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Random integer from 2 to 10.

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Random odd integer from 1 to 9.

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A random slightly bigger integer.

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1, 2 or 4.

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Random integer from 1 to 5.

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$\\var{int*int} ÷ \\var{int} \\times \\var{int} =$  $?$

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{eint*4}

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1

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$\\var{eint*2} ÷ \\var{eint/2} \\times \\var{eint}=$  $?$

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{(sint + sint + 2)*sint}

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{sint + (sint + 2)*sint}

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{(((sint + sint + 2)*sint) + (sint + (sint + 2)*sint))/2}

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$\\var{sint} + \\var{sint + 2} \\times \\var{sint} =$  $?$

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1

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{bint - 14}

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$\\var{bint - 15} - 1 \\times 0 + \\var{bint}\\div\\var{bint} =$  $?$

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Root is another way of writing a power, e.g. $\\sqrt{4} = 4^{\\frac{1}{2}}$.

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Fraction means the numerator divided by the denominator, these two can be thought of as brackets while the fraction itself is a division.

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For example,

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\\begin{align}
\\sqrt4 + \\frac{4+11}{5} &= 4^{\\frac{1}{2}} + (4+11) \\div 5 \\\\
&= 4^{\\frac{1}{2}} + 15 \\div 5 & \\text{(BRACKETS)}\\\\
&= 2 + 15 \\div 5 & \\text{(ORDINALS)} \\\\
&= 2 + 3 & \\text{(DIVISION/multiplication)} \\\\
&= 5 \\text{ .} & \\text{(ADDITION/subtraction)}
\\end{align}

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$\\displaystyle \\frac{\\var{oint}^2+ \\sqrt{\\var{eint*eint}}}{3 \\times 2 - 2 \\times 2} + (10 - 2) \\div \\var{pint} =$  []

The correct order of carrying out operations can be remembered by the mnemonic BODMAS:

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Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction

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It is important to notice that division and multiplication have the same priority - division does not have a priority over multiplication. Similarly, adition and subtraction also have the same priority. When the order is unclear, we work from left to right.

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Note that brackets have the highest priority, but when we evaluate them, we still need to follow BODMAS inside them.

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Sometimes, an alternative acronym BIDMAS (Brackets, Indices, ...) is also used.

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a)

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Division and multiplication have the same priority, so we just work from left to right. $\\var{int*int} ÷ \\var{int} = \\var{int}$ and hence

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\\\begin{align} \\var{int*int} ÷ \\var{int} \\times \\var{int} &= \\var{int} \\times \\var{int} \\\\&= \\var{int*int} \\text{.} \\end{align}\

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b)

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Similarly, $\\var{eint*2} ÷ \\var{eint/2} = 4$ and hence

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\\\begin{align} \\var{eint*2} ÷ \\var{eint/2} \\times \\var{eint} &= 4 \\times \\var{eint} \\\\&= \\var{4*eint}\\text{.} \\end{align}\

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c)

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Applying BODMAS, multiplication has a priority over addition. $\\var{sint + 2} \\times \\var{sint} = \\var{(sint + 2)*sint}$ and hence

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\\\begin{align} \\var{sint} + \\var{sint + 2} \\times \\var{sint} &= \\var{sint} + \\var{(sint + 2)*sint} \\\\&= \\var{sint + (sint + 2)*sint}\\text{.} \\end{align}\

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d)

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Applying BODMAS, multiplication and division have priority over addition and subtraction. $1 \\times 0 = 0$ and $\\var{bint}\\div\\var{bint} = 1$ so

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\\\begin{align} \\var{bint - 15} - 1 \\times 0 + \\var{bint}\\div\\var{bint} &= \\var{bint - 15} - 0 + 1 \\\\&= \\var{bint - 14}\\text{.} \\end{align}\

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e)

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Roots can be considered as powers, while fractions can be considered as a bracket divided by a bracket.

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\$\\displaystyle \\text{Numerator is considered as a bracket } (\\var{oint}^2+ \\sqrt{\\var{eint*eint}}) \\text{ and the denominator as } (3 \\times 2 - 2 \\times 2)\\text{.}\$

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Before we evaluate numerator, we calculate powers:

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\\\begin{align} \\sqrt{\\var{eint*eint}} &= \\var{eint} \\text{,} \\\\\\var{oint}^2 &= \\var{oint*oint} \\text{.} \\end{align}\

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Before we evaluate denominator we calculate multiplications:

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\\\begin{align} 3 \\times 2 &= 6 \\text{ and } \\\\ 2 \\times 2 &= 4\\text{.} \\end{align}\

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Performing addition/subtraction as the last step in evaluating numerator/denominator we get:

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\\\begin{align} (\\var{oint}^2+ \\sqrt{\\var{eint*eint}}) &= \\var{oint*oint} + \\var{eint} \\\\&= \\var{oint*oint + eint} \\\\\\text{and} \\\\(3 \\times 2 - 2 \\times 2) &= 6 - 4 \\\\&= 2 \\end{align} \

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So the fraction

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\\\begin{align} \\displaystyle \\frac{(\\var{oint}^2+ \\var{eint})}{(3 \\times 2 - 2 \\times 2)} &= \\frac{\\var{(oint*oint + eint)}}{2}\\text{.} \\end{align}\

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Evaluating the final bracket we get:

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\$(10 - 2) = 8\\text{.}\$

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As we evaluated all brackets, we can continue with:

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\$\\displaystyle \\frac{\\var{oint}^2+ \\sqrt{\\var{eint*eint}}}{3 \\times 2 - 2 \\times 2} + (10 - 2) \\div \\var{pint} = \\frac{\\var{(oint*oint + eint)}}{2} + 8 \\div \\var{pint} \$

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Now, division has a priority over addition so since $\\frac{\\var{(oint*oint + eint)}}{2} = \\var{(oint*oint + eint)/2}$ and $8 \\div \\var{pint} = \\var{8/pint}$:

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\\\begin{align} \\frac{\\var{(oint*oint + eint)}}{2} + 8 \\div \\var{pint} &= \\var{(oint*oint + eint)/2} + \\var{8/pint} \\\\&= \\var{(oint*oint + eint)/2 + 8/pint}\\text{.} \\end{align}\

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Wrong order of solving operations can often lead to incorrect answers.  Therefore, the order in which we carry out a calculation is important.

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BODMAS is a mnemonic which tells us the correct order in which operations should be carried out:

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Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction

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Apply BODMAS and try to solve these calculations.