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Numbas Worksheet

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$\\simplify[std]{{a}y + {b}x = {c} + {d}xy}\\;$

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$y =$ [[0]]

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You can click on \"Show steps\" for more information, but you will lose one mark if you do so.

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To re-arrange $ay + bx = c + dxy$ we should first collect all of the terms involving $y$ to the one side

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$ay - dxy = c - bx$

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we should then factorize out $y$ to find

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$y(a-dx) = c - bx$

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and then divide by $a-dx$ to get $y$ on its own

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$y = \\frac{c - bx}{a - dx}$

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Another transposition question.

\n

rebalmaths

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Rearrange the following equation to make $y$ the subject. 

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\n

$y = $ [[0]]

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It may be helpful to factor out y. For example: 

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\\[\\simplify{x^{{n1}}*y+{n2}*y*x - {n3}*y}=y(\\simplify{x^{{n1}} + {n2}*x - {n3}})\\]

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Here, we first have to collect all terms involving $y$ on the same side. Hence, we get:

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\\[\\simplify{x^{{n1}}*y+{n2}*y*x - {n3}*y} = \\var{n4}\\]

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We then spot that $y$ appears exactly once in each term on the left, so factorise:

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\\[y(\\simplify{x^{{n1}} + {n2}*x - {n3}}) = \\var{n4}\\]

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and simple division gives the answer.

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Consider the equation:

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\\[\\simplify{x^{{n1}}*y + {n2}*y*x} = \\simplify{{n3}*y} + \\var{n4}\\]

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Re-arrange this equation to make $y$ the subject.

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Another transposition question, which requires (basic) factorisation.

\n

rebelmaths

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$S = $ [[0]]

\n

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Start by multiplying both sides by the denominator.

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For example if you have $V=\\frac{5S}{S+12}$ then multiply both sides by $(S+12)$.

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This gives:  $V \\color{red}{(S+12)}=\\frac{5S}{S+12} \\color{red}{(S+12)} $

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The $(S+12)$ terms on the right hand side cancel out to give: $V(S+12)=5S$

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Now expand out the brackets on the left hand side:  $\\color{red}{VS+12V}=5S$

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Next collect the like terms, you want to get all the terms with $S$ in them onto one side, so subtract $VS$ from both sides:

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$VS \\color{red}{-VS} + 12V=5S \\color{red}{-VS}$

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This becomes $12V=5S-VS$

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Now you can factorise the right hand side: $12V=\\color{red}{S(5-V)}$

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Finally divide both sides by $(5-V)$ to leave $S$ on its own: $\\frac{12V}{\\color{red}{5-V}}=S$

\n

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rearranging the Michelas-Menten equation to make the substrate the subject.

\n

rebelmaths

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Rearrange the following equation to make $S$ the subject:

\n

\n

\\[ V=\\frac{\\var{a}S}{S+\\var{b}}\\]

\n

\n

Note: To write a fraction you type (numerator)/(denominator).

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$x = $ [[0]]

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We wish to make $x$ the subject, so we must get $x$ on its own. At the moment, the only $x$ in the equation is in the fraction under the square root. Until the square root is gone, everything under the square root is trapped together.

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How can we get rid of the square root? Ans: Apply the inverse function of the square root (i.e. do the opposite of taking the square root) which is to square.

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Remember, if we wish to keep the equation balanced i.e. keep both sides equal, we must always do the same thing to both sides. So, if we square the right hand side in order to get rid of the square root, we must also square the left hand side:

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\\[ \\begin{align*} p^{\\color{red}{2}} & =  \\left( \\sqrt{\\frac{\\var{a}+\\var{b}x}{\\var{c}}} \\right)^{\\color{red}{2}} \\\\ \\Rightarrow \\  p^2 & = \\frac{\\var{a}+\\var{b}x}{\\var{c}}\\end{align*}\\]

\n

Now looking at the right hand side, we have a fraction. Everything on top of the line of a fraction is trapped together and everything under the line of a fraction is trapped together until the fraction is gone. So we need to get rid of the fraction. We do this by multiplying both sides by the denominator, $\\var{c}$ (since multiplication is the opposite of division). This gives:

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\\[ \\begin{align*}  p^2 \\color{red}{\\times \\var{c}} & =\\frac{\\var{a}+\\var{b}x}{\\var{c}} \\color{red}{\\times \\var{c}}\\\\  \\Rightarrow \\ \\var{c} p^2  & = \\var{a} + \\var{b}x \\end{align*} \\]

\n

\n

Next, we subtract $\\var{a}$ from both sides to get rid of it from the right hand side:

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\\[ \\begin{align*} \\var{c} p^2  \\color{red}{- \\var{a}} & = \\var{a} + \\var{b}x \\color{red}{- \\var{a}}\\\\ \\Rightarrow \\ \\var{c} p^2 - \\var{a} & = \\var{b} x \\end{align*} \\]

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Finally, we divide both sides by $\\var{b}$ to get $x$ on its own on the right hand side (since division is the opposite of multiplication). This gives:

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\\[ \\begin{align*} \\frac{\\var{c} p^2 - \\var{a}}{\\color{red}{\\var{b}}} & = \\frac{\\var{b} x}{\\color{red}{\\var{b}}}\\\\ \\Rightarrow \\ \\frac{\\var{c} p^2 - \\var{a}}{\\var{b}} & = x\\end{align*} \\]

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Make $x$ the subject of the following formula: 

\n

\\[ p=\\sqrt{\\frac{\\var{a}+\\var{b}x}{\\var{c}}} \\]

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Transpose the following equation to find $B$

\n

\\[ \\frac{1}{A} = \\frac{1}{B} + \\frac{1}{C} \\]

\n

Note: When inputting an expression like $AB$, you need to input $A * B$.

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$B = $ [[0]]

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We need to get $B$ on its own. There are a number of different ways you might do this. This is just one possible solution:

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At the moment, there is a single $B$ in the first fraction on the right hand side. Therefore we need to get rid of that fraction. Very often it can make calculations easier if there are no fractions at all, and it is often as easy to get rid of all fractions as to get rid of just one fraction. That is the case here. If we multiply both sides by $ABC$, we can get rid of all fractions in our equation:

\n

\\[  \\begin{align*} \\frac{1}{A} \\color{red}{\\times ABC} & = \\frac{1}{B} \\color{red}{\\times ABC} + \\frac{1}{C} \\color{red}{\\times ABC}\\\\ BC & = AC + AB \\end{align*} \\]

\n

Since we wish to get $B$ on its own, we now make sure every term that has a $B$ is on the one side of the equals sign and every term without a $B$ is on the other side. If we subtract $AB$ from both sides we get:

\n

\\[ \\begin{align*} BC \\color{red}{- AB} & = AC + AB \\color{red}{- AB}\\\\ BC - AB & = AC  \\end{align*} \\]

\n

Next, factorising the left hand side:

\n

\\[  \\color{red}{B(C - A)} = AC \\]

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Finally, dividing both sides by $C - A$:

\n

\\[ \\begin{align*} \\frac{B(C - A)}{\\color{red}{C - A}} & = \\frac{AC}{\\color{red}{C - A}}\\\\ B & = \\frac{AC}{C - A} \\end{align*} \\]

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Solve the following equations for x. Enter your answer as a fraction or an integer, do not enter a decimal.

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$\\displaystyle \\var{a}x + \\var{b} = \\var{c}x + \\var{d}$

\n

$x=\\;$ [[0]]

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$\\displaystyle \\var{a1}x + \\var{b1} = \\var{c1}x + \\var{d1}$

\n

$x=\\;$ [[0]]

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$\\displaystyle \\var{a2}(\\var{b2}x + \\var{c2}) = \\var{d2}(\\var{e2}x + \\var{f2})$

\n

$x=\\;$ [[0]]

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$\\displaystyle \\frac{\\var{m}}{x}+\\var{n}=\\frac{\\var{s}}{x}+\\var{t}$

\n

$x=\\;$[[0]]

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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 $a$.

\n

\n

To write $a^{2}$, type a^2. For fractional powers, e.g. $a^{2/3}$, write a^(2/3).

", "advice": "

a)

\n

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

\n

Using this rule, 

\n

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

\n

b)

\n

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

\n

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

\n

To do this, write the product out explicitly, as

\n

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

\n

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

\n

Therefore:

\n

\\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}

\n

c)

\n

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

\n

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

\n

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

\n

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

\n

Using rules of indices,

\n

\\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}

\n

Therefore,

\n

\\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}

\n

Alternatively, 

\n

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

\n

\\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}

\n

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

\n

\\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}

\n

d)

\n

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

\n

To do this we use the rules:

\n

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

\n

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

\n

We can then expand the equation as

\n

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

\n

Then using the rule of indices mentioned previously,

\n

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

\n

e)

\n

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

\n

To do this we use the rules:

\n

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

\n

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

\n

We can expand the expression as follows:

\n

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

\n

f)

\n

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

\n

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

\n

\\[
\\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 part c

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Used in parts b,d and f

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Used in parts a,c and e

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Used in parts b and d

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Used in parts b,d and f

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Used in part c

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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 $a$.

\n

\n

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

\n

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

\n

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

\n

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

\n

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You could use one of the following rules:

\n

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

\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 $a$.

\n

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

\n

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Use the rules:

\n

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

\n

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

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

\n

$\\sqrt[\\var{d}]{a} =$ [[0]].

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

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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 $a$.

\n

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

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

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

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Simple Interest, $I=P\\times t\\times r$

\n

$I=\\var{initial}\\times \\var{n}\\times \\var{p}/100$

\n

$I=\\var{interest}$

\n

Adding the interest to the initial amount gives €$\\var{answer1}$.

\n

\n

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An investor puts €$\\var{initial}$ in a savings account that pays $\\var{p}\\%$ simple interest at the end of each year. How much would the investor have after $\\var{n}$ years?

\n

Give your answer to $2$ decimal places.

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Simple interest.

\n

rebelmaths

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The compound interest formula is: $\\ A = P(1+i)^n $

\n

Part (a)

\n

P represents the principal sum invested , so in this example it is €$\\var{P}$.

\n

Part (b)

\n

A represents the amount in the deposit account after $\\var{n}$ years, so in this example it is €$\\var{A}$.

\n

Part (c)

\n

n represents the number of compounding periods , so in this example it is $\\var{n}$ years.

\n

Part(d)

\n

Using the compound interest formula:

\n

$A=P(1+i)^n$

\n

$\\var{A}=\\var{P}(1+i)^\\var{n}$

\n

We need to rearrange the equation to find the value of $i$.

\n

$\\frac{\\var{A}}{\\var{P}}=(1+i)^\\var{n}$

\n

$\\var{ratio}=(1+i)^\\var{n}$

\n

$\\sqrt[\\var{n}]{\\var{ratio}}=1+i$

\n

$\\var{intplus}=1+i$

\n

$i=\\var{int}$ so the annual interest rate is $\\var{perc}$%.

", "rulesets": {}, "parts": [{"prompt": "

What is the value of P?

\n

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "P+0.0001", "minValue": "P-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the value of A?

\n

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "A+0.0001", "minValue": "A-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the value of n?

\n

\n

[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "n+0.0001", "minValue": "n-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the interest rate per annum?

\n

Please give your answer as a percentage correct to 2 decimal places.

\n

\n

[[0]]%

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A lump sum of €$\\var{P}$ is deposited into a savings account that pays compound interest for $\\var{n}$ years. If no withdrawals are made from the account, then the amount that the lump sum will have grown to is €$\\var{A}$.

\n

The compound interest formula is:

\n

$\\ A = P(1+i)^n $

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"A": {"definition": "precround(P*(1+int)^n,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "A", "description": ""}, "perc": {"definition": "random(1.5..5.5 #0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "ratio": {"definition": "A/P", "templateType": "anything", "group": "Ungrouped variables", "name": "ratio", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "intplus": {"definition": "ratio^(1/n)", "templateType": "anything", "group": "Ungrouped variables", "name": "intplus", "description": ""}, "n": {"definition": "random(2..6 #1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "P": {"definition": "random(1000..6000 #500)", "templateType": "anything", "group": "Ungrouped variables", "name": "P", "description": ""}}, "metadata": {"description": "

Calculate the annual interest rate for a savings account where A, P and n are given.

\n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Exam version Savings compound interest 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "variables": {"perc2": {"group": "Ungrouped variables", "name": "perc2", "definition": "random(5.0..5.8#0.1)", "description": "", "templateType": "anything"}, "int2": {"group": "Ungrouped variables", "name": "int2", "definition": "perc2/36500", "description": "", "templateType": "anything"}, "A1": {"group": "Ungrouped variables", "name": "A1", "definition": "precround(P*(1+int1)^n1,2)", "description": "", "templateType": "anything"}, "P": {"group": "Ungrouped variables", "name": "P", "definition": "random(2000..5000#100)", "description": "", "templateType": "anything"}, "n1": {"group": "Ungrouped variables", "name": "n1", "definition": "random(2..4#1)", "description": "", "templateType": "anything"}, "int1": {"group": "Ungrouped variables", "name": "int1", "definition": "perc1/100", "description": "", "templateType": "anything"}, "perc1": {"group": "Ungrouped variables", "name": "perc1", "definition": "random(5.2..6#0.1)", "description": "", "templateType": "anything"}, "int3": {"group": "Ungrouped variables", "name": "int3", "definition": "perc2/365", "description": "", "templateType": "anything"}, "A2": {"group": "Ungrouped variables", "name": "A2", "definition": "precround(P*(1+int2)^n2,2)", "description": "", "templateType": "anything"}, "n2": {"group": "Ungrouped variables", "name": "n2", "definition": "n1*365", "description": "", "templateType": "anything"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Compare two savings accounts with different interest rates.

\n

rebelmaths

"}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["perc1", "perc2", "int1", "int2", "n1", "n2", "P", "A1", "A2", "int3"], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "functions": {}, "tags": [], "rulesets": {}, "parts": [{"showCorrectAnswer": true, "variableReplacements": [], "prompt": "

Suppose that the potential customer chooses Bank A.

\n

What is the value of $n$?

\n

[[0]]

\n

What is the value of $i$?

\n

[[1]]

\n

What is the value of $A$?  Please give your answer to the nearest cent.

\n

€[[2]]

", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "n1-0.00001", "marks": "0.5", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "n1+0.00001", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "int1-0.0001", "marks": "0.5", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "int1+0.0001", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "A1-0.01", "marks": "2", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "A1+0.01", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}]}, {"showCorrectAnswer": true, "variableReplacements": [], "prompt": "

Suppose that the potential customer chooses Bank B.

\n

What is the value of $n$?  

\n

[[0]]

\n

What is the value of $i$?  Please include all the decimal places in your answer.

\n

[[1]]

\n

\n

What is the value of $A$?  Please give your answer to the nearest cent.

\n

€[[2]]

\n

", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "n2-0.00001", "marks": "0.5", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "n2+0.00001", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "int2-0.00001", "marks": "0.5", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "int2+0.00001", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "A2-0.05", "marks": "2", "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "A2+0.05", "showFeedbackIcon": true, "correctAnswerStyle": "plain"}]}], "statement": "

Two rival high street banks offer customers a new deposit account.

\n

Bank A offers an account that earns interest at a rate of $\\var{perc1}$% per annum where interest is compounded annually.

\n

Bank B offers an account that earns interest at a nominal rate of $\\var{perc2}$% per annum where interest is compounded daily.

\n

Suppose that a potential customer has €$\\var{P}$ to invest for $\\var{n1}$ years.    

\n

The compound interest formula is:

\n

$\\ A = P(1+i)^n $

\n

\n

You may assume that there are 365 days per annum.

\n

                                         

", "advice": "

The compound interest formula is: $\\ A = P(1+i)^n $

\n

Part (a)

\n

n represents the number of compounding periods , so for Bank A it is $\\var{n1}$ years.

\n

i represents the rate of compound interest, for Bank A, the annual interest rate is $\\var{perc1}$% so i is $\\frac {\\var{perc1}} {100}=\\var{int1}$.

\n

The total amount saved after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

\n

$A=P(1+i)^n$

\n

$A=\\var{P} \\times(1+\\var{int1})^\\var{n1}=\\var{A1}$

\n

Part(b)

\n

n represents the number of compounding periods. For Bank B, interest is compounded daily for $\\var{n1}$ years so there are a total of $365 \\times \\var{n1} =\\var{n2}$ compounding periods.

\n

i represents the rate of compound interest. For Bank B, the interest rate is ${\\var{perc2}}$% per annum compounded daily.

\n

The interest rate per day is $\\frac{\\var{perc2}}{365}=\\var{int3}$%

\n

Therefore $i=\\frac{\\var{int3}}{100}=\\var{int2}$

\n

The amount saved after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

\n

$A=P(1+i)^n$

\n

$A=\\var{P} \\times(1+\\var{int2})^\\var{n2}=\\var{A2}$

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