// Numbas version: finer_feedback_settings {"name": "Summer Bridge Course", "metadata": {"description": "

This is for the Summer Bridge Course in Mathematics . This exam will test their previous knowledge.

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This question aims to test understanding and ability to use the laws of indices.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.

", "advice": "

a)

Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.

Using this rule,

\\[
\\begin{align}
a^\\var{x} \\times a^\\var{y}\\ &= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\
&= a^\\var{x+y}.
\\end{align}
\\]

b)

We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.

Notice there is a constant in front of each of the terms.

To do this, write the product out explicitly, as

\\[\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\\]

We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.

Therefore:

\\begin{align}
\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&= \\var{c*d} \\times a^\\var{p+q} \\\\
&= \\simplify{{c*d}*a^{p+q}}.
\\end{align}

c)

Here we are using: $a^m \\div a^n = a^{m-n}$.

We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.

To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as

\\[\\simplify{{b}*a^{x}/({g}*a^{y})} = \\simplify{{b}/{g}} \\times \\simplify{a^{x}/(a^{y})}.\\]

Using rules of indices,

\\begin{align} \\frac{a^\\var{x}}{a^\\var{y}} &= a^\\var{x} \\div a^\\var{y}\\\\
&= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\
&= a^\\var{x-y}.
\\end{align}

Therefore,

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

Alternatively,

Using the rule of indices: $a^{-m} = \\displaystyle\\frac{1}{a^{m}}$, we can rewrite the question as:

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\frac{a^\\var{x}}{a^\\var{y}}\\\\
&= \\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}}.
\\end{align}

And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:

\\begin{align}
\\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}} &= \\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

d)

The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.

To do this we use the rules:

\\[(a^{m})^{n} = a^{mn},\\]

\\[(ab)^m = a^mb^m.\\]

We can then expand the equation as

\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]

Then using the rule of indices mentioned previously,

\\[
\\begin{align}
(\\simplify{{c}*a^{p}})^{\\var{q}}&= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\
&= \\simplify{{c}^{q}*a^{p*q}}.
\\end{align}
\\]

e)

The question asks us to simplify $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.

To do this we use the rules:

\\[a^\\frac{1}{m} = \\sqrt[m]{a},\\]

\\[(ab)^m = a^mb^m.\\]

We can expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{d}]{a} &= (\\simplify{a})^\\frac{1}{\\var{d}}\\\\
&= a^\\frac{1}{\\var{d}}.
\\end{align}
\\]

f)

The question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.

Here, we use the rule of indices: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$, allowing us to expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{c}]{\\simplify{a^{q}}} &= \\simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\\\
&= \\simplify[fractionnumbers,all]{a^{{q}/{c}}}.
\\end{align}
\\]

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Used in parts a,c and f

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Write $a^{\\var{x}} \\times a^{\\var{y}}$ as a single power of $a$.

$a^{\\var{x}} \\times a^{\\var{y}} =$ [[0]].

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Use the rule: $a^m \\times a^n = a^{m+n}$.

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Write $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$ as an integer multiplied by a single power of $a$.

$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$ [[0]].

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Write $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$ as a number multiplied by a single power of $a$.

$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$ [[0]].

You could use one of the following rules:

$a^m \\div a^n = a^{m-n}$.

$a^{-m} = \\displaystyle\\frac{1}{a^m}$.

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Write $(\\simplify{{c}*a^{p}})^{\\var{q}}$ as an integer multiplied by a single power of $a$.

$(\\simplify{{c}*a^{p}})^{\\var{q}} =$ [[0]].

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Use the rules:

$(ab)^m = a^mb^m$.

$(a^m)^n = a^{mn}$.

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Write $\\sqrt[\\var{d}]{a}$ as a single power of $a$.

$\\sqrt[\\var{d}]{a} =$ [[0]].

Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.

You must input your answer as a single power of a.

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Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.

$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].

Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.

You must input your answer as a single power of a.

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Identify well-known fractional equivalents of decimals. Convert obscure decimals and recurring decimals into fractions.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "

a)

To convert a decimal into a fraction, firstly place the decimal over $1$ as a fraction, and then multiply both the numerator and denominator by $10$ for however many decimal places the decimal has. For example, if the decimal was $0.1$, you would multiply the fraction by $10$ as there is one decimal place. If the decimal was $0.01$, you would multiply it by $100$, as there are two decimal places.

$\\var{a}$

\\[
\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}
\\]

ii)

$\\var{b}$

\\[
\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}
\\]

iii)

$\\var{d}$

\\[
\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}
\\]

iv)

$0.\\dot{\\var{c}}$

To convert a recurring decimal to a fraction, the first step is to set up a simple equation where

\\[
x=0.\\dot{\\var{c}}\\text{.}
\\]

By multiplying both sides by $10$, we can gain another simple equation where

\\[
10x=\\var{c}.\\dot{\\var{c}}\\text{.}
\\]

By subtracting one equation from the other, we can find the fraction equivalent of the recurring decimal.

\\[
\\begin{align}
&&\\var{c}.\\dot{\\var{c}}&={10}x\\\\
-&&{0.\\dot{\\var{c}}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}\\\\
&&{\\var{c}}&=9x\\\\
\\\\
&&\\frac{\\var{c}}{9}&=x
\\end{align}
\\]

$\\displaystyle\\frac{\\var{c}}{9}$ simplifies to $\\simplify{{{c}}/{9}}$ by dividing by $3$ and therefore, $0.\\dot{\\var{c}}=\\simplify{{{c}}/{9}}$ in its fractional form.}

b)

$\\displaystyle\\var{f}$

\\[
\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}
\\]

From this, we can look to see if we can cancel down the fraction by finding the highest common divisor between the numerator and denominator. This is $\\var{mygcd}$.

Therefore, it is not possible to simplify the answer any further and the final answer is

Simplifying by this amount gives the final answer

\\[\\frac{\\var{f3}}{\\var{f4}}.\\]

c)

$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$

To convert a recurring decimal to a fraction, the first step is to set up a simple equation where,

$x=\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$

By multiplying both sides by $100$ to isolate the recurring section on the left hand side of the decimal point, we can gain another simple equation

$100x=\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}.$

Now that we have two equations in terms of $x$, we can subtract one from the other and solve to get a value of $x$.

\\[
\\begin{align}
&&\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}&=100x\\\\
-&&\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}
\\\\
&&{{\\var{h}}\\var{j}\\var{k-h}}&=99x\\\\
\\\\
&&\\frac{\\var{numerator}}{\\var{g}}&=x\\text{.}\\\\
\\end{align}
\\]

From this, we should look to see if it is possible to simplify by finding the greatest common divisor of the numerator and the denominator. The greatest common divisor is $\\var{gcd1 }.$

Therefore, it is not possible to simplify and so

Simplifying by this value gives the fraction $\\displaystyle\\simplify{{{numerator}}/{g}}$ and so

\\[
\\begin{align}
\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ in its fractional form.}\\\\
\\end{align}
\\]

", "statement": "

Fractions can be equivalently represented as decimals and vice versa. One form may be more useful in a context than another and it is useful to practise how to change between them.

Have a go at these questions involving fractions and decimals, remembering to write your answer in its simplest form.

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Express these common decimals as their fraction equivalent.

$\\var{a}=$ [[0]] [[1]]

ii)

$\\var{b}=$ [[2]] [[3]]

iii)

$\\var{d}=$ [[6]] [[7]]

iv)

$0.\\dot{\\var{c}}=$ [[4]] [[5]]

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Convert this decimal to a fraction, giving your answer in its simplest form.

$\\displaystyle\\var{f} = $ [[0]] [[1]]

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Convert these decimals to a fraction, giving your answer in its simplest form.

ii)

$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}} = $ [[0]] [[1]]

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Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions.

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a)

When faced with dividing fractions, it much easier to switch one of the fractions around and multiply them together instead of divide them.

\\[ \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}} \\right) \\equiv \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\times\\frac{\\var{j_coprime}}{\\var{h_coprime}} \\right) = \\frac{\\var{fj}}{\\var{gh}} \\]

Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd1}$.

This gives a final answer of $\\displaystyle\\simplify{{fj}/{gh}}$.

b)

\\[ \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}} \\equiv \\left( \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\times\\frac{\\var{j1_coprime}}{\\var{h1_coprime}} \\right)=\\frac{\\var{f1j1}}{\\var{g1h1}} \\]

Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd2}$.

This gives a final answer of $\\displaystyle\\simplify{{f1j1}/{g1h1}}$.

c)

\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}} \\]

The first thing to do is to change the mixed numbers into improper fractions.

An improper fraction is a fraction where the numerator is greater than the denominator. To change a mixed fraction to an improper fraction, multiply the integer part of the mixed number by the denominator, and add it to the existing numerator to make the new numerator of the improper fraction. The denominator will stay the same.

\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\equiv\\frac{(\\var{f3}\\times\\var{h3_coprime})+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3}+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}} \\]

\\[ {\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}\\equiv\\frac{(\\var{f4}\\times\\var{h4_coprime})+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4}+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]

We now have our mixed numbers as improper fractions.

\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]

Now, use the same method as in parts a) and b) to divide by switching around one fraction and changing the division symbol to multiplication.

\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}}\\equiv\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\times\\frac{\\var{h4_coprime}}{\\var{f4h4+g4_coprime}}=\\frac{\\var{num}}{\\var{denom}} \\]

Finally, the last thing to do is to simplify your answer down by finding the highest common divisor in the numerator and denominator, which in this case is $\\var{gcd3}$.

By doing this, you will get a final answer of

\\[ \\simplify{{num}/{denom}} \\]

d)

\\[ \\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} \\]

Consider the denominator first, as following the rules of BODMAS, you should address brackets first.

You need to get a common denominator for both terms on the denominator, like this:

\\[ \\var{b}\\times\\frac{\\var{d}}{\\var{d}} = \\frac{\\var{bd}}{\\var{d}} \\]

This now allows you to complete the addition or subtraction as both terms have a common denominator.

\\[ {\\simplify[all,!collectNumbers]{{bd}/{d}-{c}/{d}}} = \\frac{\\var{bd_c}}{\\var{d}} \\]

This means that the expression is now:

\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}} \\]

Dealing with this requires a bit of manipulation. You have to change the divisor of the denominator to be a mulitplier of the numerator. The denominator, ${\\var{bd_c}}$ was being divided by ${\\var{d}}$ but by flipping it around, the numerator, ${\\var{a}}$ will be mulitplied by ${\\var{d}}$. The value of the expression remains the same.

\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}}\\equiv \\frac{(\\var{a})\\times(\\var{d})}{\\var{bd_c}}= \\frac{\\var{ad}}{\\var{bd_c}} \\]

From this, you can try to cancel the expression down by finding the highest common factor of the numerator and denominator, to give a final answer of

\\[ \\simplify{{ad}/{bd_c}} \\]

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Evaluate the following sums involving division of fractions. Simplify your answers where possible.

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$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}}=$ [[0]] [[1]]

$\\displaystyle\\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}}=$ [[0]] [[1]]

$\\displaystyle{\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}=$ [[0]] [[1]]

$\\displaystyle\\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} =$ [[0]] [[1]]

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PART C

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PART C

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PART A

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PART B

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PART A

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PART C

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PART A

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Variable a times variable d.

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a)

As this question involves a number greater than $1$ before the $x^2$ value it has a factorised form $(ax+b)(cx+d)$.

To find $a$ and $c$, we need to consider the factors of $\\var{a*c}$.

We are already given that one of them is $\\var{a}$, so we know that the other one must be $\\var{c}$.

This means our factorised equation must take the form

\\[(\\var{a}x+b)(\\var{c}x+d)=0\\text{.}\\]

This expands to

\\[ \\simplify{ {a*c}x^2 + ({a}*d+{c}*b)x + a*b} \\]

So we must find two numbers which add together to make $\\var{a*d+b*c}$, and multiply together to make $\\var{b*d}$.

Therefore $b$ and $d$ must satisfy

\\begin{align}
b \\times d &=\\var{b*d}\\\\
\\simplify{{a}d+{c}b} &= \\var{a*d+b*c}\\text{.}
\\end{align}

$b = \\var{b}$ and $d = \\var{d}$ satisfy these equations:

\\begin{align}
\\var{b} \\times \\var{d} &=\\var{b*d}\\\\
\\simplify[]{ {a}*{d} + {b}*{c} } &= \\var{a*d+b*c}
\\end{align}

So the factorised form of the equation is

\\[ \\simplify{({a}x+{b})({c}x+{d}) = 0} \\text{.}\\]

b)

$\\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\\var{a}x+\\var{b} = 0$ or $\\var{c}x+ \\var{d} = 0$.

So the roots of the equation are $\\var[fractionnumbers]{-b/a}$ and $\\var[fractionnumbers]{-d/c}$.

", "statement": "", "variables": {"b": {"templateType": "anything", "name": "b", "description": "

$b$ in $(ax+b)(cx+d)$

", "group": "last q", "definition": "random(-5..5 except 0)"}, "c": {"templateType": "anything", "name": "c", "description": "

$c$ in $(ax+b)(cx+d)$

", "group": "last q", "definition": "random(2..8 except a)"}, "a": {"templateType": "anything", "name": "a", "description": "

$a$ in $(ax+b)(cx+d)$

", "group": "last q", "definition": "random(2..3)"}, "roots": {"templateType": "anything", "name": "roots", "description": "

The roots of the equation

", "group": "last q", "definition": "sort([-b/a,-d/c])"}, "d": {"templateType": "anything", "name": "d", "description": "

$d$ in $(ax+b)(cx+d)$

", "group": "last q", "definition": "random(-8..8 except 0)"}}, "tags": ["coefficient of x^2 greater than 1", "Factorisation", "factorisation", "factorising", "factorising quadratic equations", "Factorising quadratic equations", "factorising quadratic equations with x^2 coefficients greater than 1", "taxonomy"], "ungrouped_variables": [], "functions": {}, "rulesets": {}, "metadata": {"description": "

Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [{"variables": ["a", "b", "c", "d", "roots"], "name": "last q"}], "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "b", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "b", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "c", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "c", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "d", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "d", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

Factorise the equation

$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\\text{.}$

$(\\var{a}x+\\phantom{.}$[[0]]$) ($[[1]]$x+\\phantom{.}$[[2]]$)\\; = 0$

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Write down the roots of the equation above.

Input your answer as $x_1$ and $x_2$, where $x_1<x_2$.

$x_1=$ [[0]]

$x_2=$ [[1]]

", "variableReplacements": [], "showFeedbackIcon": true}]}, {"name": "Fraction multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}], "advice": "

a)

To multiply $\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$, address the numerators and denominators separately.

Multiply the numerators across both fractions.

$\\var{a_coprime}\\times\\var{b_coprime}=\\var{ab}$,

and then multiply the denominators across both fractions.

$\\var{c_coprime}\\times\\var{d_coprime}=\\var{cd}$.

The values of the multiplied numerators and denominators will be the numerator and denominator of the new fraction: $\\displaystyle\\frac{\\var{ab}}{\\var{cd}}$.

This answer may need simplifying down, and to do this, find the greatest common divisor in both the numerator and denominator and divide by this number.

The greatest common divisor of $\\var{ab}$ and $\\var{cd}$ is $\\var{gcd}$.

By using $\\var{gcd}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{ab}/{cd}}$.

b)

To multiply $\\displaystyle\\simplify{{k_coprime}/{j_coprime}}\\times\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}$, we first need to change the mixed number term into an improper fraction.

To do this, we need to multiply $(\\var{f}\\times\\var{h_coprime}=\\var{fh})$ and add it to what was already on the numerator of the fraction, $\\var{g_coprime}$.

$\\displaystyle\\frac{(\\var{fh}+\\var{g_coprime})}{\\var{h_coprime}}= \\displaystyle\\frac{\\var{numif}}{\\var{h_coprime}}$.

Next, we multiply the numerators and denominators across both fractions separately, as done in part a).

$\\var{k_coprime}\\times\\var{numif} = \\var{num}$,

$\\var{j_coprime}\\times\\var{h_coprime}=\\var{denom}$.

This gives the unsimplified version of the new fraction $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$.

To simplify, find the greatest common divisor in both the numerator and denominator and divide by this number.

The greatest common divisor of $\\var{num}$ and $\\var{denom}$ is $\\var{gcdb}$.

By using $\\var{gcdb}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{num}/{denom}}$.

c)

To square a fraction means to multiply the fraction by itself. To do this, multiply the numerators and denominators across individually.

$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2=\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\times\\frac{\\var{l_coprime}}{\\var{m_coprime}}=\\frac{\\var{l_coprime^2}}{\\var{m_coprime^2}}.$

From this, we should look if it is possible to simplify by finding the highest common divisor of $\\var{l_coprime^2}$ and $\\var{m_coprime^2}.$

The greatest common divisor is $\\var{gcd_lcmc}$.

Therefore, it is not possible to simplify this further, and the final answer is

By simplifying with this value, the final answer is

$\\displaystyle\\frac{\\var{l_coprime2}}{\\var{m_coprime2}}$.

d)

Helen was on holiday for $28$ days and spent $\\displaystyle\\frac{\\var{aa}}{7}$ of her time in Spain.

$\\displaystyle\\frac{\\var{aa}}{7}\\times\\frac{28}{1}=\\frac{\\var{bb}}{7}=\\var{cc}$ days in Spain.

Whilst in Spain, she spends $\\displaystyle\\frac{\\var{dd}}{4}$ of her time in Barcelona.

$\\displaystyle\\frac{\\var{dd}}{4}\\times\\frac{\\var{cc}}{1}=\\frac{\\var{ddcc}}{4}=\\var{ee}$ days in Barcelona.

", "statement": "

Evaluate the following multiplications, giving each fraction in its simplest form.

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Random number between 1 and 20

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Random number between 1 and 20.

", "definition": "random(1..7#1)"}, "cd": {"name": "cd", "group": "Part a", "templateType": "anything", "description": "

Variable c times variable d.

", "definition": "c_coprime*d_coprime"}, "a": {"name": "a", "group": "Part a", "templateType": "anything", "description": "

Random number from 1 to 12.

", "definition": "random(2..12 except c)"}, "d": {"name": "d", "group": "Part a", "templateType": "randrange", "description": "

Random number from 1 to 12.

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Numerator of the improper fraction converted from a mixed number.

", "definition": "(f*h_coprime)+g_coprime"}, "gcd_gh": {"name": "gcd_gh", "group": "Part b", "templateType": "anything", "description": "", "definition": "gcd(g,h)"}, "fh": {"name": "fh", "group": "Part b", "templateType": "anything", "description": "

Variable f times variable h

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Random number between 1 and 4 - integer part of the mixed number.

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Random number from 1 to 12.

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PART A

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Denominator of new fraction.

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Random number between 1 and 20.

", "definition": "random(7..10#1)"}, "num": {"name": "num", "group": "Part b", "templateType": "anything", "description": "

Numerator of gap 0

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gcd of the numerator of the improper fraction

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Random number from 1 to 12.

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Random number between 1 and 20

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Variable a times variable b

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Several problems involving the multiplication of fractions, with increasingly difficult examples, including a mixed fraction and a squared fraction. The final part is a word problem.

", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{ab}/{gcd}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{ab}/{gcd}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{cd}/{gcd}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{cd}/{gcd}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "

$\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$ = [[0]] [[1]]

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$\\displaystyle\\simplify{{k}/{j}}\\times\\var{f}\\frac{\\var{g}}{\\var{h}}$ = [[0]] [[1]]

", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "l_coprime2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "l_coprime2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "m_coprime2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "m_coprime2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "

$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2= $ [[0]] [[1]]

Helen went on holiday in Europe. She spent $\\displaystyle\\frac{\\var{aa}}{7}$ of her time on holiday in Spain. Whilst in Spain, she spent $\\displaystyle\\frac{\\var{dd}}{4}$ of her time in Barcelona.

If her holiday lasted for $28$ days, how many days was she in Barcelona?

Helen was in Barcelona for [[0]] days.

", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}]}, {"name": "Select the fraction not equivalent to the others - large denominators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "advice": "

To find the odd fraction out, reduce each fraction to lowest terms.

\\begin{align}
\\frac{\\var{osix}}{\\var{psix}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{six}}{\\var{six}} \\\\[0.5em]
\\frac{\\var{oseven}}{\\var{pseven}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{seven}}{\\var{seven}} \\\\[0.5em]
\\frac{\\var{oeight}}{\\var{peight}}=\\displaystyle\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{eight}}{\\var{eight}} \\\\[0.5em]
\\frac{\\var{onine}}{\\var{pnine}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{nine}}{\\var{nine}} \\\\[0.5em]
\\frac{\\var{oten}}{\\var{pten}} = \\frac{\\var{oten/gcd(oten,pten)}}{\\var{pten/gcd(oten,pten)}} \\times \\frac{\\var{gcd(oten,pten)}}{\\var{gcd(oten,pten)}}
\\end{align}

The odd fraction out is $\\displaystyle\\frac{\\var{oten}}{\\var{pten}}$.

", "statement": "", "variables": {"five": {"name": "five", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..7 except one except two except three except four)"}, "o": {"name": "o", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..7 except p except m)"}, "six": {"name": "six", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..10 except one except two except three except four except five)"}, "oten": {"name": "oten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*ten+random(-6..6 except 0)"}, "pten": {"name": "pten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*ten+random(-2..2 except 0)"}, "m": {"name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": "

Random number between 1 and 10

", "definition": "random(1..7)"}, "pseven": {"name": "pseven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*seven"}, "eight": {"name": "eight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(6..15 except one except two except three except four except five except six except seven)"}, "nine": {"name": "nine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(4..12 except one except two except three except four except five except six except seven except eight)"}, "three": {"name": "three", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..6 except one except two)"}, "psix": {"name": "psix", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*six"}, "ten": {"name": "ten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..10 except one except two except three except four except five except six except seven except eight except nine)"}, "four": {"name": "four", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(5..20 except one except two except three)"}, "two": {"name": "two", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..15 except one)"}, "peight": {"name": "peight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*eight"}, "o_coprime": {"name": "o_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o/gcd_op"}, "osix": {"name": "osix", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*six"}, "pnine": {"name": "pnine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*nine"}, "seven": {"name": "seven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..6 except one except two except three except four except five except six)"}, "n": {"name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(7..15 except m)"}, "onine": {"name": "onine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*nine"}, "one": {"name": "one", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..10)"}, "p_coprime": {"name": "p_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p/gcd_op"}, "gcd_op": {"name": "gcd_op", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "gcd(o,p)"}, "oeight": {"name": "oeight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*eight"}, "oseven": {"name": "oseven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*seven"}, "p": {"name": "p", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(7..15 except n)"}}, "tags": ["equivalent fractions", "fractions", "Fractions", "taxonomy"], "ungrouped_variables": ["m", "n", "one", "two", "three", "four", "five", "o", "p", "gcd_op", "o_coprime", "p_coprime", "six", "osix", "psix", "seven", "oseven", "pseven", "eight", "oeight", "peight", "nine", "onine", "pnine", "ten", "oten", "pten"], "functions": {}, "preamble": {"js": "", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "

Given five fractions, identify the odd fraction out. The denominators are mainly two or three digits long.

", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "prompt": "

From the options below, select the fraction which is not equivalent to the others.

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$\\displaystyle\\frac{\\var{osix}}{\\var{psix}}$

", "

$\\displaystyle\\frac{\\var{oseven}}{\\var{pseven}}$

", "

$\\displaystyle\\frac{\\var{oeight}}{\\var{peight}}$

", "

$\\displaystyle\\frac{\\var{onine}}{\\var{pnine}}$

", "

$\\displaystyle\\frac{\\var{oten}}{\\var{pten}}$

"], "showFeedbackIcon": true, "minMarks": 0}]}, {"name": "AJAY's copy of Expand brackets and collect like terms", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "AJAY OTTA", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3976/"}], "tags": [], "metadata": {"description": "

Eight expressions, of increasing complexity. The student must simplify them by expanding brackets and collecting like terms.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

For each expression below, collect like terms and expand brackets.

The * symbol is required between algebraic symbols, e.g. $5ab^2$ should be written 5*a*b^2.

", "advice": "

When simplifying expressions, only terms of the same type or like terms can be added together.

Algebraic symbols or letters can be added together provided that they are raised to the same power. For example, we can add $x^2+x^2=2x^2$, but we cannot collect both $x^2$ and $x$ into one term.

a)

\\[
\\begin{align}
\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x&=(\\var{c[0]}+\\var{c[1]}+\\var{c[2]})x\\\\
&=\\simplify{({c[0]}+{c[1]}+{c[2]})}x
\\end{align}
\\]

b)

\\[
\\begin{align}
\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}&=(\\var{a[1]}+\\var{a[2]})x^2+(\\var{a[3]}+\\var{a[4]})x +\\var{a[0]}\\\\
&=\\simplify{({a[1]}+{a[2]})}x^2+\\simplify{({a[3]}+{a[4]})}x+\\var{a[0]}
\\end{align}
\\]

c)

\\[
\\begin{align}
\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5&=(\\var{b[0]}+\\var{b[1]}+\\var{b[2]}+\\var{b[4]}+\\var{b[3]})y^5\\\\
&=\\simplify{({b[1]}+{b[2]}+{b[3]}+{b[4]}+{b[0]})}y^5
\\end{align}
\\]

d)

\\[
\\begin{align}
\\var{d[0]}ab+\\var{d[1]}abc+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c+\\var{d[5]}abc
&=(\\var{d[1]}+\\var{d[5]})abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c\\\\
&=\\simplify{{d[1]}+{d[5]}}abc+\\var{d[0]}ab+\\var{d[2]}a+\\var{d[3]}b+\\var{d[4]}c
\\end{align}
\\]

e)

\\[
\\begin{align}
\\var{f[0]}a^2b+\\var{f[1]}ab^2+\\var{f[2]}ab+\\var{f[3]}a^2b+\\var{f[4]}ab^2
&=(\\var{f[0]}+\\var{f[3]})a^2b+(\\var{f[1]}+\\var{f[4]})ab^2+\\var{f[2]}ab\\\\
&=\\simplify{{f[0]}+{f[3]}}a^2b+\\simplify{{f[1]}+{f[4]}}ab^2+\\var{f[2]}ab
\\end{align}
\\]

f)

\\[
\\begin{align}
\\var{g[0]}(\\var{g[1]}x+\\var{g[2]}y)+\\var{g[4]}x+\\var{g[5]}y
&=(\\var{g[0]}\\times \\var{g[1]}+\\var{g[4]})x+(\\var{g[0]} \\times\\var{g[2]}+\\var{g[5]})y\\\\
&=(\\simplify{{g[0]}*{g[1]}}+\\var{g[4]})x+(\\simplify{{g[0]}*{g[2]}}+\\var{g[5]})y\\\\
&=\\simplify{{g[0]}*{g[1]}+{g[4]}}x+\\simplify{{g[0]}*{g[2]}+{g[5]}}y
\\end{align}
\\]

g)

\\[
\\begin{align}
\\var{h[0]}x(\\var{h[1]}x+\\var{h[2]}z)+\\var{h[3]}x+\\var{h[6]}z+\\var{h[4]}x^2+\\var{h[5]}z^2
&=(\\simplify[]{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=(\\simplify{{h[0]}{h[1]}}+\\var{h[4]})x^2+(\\simplify[]{{h[0]}{h[2]}})zx+\\var{h[3]}x+\\var{h[5]}z^2+\\var{h[6]}z\\\\
&=\\simplify{{h[0]}*{h[1]}+{h[4]}}x^2+\\simplify{{h[0]}*{h[2]}}zx+\\simplify{{h[3]}x+{h[5]}}z^2+\\var{h[6]}z
\\end{align}
\\]

h)

\\[
\\begin{align}
\\var{j[0]}(\\var{j[1]}x-\\var{j[2]}y)+\\var{j[3]}(\\var{j[4]}x-\\var{j[5]}y)+\\var{j[6]}(\\var{j[7]}x-\\var{j[8]}y)
&= (\\simplify[]{{j[0]}{j[1]}}+\\simplify[]{{j[3]}{j[4]}}+\\simplify[]{{j[6]}{j[7]}})x-(\\simplify[]{{j[0]}{j[2]}}+\\simplify[]{{j[3]}{j[5]}}+\\simplify[]{{j[6]}{j[8]}})y\\\\
&= (\\simplify{{j[0]}{j[1]}}+\\simplify{{j[3]}{j[4]}}+\\simplify{{j[6]}{j[7]}})x-(\\simplify{{j[0]}{j[2]}}+\\simplify{{j[3]}{j[5]}}+\\simplify{{j[6]}{j[8]}})y\\\\
&= \\simplify{({j[0]}*{j[1]}+{j[4]*j[3]}+{j[6]}*{j[7]})x}-\\simplify{({j[0]}*{j[2]}+{j[5]}{j[3]}+{j[6]}*{j[8]})y}
\\end{align}
\\]

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random variables for part 1

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$\\var{c[0]}x+\\var{c[1]}x+\\var{c[2]}x=$ [[0]]

You must collect like terms to fully simplify.

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$\\var{a[1]}x^2+\\var{a[2]}x^2+\\var{a[3]}x+\\var{a[4]}x +\\var{a[0]}=$ [[0]]

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$\\var{b[0]}y^5+\\var{b[1]}y^5+\\var{b[2]}y^5+\\var{b[4]}y^5+\\var{b[3]}y^5=$ [[0]]