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Given a set K = 1, 2, ..., 11, find (a) prime numbers, (b) probability of selecting a non-prime number, and (c) factorize (x+y)(2m-n) - m(x+y).

Mathematics

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3 parts
a

{2,3,5,7,11}\{2, 3, 5, 7, 11\}

611\frac{6}{11}

b

(x+y)(m-n)

c

GH¢ 365.00\text{GH¢ } 365.00

Step-by-step solution

10 steps

Part a· 4 steps

  1. List all elements in set KK.

    The set K={1,2,3,4,5,6,7,8,9,10,11}K = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}.

    a)(i)

  2. Identify prime numbers in KK.

    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
    The prime numbers in KK are 2,3,5,7,112, 3, 5, 7, 11.
    The list of prime numbers in KK is:

    {2,3,5,7,11}\boxed{\{2, 3, 5, 7, 11\}}

    a)(ii)

  3. Determine the total number of elements and the number of non-prime numbers in KK.

    Total number of elements in KK is 11.
    Number of prime numbers in KK is 5.
    Number of non-prime numbers = Total elements - Number of prime numbers = 11−5=611 - 5 = 6.
    The non-prime numbers are {1,4,6,8,9,10}\{1, 4, 6, 8, 9, 10\}.

  4. Calculate the probability that a number selected at random from KK is not a prime number.

    Probability (not prime) = Number of non-prime numbersTotal number of elements\frac{\text{Number of non-prime numbers}}{\text{Total number of elements}}

    P(not prime)=611P(\text{not prime}) = \frac{6}{11}

    The probability is:

    611\boxed{\frac{6}{11}}

Part b(x+y)(m-n)

  1. Write the expression to be factorized.

    (x+y)(2m−n)−m(x+y)(x+y)(2m-n) - m(x+y)

  2. Identify the common factor and factor it out.

    The common factor is (x+y)(x+y).

    (x+y)[(2m−n)−m](x+y)[(2m-n) - m]

  3. Simplify the expression inside the square brackets.

    (x+y)[2m−n−m](x+y)[2m - n - m] (x+y)[m−n](x+y)[m - n]

    The completely factorized expression is:

    (x+y)(m−n)\boxed{(x+y)(m-n)}

Part c

GH¢ 365.00\text{GH¢ } 365.00

  1. Calculate the number of additional sheets typed.

    Total sheets typed = 35.
    First 8 sheets.
    Additional sheets = Total sheets - First 8 sheets = 35−8=2735 - 8 = 27 sheets.

  2. Calculate the cost for the additional sheets.

    Charge per additional sheet = GH¢ 11.00.
    Cost for additional sheets = 27 sheets×GH¢ 11.00/sheet=GH¢ 297.0027 \text{ sheets} \times \text{GH¢ } 11.00/\text{sheet} = \text{GH¢ } 297.00.

  3. Calculate the total amount earned.

    Charge for the first 8 sheets = GH¢ 68.00.
    Total amount earned = Charge for first 8 sheets + Cost for additional sheets
    Total amount earned = GH¢ 68.00+GH¢ 297.00=GH¢ 365.0068.00 + \text{GH¢ } 297.00 = \text{GH¢ } 365.00.
    The total amount earned is:

    GH¢ 365.00\boxed{\text{GH¢ } 365.00}
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