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Solve the following integer and common fraction problems showing all steps: -23 - (-18), 2(8-3+4), 33 + 42 - 52, sqrt(9) + 16 + 7, (-6+3(-4)-3(4))/((2+3)(-3)), and 10 divided by 5/8.

Mathematics

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3. INTEGERS

Part 3.1· 2 steps

Calculate the following and show all the steps: −23−(−18)-23 - (-18)

  1. Simplify the double negative.

    −23−(−18)=−23+18-23 - (-18) = -23 + 18
  2. Perform the addition.

    =−5= -5

    The result is −5\boxed{\mathbf{-5}}.

Part 3.2· 2 steps

2(8−3+4)2(8-3+4)

  1. Perform the operations inside the parenthesis.

    2(8−3+4)=2(5+4)2(8-3+4) = 2(5+4) =2(9)= 2(9)
  2. Perform the multiplication.

    =18= 18

    The result is 18\boxed{\mathbf{18}}.

Part 3.3· 3 steps

33+42−523^3 + 4^2 - 5^2

  1. Calculate the powers.

    33=273^3 = 27 42=164^2 = 16 52=255^2 = 25
  2. Substitute the values back into the expression.

    27+16−2527 + 16 - 25
  3. Perform addition and subtraction from left to right.

    43−25=1843 - 25 = 18

    The result is 18\boxed{\mathbf{18}}.

Part 3.4· 2 steps

9+16+7\sqrt{9} + 16 + 7

  1. Calculate the square root.

    9=3\sqrt{9} = 3
  2. Perform the additions from left to right.

    3+16+7=19+73 + 16 + 7 = 19 + 7 =26= 26

    The result is 26\boxed{\mathbf{26}}.

Part 3.5· 3 steps

−6+3(−4)−3(4)(2+3)(−3)\frac{-6+3(-4)-3(4)}{(2+3)(-3)}

  1. Calculate the numerator.

    −6+3(−4)−3(4)=−6−12−12-6 + 3(-4) - 3(4) = -6 - 12 - 12 =−30= -30
  2. Calculate the denominator.

    (2+3)(−3)=(5)(−3)(2+3)(-3) = (5)(-3) =−15= -15
  3. Divide the numerator by the denominator.

    −30−15=2\frac{-30}{-15} = 2

    The result is 2\boxed{\mathbf{2}}.

    4. COMMON FRACTIONS

Part 4.1· 3 steps

Calculate: 10÷5810 \div \frac{5}{8}

  1. Convert division to multiplication by the reciprocal of the fraction.

    10÷58=10×8510 \div \frac{5}{8} = 10 \times \frac{8}{5}
  2. Multiply the numbers.

    =10×85= \frac{10 \times 8}{5} =805= \frac{80}{5}
  3. Simplify the fraction.

    =16= 16

    The result is 16\boxed{\mathbf{16}}.

Part 4.2· 2 steps

Simplify: 2549\sqrt{\frac{25}{49}}

  1. Take the square root of the numerator and the denominator separately.

    2549=2549\sqrt{\frac{25}{49}} = \frac{\sqrt{25}}{\sqrt{49}}
  2. Calculate the square roots.

    =57= \frac{5}{7}

    The simplified expression is 57\boxed{\mathbf{\frac{5}{7}}}.

Part 4.3· 3 steps

Simplify: 8273+(12)2\sqrt[3]{\frac{8}{27}} + \left(\frac{1}{2}\right)^2

  1. Calculate the cube root of the first term.

    8273=83273=23\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}
  2. Calculate the square of the second term.

    (12)2=1222=14\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4}
  3. Add the two fractions.

    Find a common denominator, which is 12.

    23+14=2×43×4+1×34×3\frac{2}{3} + \frac{1}{4} = \frac{2 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} =812+312= \frac{8}{12} + \frac{3}{12} =8+312= \frac{8+3}{12} =1112= \frac{11}{12}

    The simplified expression is 1112\boxed{\mathbf{\frac{11}{12}}}.

Part 4.4· 2 steps

Determine the percentage decrease if a chocolate that cost R10 is marked down to R7.

  1. Calculate the amount of decrease.

    Decrease=Original Price−New Price\text{Decrease} = \text{Original Price} - \text{New Price} Decrease=R10−R7=R3\text{Decrease} = \text{R}10 - \text{R}7 = \text{R}3
  2. Calculate the percentage decrease using the formula

    DecreaseOriginal Price×100%\frac{\text{Decrease}}{\text{Original Price}} \times 100\%.

    Percentage Decrease=R3R10×100%\text{Percentage Decrease} = \frac{\text{R}3}{\text{R}10} \times 100\% =0.3×100%= 0.3 \times 100\% =30%= 30\%

    The percentage decrease is 30%\boxed{\mathbf{30\%}}.

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