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What are the answers to problems 30a, 30b, 31a, and 31b?

Mathematics

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30. a)

  1. Add the numbers.

    22+4=2622 + 4 = 26

  2. Find the remainder when 26 is divided by 7.

    26÷7=3 with a remainder of 526 \div 7 = 3 \text{ with a remainder of } 5

    So, 26≡5(mod7)26 \equiv 5 \pmod 7.
    The value is 5\boxed{\textbf{5}}.

    30. b)

  3. Find the remainder of 27 when divided by 7.

    27÷7=3 with a remainder of 627 \div 7 = 3 \text{ with a remainder of } 6

    This means that 27 days from now will be the same day as 6 days from now.

  4. Count 6 days forward from Thursday.

    Thursday + 1 day = Friday
    Thursday + 2 days = Saturday
    Thursday + 3 days = Sunday
    Thursday + 4 days = Monday
    Thursday + 5 days = Tuesday
    Thursday + 6 days = Wednesday
    The day will be Wednesday\boxed{\textbf{Wednesday}}.

    31. a)

  5. Identify the common factor and apply the distributive property a×b+a×c=a×(b+c)a \times b + a \times c = a \times (b + c).

    (34×70)+(30×34)=34×(70+30)(34 \times 70) + (30 \times 34) = 34 \times (70 + 30)

  6. Perform the addition inside the parenthesis.

    34×(70+30)=34×10034 \times (70 + 30) = 34 \times 100

  7. Perform the multiplication.

    34×100=340034 \times 100 = 3400

    The value is 3400\boxed{\textbf{3400}}.

    31. b)

  8. Substitute the given values a=12a = \frac{1}{2}, b=4b = 4, and c=8c = 8 into the expression a(b+c)a(b+c).

    a(b+c)=12(4+8)a(b+c) = \frac{1}{2}(4+8)

  9. Perform the addition inside the parenthesis.

    12(4+8)=12(12)\frac{1}{2}(4+8) = \frac{1}{2}(12)

  10. Perform the multiplication.

    12×12=122=6\frac{1}{2} \times 12 = \frac{12}{2} = 6

    The value is 6\boxed{\textbf{6}}.

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