// Numbas version: exam_results_page_options {"name": "Ian's copy of Using BODMAS to evaluate arithmetic expressions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "tags": ["BODMAS", "bodmas", "taxonomy"], "extensions": [], "name": "Ian's copy of Using BODMAS to evaluate arithmetic expressions", "variable_groups": [], "ungrouped_variables": ["int", "sint", "eint", "oint", "pint", "bint"], "rulesets": {}, "variables": {"sint": {"definition": "random(2..6)", "templateType": "anything", "name": "sint", "description": "

Random integer from 1 to 5.

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Random even integer from 2 to 10.

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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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Use the BODMAS rule to determine the order in which to evaluate some arithmetic expressions.

"}, "advice": "

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

\n
Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction
\n

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.

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

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

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

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

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

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

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

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

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

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

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

\\[\\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{.}\\]

Before we evaluate numerator, we calculate powers:

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

Before we evaluate denominator we calculate multiplications:

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

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

\\[ \\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} \\]

So the fraction

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

Evaluating the final bracket we get:

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

As we evaluated all brackets, we can continue with:

\\[\\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} \\]

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

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

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{int*int}

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{int}

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

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{eint*4}

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{random(2..40 except eint*4)}

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

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

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", "

{bint - 14}

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$\\displaystyle \\frac{\\var{oint}^2+ \\sqrt{\\var{eint*eint}}}{3 \\times 2 - 2 \\times 2} + (10 - 2) \\div \\var{pint} =$ [[0]]

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

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

For example,

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

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

\n
Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction
\n

Apply BODMAS and try to solve these calculations.

", "type": "question", "contributors": [{"name": "Ian Loasby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/890/"}]}]}], "contributors": [{"name": "Ian Loasby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/890/"}]}