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Financial Maths
NOTE: Clicking an entry in the right column will take you directly to that question!
(BLUE = Paper 1, RED = Paper 2)
| Since 2015, the following subtopic... | Has explicitly appeared in... |
|---|---|
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Purchases & Bills |
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Currency Exchange |
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Income Calculations |
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Compound Interest |
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Depreciation |
2022 Paper 1 Question 3
Joe, Émile, and Wei are all PAYE workers.
Each of them has an annual tax credit of €3300.
Their tax rates and bands are shown in the table below.
Assume that no other deductions are made from their income.
| Annual Income | Tax Rate |
|---|---|
|
First €\(35{,}300\) |
\(20\%\) |
|
Balance |
\(40\%\) |
(a) Joe’s gross annual income is €\(27{,}500\). Joe only pays tax at the lower rate.
Work out Joe’s net annual income.
\(25{,}300\mbox{ euro}\)
Income Tax
\begin{align}27{,}500\times0.2=5{,}500\mbox{ euro}\end{align}
\[\,\]
Tax Due
\begin{align}5{,}500-3{,}300=2{,}200\mbox{ euro}\end{align}
\[\,\]
Net Income
\begin{align}27{,}500-2{,}200=25{,}300\mbox{ euro}\end{align}
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(b) Émile’s gross annual income is €\(43{,}450\).
Work out Émile’s net annual income.
\(36{,}430\mbox{ euro}\)
Income Tax
\begin{align}35{,}300\times0.2+(43{,}450-35{,}300)\times0.4=10{,}320\mbox{ euro}\end{align}
\[\,\]
Tax Due
\begin{align}10{,}320-3{,}300=7{,}020\mbox{ euro}\end{align}
\[\,\]
Net Income
\begin{align}43{,}450-7{,}020=36{,}430\mbox{ euro}\end{align}
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(c) Wei’s gross annual income is over €\(35{,}300\), so she pays tax at both rates.
Wei is looking for a pay rise.
She wants her net income to increase by €\(80\) each month.
Work out how much her gross annual income will need to increase by, in order for this to happen.
\(1{,}600\mbox{ euro}\)
Net Annual Increase
\begin{align}80\times12=960\mbox{ euro}\end{align}
\[\,\]
Gross Annual Increase
\begin{align}\frac{960}{60}\times100=1{,}600\mbox{ euro}\end{align}
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2022 Paper 1 Question 9(a) - (d)
Brian buys a new car.
The graph below represents a model that can be used to predict the value of this car, \(V\), for the next
number of years. This model assumes that the value of the car reduces (depreciates) by a fixed percentage each year.
(a)
(i) Use the graph to write down \(V(0)\), the initial value of Brian’s car, and \(V(1)\), the value of Brian’s car after \(1\) year.
(ii) Show that the value of the car will reduce by \(20\%\) in its first year, according to this model.
(i) \(V(0)=30{,}000\mbox{ euro}\) and \(V(1)=24{,}000\mbox{ euro}\)
(ii) The answer is already in the question!
(i)
\begin{align}V(0)=30{,}000\mbox{ euro} && V(1)=24{,}000\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{30{,}000-24{,}000}{30{,}000}\times 100=20\%\end{align}
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(b)
(i) Based on this model, write a formula for \(V(t)\), the value of Brian’s car after \(t\) years, in terms of the age of the car (\(t\)).
Use the fact that the value decreases by \(20\%\) each year.
(ii) Hence, or otherwise, work out the value of Brian’s car after \(4\) years, according to this model. Show your working out.
(i) \(V(t)=30{,}000(0.8)^t\)
(ii) \(12{,}288\mbox{ euro}\)
(i)
\begin{align}V(t)=30{,}000(0.8)^t\end{align}
(ii)
\begin{align}V(4)&=30{,}000(0.8)^4\\&=12{,}288\mbox{ euro}\end{align}
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(c) The graph from part (a) is shown again below.
A different (linear) model assumes that the value of the car reduces (depreciates) by a fixed amount each year. The value of the car will also reduce by \(20\%\) in its first year, according to this model.
(i) Draw a line on the diagram above, passing through the first two points on the graph
with whole-number values of \(t\) (\(t=0\) and \(t=1\)). Continue your line until it reaches the horizontal axis.
(ii) Hence, or otherwise, estimate \(T\), the age of Brian’s car when its value would be €\(0\), according to this new model.
(i)
(ii) \(T=5\mbox{ years}\)
(i)
(ii) \(T=5\mbox{ years}\)
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(d) Eva buys a new car that has a price of €\(19{,}445\).
She pays \(30\%\) of this price as a deposit and makes repayments of €\(206.97\) each month for the following 3 years. At the end of the \(3\) years, she pays an additional lump sum of €\(7{,}389\).
Work out the total cost of the car for Eva.
\(20{,}673.42\mbox{ euro}\)
\begin{align}0.3\times19{,}445+36\times206.97+7{,}389=20{,}673.42\mbox{ euro}\end{align}
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2022 Paper 1 Question 10(c)
(c)
(i) Keith buys a new hurl. It usually costs €\(33\).
Keith gets a student discount of \(15\%\).
Work out the price Keith pays for the hurl.
(ii) Keith also buys a jersey. This costs €\(49.50\), including VAT at \(23\%\).
Work out the VAT on this jersey. Give your answer correct to the nearest cent.
(i) \(28.05\mbox{ euro}\)
(ii) \(9.26\mbox{ euro}\)
(i)
\begin{align}33\times0.85=28.05\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{49.50}{123}\times23\approx9.26\mbox{ euro}\end{align}
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2021 Paper 1 Question 1
(a) A television costs €\(380\) before VAT at \(21\%\) has been added.
Find the cost of the television after VAT has been added.
\(459.80\mbox{ euro}\)
\begin{align}380\times1.21=459.80\mbox{ euro}\end{align}
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(b) When VAT at \(21\%\) is included, the price of a laptop increases by €\(130.20\).
Find the total cost of the laptop including VAT.
\(750.20\mbox{ euro}\)
\begin{align}\frac{130.20}{0.21}+130.20=750.20\mbox{ euro}\end{align}
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(c) A printer is priced at €\(290.40\) including VAT at \(21\%\).
Find how much VAT is included in the price of this printer.
\(50.40\mbox{ euro}\)
\begin{align}290.40\times\frac{21}{121}=50.40\mbox{ euro}\end{align}
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(d) On September 1, 2020 the Standard Rate of VAT in Ireland was reduced from \(23\%\) to \(21\%\).
A company bought \(30\) computers in September, all at the same price.
The company calculated that it saved €\(336\) due to the reduction in the VAT rate.
Find the price of one computer before VAT had been added.
\(560\mbox{ euro}\)
\begin{align}\frac{100}{23-21}\times\frac{336}{30}=560\mbox{ euro}\end{align}
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2021 Paper 1 Question 7
The rates and thresholds of the Universal Social Charge (USC) in Ireland (excluding the top rate) during \(2020\) are given in the table below.
| Annual Income | Rate |
|---|---|
|
First €\(12{,}012\) |
\(0.5\%\) |
|
Next €\(8{,}472\) |
\(2\%\) |
|
Next €\(49{,}560\) |
\(5%\) |
|
Balance |
(Top Rate) |
(a) At what level of annual income does a worker start paying the top rate of USC?
\(70{,}044\mbox{ euro}\)
\begin{align}12{,}012+8{,}472+49{,}560=70{,}044\mbox{ euro}\end{align}
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(b) How much USC will a worker have paid in total if they pay the maximum amount due at each of the first three rates?
\(2{,}459.70\mbox{ euro}\)
\begin{align}\frac{0.05}{100}\times12{,}012+\frac{2}{100}\times8{,}472+\frac{4.5}{100}\times49{,}560=2{,}459.70\mbox{ euro}\end{align}
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(c) John’s annual income is €\(54{,}800\). Find the amount of USC he will pay.
\(1{,}773.72\mbox{ euro}\)
\begin{align}\frac{0.05}{100}\times12{,}012+\frac{2}{100}\times8{,}472+\frac{4.5}{100}\times(54{,}800-12{,}012-8{,}472)=1{,}773.72\mbox{ euro}\end{align}
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(d) Mary pays €\(1602.72\) in USC in \(2020\). Find her annual income.
\(51{,}000\mbox{ euro}\)
\begin{align}1{,}602.72-\frac{0.05}{100}\times12{,}012-\frac{2}{100}\times8{,}472=1{,}373.22\mbox{ euro}\end{align}
\begin{align}\downarrow\end{align}
\begin{align}\mbox{Annual Income}&=\frac{1373.22\times100}{4.5}+12{,}012+8{,}472\\&=51{,}000\mbox{ euro}\end{align}
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(e) Patrick has an annual income of €\(83{,}000\) and pays a total of €\(3496.18\) in USC.
Find the top rate of USC.
\(8\%\)
\begin{align}\frac{(3{,}496.18-2{,}459.70)\times100}{83{,}000-70{,}044}=8\%\end{align}
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2020 Paper 1 Question 1
(a) John works as part of a sales team. He earns a basic rate of €\(12.60\) per hour.
In addition to his hourly pay, he earns a commission of \(22\%\) on any sales he makes above €\(200\) each week.
(i) During a particular week, John worked \(45\) hours at the basic rate and made sales amounting to €\(350\). Find John’s gross pay for this week.
(ii) During the following week John worked \(51\) hours. This included \(3\) hours on Sunday.
If John works on a Sunday he receives \(1.5\) times the basic rate for those hours.
His gross pay for that week was €\(713.20\).
Find the amount of sales John made in that week.
(i) \(600\mbox{ euro}\)
(ii) \(435\mbox{ euro}\)
(i)
\begin{align}45\times12.60+(350-200)\times0.22=600\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{713.20-(51-3)\times12.60-3\times(12.60\times1.5)}{0.22}+200=435\mbox{ euro}\end{align}
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(b) John pays tax at the Standard Rate of \(20\%\) and at the Higher Rate of \(40\%\).
He has a weekly Tax Credit of €\(26\).
The weekly Standard Rate Cut-off Point is €\(678\).
Find John’s net income for the week where his salary was €\(713.20\).
\(589.52\mbox{ euro}\)
\begin{align}713.20-678\times0.2-(713.20-678)\times0.4+26=589.52\mbox{ euro}\end{align}
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2020 Paper 1 Question 7
(a) Pat buys a new car for €\(32{,}000\).
He trades in his old car and is given an allowance of €\(20{,}000\) by the garage.
He borrows the balance of the money from the credit union.
His fixed monthly repayment over three years is €\(44.366\) per month.
(i) How much money does Pat pay in total to the credit union for the loan?
(ii) Show the amount that Pat repays as a percentage of the amount that he borrows from the credit union is \(133.1\%\), correct to one decimal place.
(i) \(15{,}971.76\mbox{ euro}\)
(ii) The answer is already in the question!
(i)
\begin{align}443.66\times36=15{,}971.76\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{15{,}971.6}{12{,}000}\times100\approx133.1\%\end{align}
as required.
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(b) A sum of money is invested at \(r\%\) per annum compound interest for \(3\) years.
At the end of the \(3\) years the value of the investment has increased by \(33.1\%\).
Find the value of \(r\).
\(10%\)
\begin{align}(1+r)^3=1.331\end{align}
\begin{align}\downarrow\end{align}
\begin{align}1+r&=\sqrt[3]{1.331}\\&=1.1\end{align}
\begin{align}\downarrow\end{align}
\begin{align}r&=0.1\\&=10\%\end{align}
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(c)
(i) It is estimated that the value of cars depreciates at a compound rate of \(20\%\) per year.
Use this percentage to find the value of Pat’s car after three years (original price €\(32{,}000\)).
(ii) Pat’s friend Caitlín bought a new car three years ago.
Its value also depreciated by \(20\%\) per year.
It is now worth €\(17{,}920\).
Find the original value of the car.
(i) \(16{,}384\mbox{ euro}\)
(ii) \(35{,}000\mbox{ euro}\)
(i)
\begin{align}32{,}000\times0.8^3=16{,}384\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{17{,}920}{0.8^3}=35{,}000\mbox{ euro}\end{align}
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2020 Paper 2 Question 7(b)
A vertical mobile phone mast, \([DC]\), of height \(h\mbox{ m}\), is secured with two cables: \([AC]\) of length \(x\mbox{ m}\),
and \([BC]\) of length \(y\mbox{ m}\), as shown in the diagram.
The angle of elevation to the top of the mast from \(A\) is \(30^{\circ}\) and from \(B\) is \(45^{\circ}\).
(b) The two cables to secure the mast costs €\(25\) per metre. The mast itself costs €\(580\) per metre. VAT at \(23\%\) is then added in each case.
Calculate the total cost of the cables and mast after VAT is included.
\(29{,}954.19\mbox{ euro}\)
\begin{align}[25(73.2+51.8)+580(36.6)]\times1.23=29{,}954.19\mbox{ euro}\end{align}
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2019 Paper 1 Question 1
Eimear earns a gross wage of €\(40{,}000\) per annum with Company \(A\).
(a) Eimear pays income tax at a rate of \(20\%\) on income up to the standard rate cut-off point of €\(35{,}300\). She pays tax at a rate of \(40\%\) on the remainder.
She has annual tax credits of €\(1{,}650\).
Find how much income tax she pays per annum.
\(7{,}290\mbox{ euro}\)
\begin{align}35{,}300\times0.2+(40{,}000-35{,}300)\times0.4-1{,}650=7{,}290\mbox{ euro}\end{align}
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(b) Eimear pays her health insurance which costs her €\(1500\) net. Find her annual income after paying income tax and health insurance (i.e. her net annual income).
\(31{,}210\mbox{ euro}\)
\begin{align}40{,}000-7{,}290-1{,}500=31{,}210\mbox{ euro}\end{align}
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(c) Eimear is planning to change jobs. She is offered a job by Company \(B\) with a gross wage of €\(38{,}000\) and a bonus of €\(1500\) (tax free to Eimear) to be paid by the company, which she would use to pay her health insurance. Her tax rates and credits would remain the same.
Find by how much Eimear’s net annual income (after paying income tax and health insurance) will increase if she accepts the job with Company \(B\).
\(300\mbox{ euro}\)
\begin{align}[38{,}000-35{,}300\times0.2-(38{,}000-35{,}300)\times0.4+1{,}650]-31{,}210=300\mbox{ euro}\end{align}
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2019 Paper 1 Question 9(d)
(d) Another web design company in the UK offers to host and maintain her website for £\(55\) sterling per month. Avril uses the exchange rate on a particular day to find how much this would cost in euro. She finds that it would cost €\(62.70\).
Find the exchange rate for \(1\) euro on that day. Give your answer correct to 4 decimal places.
\(0.8772\mbox{ sterling}\)
\begin{align}\frac{55}{62.70}\approx0.8772\mbox{ sterling}\end{align}
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2018 Paper 1 Question 1
When Sean joined a sales company he was offered a choice of two different salary contracts.
The details of the contracts are outlined in the table below.
| \(t\) | Salary | End of year commision on total sales |
|---|---|---|
|
Contract A |
\(35{,}000\) euro |
\(2\%\) |
|
Contract B |
\(30{,}000\) euro |
\(3\%\) |
(a) Find how much Sean would earn under each contract in a year where his total sales were €\(400{,}000\).
Contract A: \(43{,}000\mbox{ euro}\)
Contract B: \(42{,}000\mbox{ euro}\)
Contract A
\begin{align}35{,}000+400{,}000\times0.02=43{,}000\mbox{ euro}\end{align}
\[\,\]
Contract B
\begin{align}30{,}000+400{,}000\times0.03=42{,}000\mbox{ euro}\end{align}
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(b) Another employee, Mary, earned €\(50{,}000\) in a particular year. She is on Contract \(A\).
Find her total sales for that year.
\(750{,}000\mbox{ euro}\)
\begin{align}\frac{50{,}000-35{,}000}{0.02}=750{,}000\mbox{ euro}\end{align}
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(c) Find the total sales for which a salesperson would earn the same amount of money under each contract.
\(500{,}000\mbox{ euro}\)
\begin{align}35{,}000+0.02x=30{,}000+0.03x\end{align}
\begin{align}\downarrow\end{align}
\begin{align}0.01x=5{,}000\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{5{,}000}{0.01}\\&=500{,}000\mbox{ euro}\end{align}
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2018 Paper 1 Question 7(e)-(f)
(e) For a particular show, adult tickets cost €\(25\) each and children’s tickets cost €\(12\) each.
Find the total income from ticket sales if \(276\) adult tickets and \(212\) children’s tickets were sold.
\(9{,}444\mbox{ euro}\)
\begin{align}276\times25+212\times12=9{,}444\mbox{ euro}\end{align}
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(f)
(i) For a different show \(752\) tickets were sold.
The ratio of adult tickets sold to children’s tickets sold is \(3:1\).
Find how many adult tickets and how many children’s tickets were sold for that show.
(ii) For the show referred to in part f(i) an adult ticket cost \(2\dfrac{1}{2}\) times as much as a children’s ticket. The income was €\(17{,}578\). Find the cost of a children’s ticket for this show.
(i) \(\mbox{Number of children’s tickets}=188\) and \(\mbox{Number of adult tickets}=564\)
(ii) \(11\mbox{ euro}\)
(i)
\begin{align}\mbox{Number of children’s tickets}&=\frac{752}{1+3}\\&=188\end{align}
\begin{align}\mbox{Number of adult tickets}&=3\times188\\&=564\end{align}
(ii)
\begin{align}188x+564(2.5x)=17{,}758\end{align}
\begin{align}\downarrow\end{align}
\begin{align}1{,}598x=17{,}578\end{align}
\begin{align}\downarrow\end{align}
\begin{align}x&=\frac{17{,}758}{1{,}598}\\&=11\mbox{ euro}\end{align}
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2017 Paper 1 Question 1
(a) A new machine is bought for €\(30000\). Its value depreciates by \(15\%\) each year for five years.
Find the value of the machine at the end of the five years.
\(13{,}311.16\mbox{ euro}\)
\begin{align}30{,}000\times0.85^5=13{,}311.16\mbox{ euro}\end{align}
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(b) A sum of money was invested for two years at \(3\%\) compound interest per year.
At the end of the two years it amounted to €\(30000\). Find the sum invested.
\(28{,}277.38\mbox{ euro}\)
\begin{align}P\times1.03^2=30{,}000\end{align}
\begin{align}\downarrow\end{align}
\begin{align}P&=\frac{30{,}000}{1.03^2}\\&=28{,}277.38\mbox{ euro}\end{align}
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(c) A company invested €\(25000\) for three years at a fixed rate of compound interest.
At the end of the three years it amounted to €\(26530.20\). Find the rate of interest.
\(2\%\)
\begin{align}25{,}000(1+i)^3=26{,}530.20\end{align}
\begin{align}\downarrow\end{align}
\begin{align}1+i&=\sqrt[3]{\frac{26{,}530.20}{25{,}000}}\\&=1.02\end{align}
\begin{align}\downarrow\end{align}
\begin{align}i&=0.02\\&=2\%\end{align}
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2017 Paper 1 Question 6
(a) A salesman earns a basic salary of €\(150\) per week. In addition, he gets commission of \(20\%\) on sales up to the value of €\(1000\) in the week and \(30\%\) commission on any sales above this.
Find his total income for a week when his total sales amount to €\(3000\).
\(950\mbox{ euro}\)
\begin{align}150+0.2\times1{,}000+0.3\times(3{,}000-1{,}000)=950\mbox{ euro}\end{align}
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(b) On a different week his total income is €\(1160\). Find his total sales for this week.
\(3{,}700\mbox{ euro}\)
\begin{align}\frac{1{,}160-150-0.2\times1{,}000}{0.3}+1{,}000=3{,}700\mbox{ euro}\end{align}
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2016 Paper 1 Question 1
Emma works part time after school at the local takeaway. She is paid a rate per hour and also receives €\(2\) for each delivery she makes.
(a) One day, she works for \(2\) hours, makes \(5\) deliveries and is paid a total of €\(28\).
Find her hourly rate of pay.
\(9\mbox{ euro}\)
\begin{align}\frac{28-5\times2}{2}=9\mbox{ euro}\end{align}
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(b) One week, she works for \(h\) hours and makes \(d\) deliveries.
Write a formula in \(h\) and \(d\) for the wage (\(w\)) she receives.
\(w=9h+2d\)
\begin{align}w=9h+2d\end{align}
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(c) Another week, she works for \(6\) hours and makes \(12\) deliveries. She also works \(5\) hours on a Sunday, at time and a half, and makes some deliveries. In total, she receives €\(161.50\) for that week. Find how many deliveries she makes on the Sunday.
\(8\)
\begin{align}9\times6+12\times2+5\times1.5\times9+2d=161.50\end{align}
\begin{align}\downarrow\end{align}
\begin{align}145.5+2d=161.50\end{align}
\begin{align}\downarrow\end{align}
\begin{align}d&=\frac{161.50-145.5}{2}\\&=8\end{align}
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2016 Paper 1 Question 6
Fiona earns a gross wage of €\(1550\) every fortnight. She pays income tax, a Universal Social Charge (USC), and Pay Related Social Insurance (PRSI) on this wage.
(a) Each fortnight, Fiona pays income tax at the rate of \(20\%\) on the first €\(1300\) she earns and \(40\%\) on the remainder. She has tax credits of €\(126\) per fortnight.
Find how much income tax she pays per fortnight
\(234\mbox{ euro}\)
\begin{align}1{,}300\times0.2+(1{,}550-1{,}300)\times0.4-126=234\mbox{ euro}\end{align}
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(b) Each fortnight, Fiona also pays USC on her gross wage. The rates are:
\(1\%\) on the first €\(462\) she earns, \(3\%\) on the next €\(214\), and \(5.5\%\) on the balance.
Find the total amount of USC she pays each fortnight.
\(59.11\mbox{ euro}\)
\begin{align}462\times0.01+214\times0.03+(1{,}550-62-214)\times0.055=59.11\mbox{ euro}\end{align}
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(c)
(i) Fiona pays PRSI amounting to €\(18\) each fortnight.
Find the sum of her fortnightly deductions.
(ii) Write the sum of her fortnightly deductions as a percentage of her gross wage.
Give your answer correct to one decimal place.
(i) \(311.11\mbox{ euro}\)
(ii) \(20.1\%\)
(i)
\begin{align}234+59.11+18=311.11\mbox{ euro}\end{align}
(ii)
\begin{align}\frac{311.11}{1{,}550}\times100\approx20.1\%\end{align}
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2016 Paper 1 Question 7
Paul has €\(8000\) that he wants to invest for a maximum of \(3\) years. His local bank is offering him two
options, Option \(1\) and Option \(2\), as shown in the table below.
| Option 1 |
|---|
|
\(2\%\) interest in Year \(1\) |
|
\(3\%\) interest in Year \(2\) |
|
\(5\%\) interest in Year \(3\) |
|
Money can be taken out at the end of Year \(1\) or Year \(2\) without penalty |
| Option 2 |
|---|
|
|