation of p left parenthesis x right parenthesis equals a x plus b.
    From the option We can see that option B is in the form of p left parenthesis x right parenthesis equals a x plus b
    So, Option B is correct.

    Identify a linear polynomial.

    Maths-General
    Explanation:
    We have to identify the linear polynomial from the given four options in the question
    Step 1 of 1:
    A linear polynomial is defined as any polynomial expressed in the form of an equation of p left parenthesis x right parenthesis equals a x plus b.
    From the option We can see that option B is in the form of p left parenthesis x right parenthesis equals a x plus b
    So, Option B is correct.
    General
    Maths-

    Find the value of x. Identify the theorem used to find the answer.

    Answer:
    • Hint:
      • Mid-point theorem:
      • According to mid-point theorem, in triangle, the line segment which joins the midpoint of two sides is parallel to third side and is equal to half of third side,
    • Step by step explanation: 
      • Given:
    From figure,
    BM = MC,
    hence M is midpoint of BC.
    BN = NA,
    hence N is midpoint of AB.
    MN = x
    AC = 24
    • Step 1:
    In straight triangle ABC text ,  end text
    The line segment MN joins midpoint of AB and BC
    So,
    According to midpoint theorem,
    MN is parallel to AC and
    MN = fraction numerator A C over denominator 2 end fraction
    MN = 24 over 2
    MN = 12
    ∴ x = 12.
    Final Answer:
    x = 12.

    Find the value of x. Identify the theorem used to find the answer.

    Maths-General
    Answer:
    • Hint:
      • Mid-point theorem:
      • According to mid-point theorem, in triangle, the line segment which joins the midpoint of two sides is parallel to third side and is equal to half of third side,
    • Step by step explanation: 
      • Given:
    From figure,
    BM = MC,
    hence M is midpoint of BC.
    BN = NA,
    hence N is midpoint of AB.
    MN = x
    AC = 24
    • Step 1:
    In straight triangle ABC text ,  end text
    The line segment MN joins midpoint of AB and BC
    So,
    According to midpoint theorem,
    MN is parallel to AC and
    MN = fraction numerator A C over denominator 2 end fraction
    MN = 24 over 2
    MN = 12
    ∴ x = 12.
    Final Answer:
    x = 12.
    parallel
    General
    Maths-

    identify a cubic polynomial.

    Explanation:
    • We have to identify the cubic polynomial from the given four options.
    Step 1 of 1:
    We know that a cubic polynomial is a polynomial with degree 3.
    Option A:
    x + 3
    The degree of this polynomial 1.
    Option B:
    x cubed plus 3 y to the power of 4
    The degree of the polynomial is 4.
    Option C:
    x squared y plus 2 y
    The degree of this polynomial is 3.
    Option D:
    x y plus 2 y squared
    The degree of this polynomial 2.

    identify a cubic polynomial.

    Maths-General
    Explanation:
    • We have to identify the cubic polynomial from the given four options.
    Step 1 of 1:
    We know that a cubic polynomial is a polynomial with degree 3.
    Option A:
    x + 3
    The degree of this polynomial 1.
    Option B:
    x cubed plus 3 y to the power of 4
    The degree of the polynomial is 4.
    Option C:
    x squared y plus 2 y
    The degree of this polynomial is 3.
    Option D:
    x y plus 2 y squared
    The degree of this polynomial 2.
    General
    Maths-

    Find the value of 𝑚 & 𝑛 to make a true statement.
    (𝑚𝑥 + 𝑛𝑦)2 = 4𝑥2 + 12𝑥𝑦 + 9𝑦2

    (mx + ny)2 can be written as (mx + ny)(mx + ny)
    (mx + ny)(mx + ny) = mx(mx + ny) + ny(mx + ny)
    =  mx(mx) +  mx(ny) + ny(mx) + ny(ny)
    = m2x2 + mnxy + mnxy + n2y2
    = m2x2 + 2mnxy + n2y2
    Now, m2x2 + 2mnxy + n2y2 = 4𝑥2 + 12𝑥𝑦 + 9𝑦2
    Comparing both sides, we get
    m2 = 4, n = 9, 2mn = 12
    So, m = +2 or -2 , n = +3 or -3
    Considering 2mn = 12, there are two combinations possible
    1. m = +2 and n = +2
    2. m = -2 and n = -2
    Final Answer:
    Hence, the values of (m, n) are (2,2) and (-2,-2).

    Find the value of 𝑚 & 𝑛 to make a true statement.
    (𝑚𝑥 + 𝑛𝑦)2 = 4𝑥2 + 12𝑥𝑦 + 9𝑦2

    Maths-General
    (mx + ny)2 can be written as (mx + ny)(mx + ny)
    (mx + ny)(mx + ny) = mx(mx + ny) + ny(mx + ny)
    =  mx(mx) +  mx(ny) + ny(mx) + ny(ny)
    = m2x2 + mnxy + mnxy + n2y2
    = m2x2 + 2mnxy + n2y2
    Now, m2x2 + 2mnxy + n2y2 = 4𝑥2 + 12𝑥𝑦 + 9𝑦2
    Comparing both sides, we get
    m2 = 4, n = 9, 2mn = 12
    So, m = +2 or -2 , n = +3 or -3
    Considering 2mn = 12, there are two combinations possible
    1. m = +2 and n = +2
    2. m = -2 and n = -2
    Final Answer:
    Hence, the values of (m, n) are (2,2) and (-2,-2).
    General
    Maths-

    37. In an academic contest correct answers earn 12 points and incorrect answers lose 5
    points. In the final round, school A starts with 165 points and gives the same number
    of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.
    i)Which equation models the scoring in the final round and the outcome of the contest

    Answer:
    • Hint:
    ○    Form equation using the given information.
    ○    Take the variable value as x or any alphabet.
    • Step by step explanation:
    ○    Given:
    correct answer = 12 points
    incorrect answers = -5 points
    School A starts with 165 points and gives the same number of correct and incorrect answers.
    School B starts with 65 points and gives no incorrect answers and the same number of                           correct answers as school A
    ○    Step 1:
    ○    Let the number of correct answers given by school A be x.
    So, the number of incorrect answers is also x.
    At school A starts with 165 points. After giving x correct and a incorrect answers points will be
    rightwards double arrow165 + 12x - 5x

    School B starts with 65 points. Schools are given the same number of correct answers as                       school A and no incorrect answers. So, their points will be
    rightwards double arrow65 + 12x
    ○    Step 2:
    ○    As both schools tied
    ∴ 165 + 12x - 5x = 65 + 12x
    • Final Answer:
    Correct option:
    Option B. 165 + 12x - 5x = 65 + 12x

    37. In an academic contest correct answers earn 12 points and incorrect answers lose 5
    points. In the final round, school A starts with 165 points and gives the same number
    of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.
    i)Which equation models the scoring in the final round and the outcome of the contest

    Maths-General
    Answer:
    • Hint:
    ○    Form equation using the given information.
    ○    Take the variable value as x or any alphabet.
    • Step by step explanation:
    ○    Given:
    correct answer = 12 points
    incorrect answers = -5 points
    School A starts with 165 points and gives the same number of correct and incorrect answers.
    School B starts with 65 points and gives no incorrect answers and the same number of                           correct answers as school A
    ○    Step 1:
    ○    Let the number of correct answers given by school A be x.
    So, the number of incorrect answers is also x.
    At school A starts with 165 points. After giving x correct and a incorrect answers points will be
    rightwards double arrow165 + 12x - 5x

    School B starts with 65 points. Schools are given the same number of correct answers as                       school A and no incorrect answers. So, their points will be
    rightwards double arrow65 + 12x
    ○    Step 2:
    ○    As both schools tied
    ∴ 165 + 12x - 5x = 65 + 12x
    • Final Answer:
    Correct option:
    Option B. 165 + 12x - 5x = 65 + 12x
    parallel
    General
    Maths-

    Find the value of x. Identify the theorem used to find the answer.

    Answer:
    • Hints:
      • Perpendicular bisector theorem
      • According to perpendicular bisector theorem, in the triangle, any point on perpendicular bisector is at equal distance from both end points of the line segment on which it is drawn.
    • Step by step explanation: 
      • Given:
    AB = 2x.
    AC = 4x – 4
    AD is perpendicular bisector at BC.
    • Step 1:
    • In  straight triangle ABC text ,  end text
    AD is perpendicular bisector.
    A is point on AD
    A is equidistant from B and C.
    So,
    AB = AC
    2x = 4x – 4
    4 = 4x – 2x
    4 = 2x
    4 over 2 equals x
    x = 2
    • Final Answer: 
                 Hence, x = 2.
    Perpendicular bisector theorem is used.

    Find the value of x. Identify the theorem used to find the answer.

    Maths-General
    Answer:
    • Hints:
      • Perpendicular bisector theorem
      • According to perpendicular bisector theorem, in the triangle, any point on perpendicular bisector is at equal distance from both end points of the line segment on which it is drawn.
    • Step by step explanation: 
      • Given:
    AB = 2x.
    AC = 4x – 4
    AD is perpendicular bisector at BC.
    • Step 1:
    • In  straight triangle ABC text ,  end text
    AD is perpendicular bisector.
    A is point on AD
    A is equidistant from B and C.
    So,
    AB = AC
    2x = 4x – 4
    4 = 4x – 2x
    4 = 2x
    4 over 2 equals x
    x = 2
    • Final Answer: 
                 Hence, x = 2.
    Perpendicular bisector theorem is used.
    General
    Maths-

    Where is the circumcentre located in any right triangle? Write a coordinate proof of this result.

     

    Answer:
    • Hints:
    • Distance between two points having coordinates (x1, y1) and (x2, y2) is given by formula:
    • Distance =square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared end root
    • Step by step explanation: 
      • Step 1:
      • Let triangle ABO,
    where,
    O = (0, 0)
    A = (2a, 0)
    B = (0, 2b).
    • Step 1:
    • Let triangle ABO, where:

    The midpoint of BC is given by,
    not stretchy rightwards double arrow fraction numerator 0 plus 2 a over denominator 2 end fraction comma fraction numerator 2 b plus 0 over denominator 2 end fraction
    not stretchy rightwards double arrow left parenthesis a comma b right parenthesis
    So, the perpendicular bisector will intersect BC at M (a, b).
    Equation of line BC is
    y minus y subscript 1 equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction open parentheses x minus x subscript 1 close parentheses
    y minus 0 equals fraction numerator 2 b minus 0 over denominator 0 minus 2 a end fraction left parenthesis x minus 2 a right parenthesis
    y equals fraction numerator 2 b over denominator negative 2 a end fraction left parenthesis x minus 2 a right parenthesis
    y equals fraction numerator negative b over denominator a end fraction left parenthesis x minus 2 a right parenthesis
    And point M (a, b) satisfy above equation
    b equals fraction numerator negative b over denominator a end fraction left parenthesis a minus 2 a right parenthesis
    b equals fraction numerator negative b over denominator a end fraction left parenthesis negative a right parenthesis
    b = b
    Hence, point M (a, b) lies on BC.
    • Final Answer: 
    Hence, circumcentre of right angle triangle lie on midpoint of hypotenuse.

    Where is the circumcentre located in any right triangle? Write a coordinate proof of this result.

     

    Maths-General
    Answer:
    • Hints:
    • Distance between two points having coordinates (x1, y1) and (x2, y2) is given by formula:
    • Distance =square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared end root
    • Step by step explanation: 
      • Step 1:
      • Let triangle ABO,
    where,
    O = (0, 0)
    A = (2a, 0)
    B = (0, 2b).
    • Step 1:
    • Let triangle ABO, where:

    The midpoint of BC is given by,
    not stretchy rightwards double arrow fraction numerator 0 plus 2 a over denominator 2 end fraction comma fraction numerator 2 b plus 0 over denominator 2 end fraction
    not stretchy rightwards double arrow left parenthesis a comma b right parenthesis
    So, the perpendicular bisector will intersect BC at M (a, b).
    Equation of line BC is
    y minus y subscript 1 equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction open parentheses x minus x subscript 1 close parentheses
    y minus 0 equals fraction numerator 2 b minus 0 over denominator 0 minus 2 a end fraction left parenthesis x minus 2 a right parenthesis
    y equals fraction numerator 2 b over denominator negative 2 a end fraction left parenthesis x minus 2 a right parenthesis
    y equals fraction numerator negative b over denominator a end fraction left parenthesis x minus 2 a right parenthesis
    And point M (a, b) satisfy above equation
    b equals fraction numerator negative b over denominator a end fraction left parenthesis a minus 2 a right parenthesis
    b equals fraction numerator negative b over denominator a end fraction left parenthesis negative a right parenthesis
    b = b
    Hence, point M (a, b) lies on BC.
    • Final Answer: 
    Hence, circumcentre of right angle triangle lie on midpoint of hypotenuse.
    General
    Maths-

    Ayush is choosing between two health clubs. Health club 1: Membership R s 22 and
    Monthly fee R s 24.50. Health club 2: Membership R s 47.00 , monthly fee R s 18.25
    . After how many months will the total cost for each health club be the same ?

    Answer:
    • Hint:
    ○    The concept used in the question is of the quadratic equation.
    • Step by step explanation:
    ○    Given:
    Health club 1: Membership R s 22 and Monthly fee R s 24.50
    Health club 2: Membership R s 47.00 , monthly fee R s 18.25.
    ○    Step 1:
    ○    Let the number of months after which the total cost is equal be x.
    So, for Health club 1:
    After x months, total cost will be =
    rightwards double arrowMembership Rs 22 + fee for x months
    rightwards double arrow 22 + 24.50x
    Health club 2:
    After x months, total cost will be =
    rightwards double arrowMembership R s 47 + fee for x months
    rightwards double arrow 47 + 18.25x
    ○    Step 2:
    ○    Equalize both costs to get number of months
    rightwards double arrow22 + 24.50x = 47 + 18.25x
    rightwards double arrow24.50x - 18.25x= 47 - 22
    rightwards double arrow6.25x = 25
    rightwards double arrowx = fraction numerator 25 over denominator 6.25 end fraction
    rightwards double arrowx = 4
    • Final Answer:
    Hence, after 4 months the total cost of each health club will be equal.

    Ayush is choosing between two health clubs. Health club 1: Membership R s 22 and
    Monthly fee R s 24.50. Health club 2: Membership R s 47.00 , monthly fee R s 18.25
    . After how many months will the total cost for each health club be the same ?

    Maths-General
    Answer:
    • Hint:
    ○    The concept used in the question is of the quadratic equation.
    • Step by step explanation:
    ○    Given:
    Health club 1: Membership R s 22 and Monthly fee R s 24.50
    Health club 2: Membership R s 47.00 , monthly fee R s 18.25.
    ○    Step 1:
    ○    Let the number of months after which the total cost is equal be x.
    So, for Health club 1:
    After x months, total cost will be =
    rightwards double arrowMembership Rs 22 + fee for x months
    rightwards double arrow 22 + 24.50x
    Health club 2:
    After x months, total cost will be =
    rightwards double arrowMembership R s 47 + fee for x months
    rightwards double arrow 47 + 18.25x
    ○    Step 2:
    ○    Equalize both costs to get number of months
    rightwards double arrow22 + 24.50x = 47 + 18.25x
    rightwards double arrow24.50x - 18.25x= 47 - 22
    rightwards double arrow6.25x = 25
    rightwards double arrowx = fraction numerator 25 over denominator 6.25 end fraction
    rightwards double arrowx = 4
    • Final Answer:
    Hence, after 4 months the total cost of each health club will be equal.
    parallel
    General
    Maths-

    Find the gradient and  y- Intercept of the line x plus 2 y equals 14

    Hint:
    Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, compare the equation with the standard form to get the slope and the y-intercept.
    Step by step solution:
    The given equation of the line is
    x + 2y = 14
    We need to convert this equation in the slope-intercept form of the line, which is
    y = mx + c
    Rewriting the given equation, we have
    2y = 14 - x
    Dividing by 2, we get
    y equals 14 over 2 minus x over 2
    Simplifying, we get
    y equals negative 1 half x plus 7
    Comparing the above equation with , we get
    straight m equals negative 1 half semicolon straight c equals 7
    Thus, we get
    Gradient = negative 1 half
    y-intercept = 7
    Note:
    We can find the slope and y-intercept directly from the general form of the equation too; slope = negative a over b  and y-intercept =c over b , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula.

    Find the gradient and  y- Intercept of the line x plus 2 y equals 14

    Maths-General
    Hint:
    Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, compare the equation with the standard form to get the slope and the y-intercept.
    Step by step solution:
    The given equation of the line is
    x + 2y = 14
    We need to convert this equation in the slope-intercept form of the line, which is
    y = mx + c
    Rewriting the given equation, we have
    2y = 14 - x
    Dividing by 2, we get
    y equals 14 over 2 minus x over 2
    Simplifying, we get
    y equals negative 1 half x plus 7
    Comparing the above equation with , we get
    straight m equals negative 1 half semicolon straight c equals 7
    Thus, we get
    Gradient = negative 1 half
    y-intercept = 7
    Note:
    We can find the slope and y-intercept directly from the general form of the equation too; slope = negative a over b  and y-intercept =c over b , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula.
    General
    Maths-

    We can express any constant  in the variable form without changing its value as

    Explanation:
    • We have given a constant
    • We have to find how we can express any constant 𝑘 in the variable form without changing its value.
    Step 1 of 1:
    We have given a constant k
    We have to find how we can express any constant 𝑘 in the variable form without changing its value
    We know that x0 = 1
    So After multiplying it with any constant, it will not change its value.
    So,kx0 is the answer
    Hence, Option C is correct.

    We can express any constant  in the variable form without changing its value as

    Maths-General
    Explanation:
    • We have given a constant
    • We have to find how we can express any constant 𝑘 in the variable form without changing its value.
    Step 1 of 1:
    We have given a constant k
    We have to find how we can express any constant 𝑘 in the variable form without changing its value
    We know that x0 = 1
    So After multiplying it with any constant, it will not change its value.
    So,kx0 is the answer
    Hence, Option C is correct.
    General
    Maths-

    x0 = ?

    x0 = ?

    Maths-General
    parallel

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