A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Mr Chan plans to buy a new car in $2031$, which costs $\$120000$ in $2021$. There is a yearly inflation rate of $0.5\%$ on the price of the car.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that the cost of the car in $2031$ with inflation is $\$126136.82$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Mr Chan comes across a savings plan in a local bank that pays $1.5\%$ interest on $31$ December of each year. In order to buy the car on $31$ December $2031$, he intends to save $\$y$ every year starting from $1$ January $2022$ for&nbsp;$10$ years with this bank without any withdrawal. On $1$ January $2022$ Mr Chan puts $\$y$ into the bank, and on the $1$ January of each subsequent year, he puts another $\$y$ into the bank.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the least value of $\$y$ correct to the nearest dollar.<span class="inline-marks">[5]</span></div></li></ol>
<p>Mr Chan's daughter, Elly, wants to buy a gift for her brother and works out a savings plan with her father. The plan is for&nbsp;$n$ months and Elly will save $\$5$ at the start of every month in a piggy bank. She saves her first $\$5$ on $1$ May $2021$. To help Elly, Mr Chan adds $\$5$ in the piggy bank on the last day of May. For the subsequent months, he puts in $\$5$ more than the previous month in the piggy bank on the last day of each month, so that he puts in $\$5$ in May, $\$10$ in June, and so on. If the gift costs $\$120$, write down an inequality involving $n$ and solve it. Find the date when Elly will save enough money in her piggy bank to buy the gift.<span class="inline-marks">[5]</span></p>

Mr Chan plans to buy a new car in $2031$, which costs $\$120000$ in $2021$. There is a yearly inflation rate of $0.5\%$ on the price of the car.

Mr Chan comes across a savings plan in a local bank that pays $1.5\%$ interest on $31$ December of each year. In order to buy the car on $31$ December $2031$, he intends to save $\$y$ every year starting from $1$ January $2022$ for $10$ years with this bank without any withdrawal. On $1$ January $2022$ Mr Chan puts $\$y$ into the bank, and on the $1$ January of each subsequent year, he puts another $\$y$ into the bank.

Mr Chan's daughter, Elly, wants to buy a gift for her brother and works out a savings plan with her father. The plan is for $n$ months and Elly will save $\$5$ at the start of every month in a piggy bank. She saves her first $\$5$ on $1$ May $2021$. To help Elly, Mr Chan adds $\$5$ in the piggy bank on the last day of May. For the subsequent months, he puts in $\$5$ more than the previous month in the piggy bank on the last day of each month, so that he puts in $\$5$ in May, $\$10$ in June, and so on. If the gift costs $\$120$, write down an inequality involving $n$ and solve it. Find the date when Elly will save enough money in her piggy bank to buy the gift.

Problems involving inequalities

Geometric Progression

Arithmetic Progression

Annuities and Amortization

2021 JPJC J2 MYE P1 Q10

Sequence and Series

<p>Mabel and Janice decided to start a $5$-year savings plan beginning in January $2026$.</p>
<p>Mabel saves using a piggy bank. At the start of January $2026$, she deposits $\$101$. Each subsequent month, she increases her deposit by $\$1$—so she deposits $$101$ in January, $\$102$ in February, $\$103$ in March, and so on, until $\$112$ in December. At the start of each new year, she resets her monthly deposit to $\$101$ in January and repeats the same pattern through December. She continues this routine from $2026$ to $2030$, inclusive.</p>
<p>Janice, on the other hand, deposits $$100$ at the start of every month into a bank account that earns $0.3$% interest per month, with interest calculated and added into the account at the end of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-6-a" data-list-item-path="item-6-a"><div class="list-item-paragraph">Show that Janice will have more money in her savings account than Mabel has in her piggy bank at the end of December $2030$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-6-b" data-list-item-path="item-6-b"><div class="list-item-paragraph">Find the month and year when Janice’s savings first exceed Mabel’s savings.<span class="inline-marks">[4]</span></div></li></ol>

Mabel and Janice decided to start a $5$-year savings plan beginning in January $2026$.

Mabel saves using a piggy bank. At the start of January $2026$, she deposits $$101$. Each subsequent month, she increases her deposit by $$1$—so she deposits $$101$ in January, $$102$ in February, $$103$ in March, and so on, until $$112$ in December. At the start of each new year, she resets her monthly deposit to $$101$ in January and repeats the same pattern through December. She continues this routine from $2026$ to $2030$, inclusive.

Janice, on the other hand, deposits $$100$ at the start of every month into a bank account that earns $0.3$% interest per month, with interest calculated and added into the account at the end of each month.

Show that Janice will have more money in her savings account than Mabel has in her piggy bank at the end of December $2030$.

Find the month and year when Janice’s savings first exceed Mabel’s savings.

Inter-sequence Relationships

2025 TJC P1 Q7

<p>Mr Huat is saving for a car that he plans to buy in the future. He needs to save a minimum of $$150,000$. A savings plan allows him to deposit $$6,000$ into a savings account on the first day of every month. At the end of each month, the total amount in the savings account (including interest) is increased by $3\%$. Mr Huat makes an initial deposit on $1$ January $2026$.</p>
<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph">Show that the total amount, correct to $2$ decimal places, in the savings account at the end of December $2026$ is $$87,706.74$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-paragraph">Find the month and year in which the total amount in the savings account will first exceed $$150,000$. Explain whether this occurs on the first or last day of the month.<span class="inline-marks">[6]</span></div></li></ol>
<p>Mr Huat can also choose to save regularly in a different savings account which offers no interest. He makes an initial deposit on $1$ January $2026$ of $$6,000$. On the first day of each subsequent month, he deposits $$d$ more than he deposited in the previous month.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c-2" data-list-item-path="item-c-2"><div class="list-item-paragraph">Find, in terms of $d$, the total amount in the savings account at the end of December $2026$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d-2" data-list-item-path="item-d-2"><div class="list-item-paragraph">Hence, find the minimum value of $d$ such that Mr Huat can buy the car by end of December $2026$. Give your answer correct to the nearest integer.<span class="inline-marks">[2]</span></div></li></ol>

Mr Huat is saving for a car that he plans to buy in the future. He needs to save a minimum of $$150,000$. A savings plan allows him to deposit $$6,000$ into a savings account on the first day of every month. At the end of each month, the total amount in the savings account (including interest) is increased by $3\%$. Mr Huat makes an initial deposit on $1$ January $2026$.

Mr Huat can also choose to save regularly in a different savings account which offers no interest. He makes an initial deposit on $1$ January $2026$ of $$6,000$. On the first day of each subsequent month, he deposits $$d$ more than he deposited in the previous month.

Find, in terms of $d$, the total amount in the savings account at the end of December $2026$.

Hence, find the minimum value of $d$ such that Mr Huat can buy the car by end of December $2026$. Give your answer correct to the nearest integer.

2025 CJC P1 Q10

<p>Sarah, a Singaporean Citizen, started a $4$-year undergraduate Computing course at the National University of Singapore (NUS) in August $2021$. The annual tuition fee, after the MOE Tuition Grant, is $$8250$. Sarah took a Tuition Fee Loan (TFL) covering $90$% of her tuition fees. This loan is interest-free during her course of study.</p>
<p>Sarah decided to set aside $$x$ per month from August $2021$. This amount is increased by $\$2$ each subsequent month. (Sarah set aside $$x$ in August $2021$, $$(x + 2)$ in September $2021$, ...) With the amount set aside, she repaid one-fifth of her total TFL at the end of July $2025$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $x$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Upon Sarah's graduation, the TFL's interest rate becomes effective at a fixed rate of $0.4$% per month, calculated on the outstanding balance at the end of the month. The interest is first charged on $31$st August $2025$.</p>
<p>Sarah plans to make monthly repayments of $$y$ to the bank on the $1$st of every month, starting from $1$st September $2025$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the amount of outstanding loan after Sarah's first monthly repayment.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the amount of outstanding loan after Sarah's $n$th monthly repayment is $23760(1.004)^n - 250y(1.004^n - 1)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sarah wants to pay off her remaining loan by the end of $3$ years. Find the value of $y$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Sarah decided on a different repayment strategy. For the first $24$ months, she repays $$500$ per month. After these $24$ months, she doubles her monthly repayment to $\$1000$.</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Determine the month and year Sarah will clear her loan and calculate the amount of her final repayment.</span><span class="inline-marks">[4]</span></div></li></ol>

Sarah, a Singaporean Citizen, started a $4$-year undergraduate Computing course at the National University of Singapore (NUS) in August $2021$. The annual tuition fee, after the MOE Tuition Grant, is $$8250$. Sarah took a Tuition Fee Loan (TFL) covering $90$% of her tuition fees. This loan is interest-free during her course of study.

Sarah decided to set aside $$x$ per month from August $2021$. This amount is increased by $$2$ each subsequent month. (Sarah set aside $$x$ in August $2021$, $$(x + 2)$ in September $2021$, ...) With the amount set aside, she repaid one-fifth of her total TFL at the end of July $2025$.

Upon Sarah's graduation, the TFL's interest rate becomes effective at a fixed rate of $0.4$% per month, calculated on the outstanding balance at the end of the month. The interest is first charged on $31$st August $2025$.

Sarah plans to make monthly repayments of $$y$ to the bank on the $1$st of every month, starting from $1$st September $2025$.

Find the amount of outstanding loan after Sarah's first monthly repayment.

Show that the amount of outstanding loan after Sarah's $n$th monthly repayment is $23760(1.004)^n - 250y(1.004^n - 1)$.

Sarah wants to pay off her remaining loan by the end of $3$ years. Find the value of $y$.

Sarah decided on a different repayment strategy. For the first $24$ months, she repays $$500$ per month. After these $24$ months, she doubles her monthly repayment to $$1000$.

Determine the month and year Sarah will clear her loan and calculate the amount of her final repayment.

2025 ACJC P1 Q12

<p>At the beginning of January 2025, James took a housing loan of ${\$550\text{ }000}$ from National Housing Board (NHB) that charges ${2.60\%}$ at the end of every year. James pays back $\$3900$ in the middle of every month, starting from January 2025.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the amount that James owes NDB at the start of $n$ year is $1\text{ 846 800}-1\text{ }296\text{ }800{{\left( 1.026 \right)}^{n-1}}$.</div></li><li><div class="list-item-paragraph">Determine the earliest month and year he will fully repay his loan and the amount he of his final instalment.</div></li><li><div class="list-item-paragraph">Calculate the total interest he will have paid upon completing the loan repayment.</div></li></ol>

At the beginning of January 2025, James took a housing loan of ${\$550\text{ }000}$ from National Housing Board (NHB) that charges ${2.60\%}$ at the end of every year. James pays back $\$3900$ in the middle of every month, starting from January 2025.

Worked Example 3

<p>A fund is started at $\$1000$ and compound interest is reckoned at $4\%$ per annum.</p>
<p>If withdrawals of $\$50$ are made at the beginning of each of the subsequent years, show that the amount in the fund at the beginning of the ${(n+1)^{\mathrm{th}}}$ year is $\$$ $1250\left[1 - \frac{1}{5}(1.04)^n\right]$</p>
<p>Find the amount in the fund to the nearest dollar after twenty years, and also how many years it will last.</p>

A fund is started at $\$1000$ and compound interest is reckoned at $4\%$ per annum.

If withdrawals of $\$50$ are made at the beginning of each of the subsequent years, show that the amount in the fund at the beginning of the ${(n+1)^{\mathrm{th}}}$ year is $\$$ $1250\left[1 - \frac{1}{5}(1.04)^n\right]$

Find the amount in the fund to the nearest dollar after twenty years, and also how many years it will last.

Compound Interest with regular writhdrawals

<p>In January 2025, Samantha obtained a housing loan of $\$600\;000$ from the National Housing Trust (NHT), which applies an annual interest rate of $3\%$ at the end of each year. Every middle of the month, starting from January 2025, Samantha begins repaying an instalment of $\$4500$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Demonstrate that the amount Samantha owes NHT at the start of the $n$ year can be expressed as&nbsp;$1,854,000 - 1,254,000(1.03)^{n-1}$.</div></li><li><div class="list-item-paragraph">Determine the earliest month and year in which she will fully repay her loan and the amount of her final instalment.</div></li><li><div class="list-item-paragraph">Calculate the total interest she will have paid upon completing the loan repayment.</div></li></ol>

In January 2025, Samantha obtained a housing loan of $\$600\;000$ from the National Housing Trust (NHT), which applies an annual interest rate of $3\%$ at the end of each year. Every middle of the month, starting from January 2025, Samantha begins repaying an instalment of $\$4500$.

<p>At the beginning of January 2025, James took a housing loan of $\$550\;{000}$ from National Housing Board (NHB) that charges $2.60\%$ at the end of every year. James pays back $\$3900$ in the middle of every month, starting from January 2025.</p>
<ol class="abc-list"><li data-list-item-id="item-1761911484169-zxzlfqhm9" data-list-item-path="item-1761911484169-zxzlfqhm9"><div class="list-item-paragraph">Show that the amount that James owes NHB at the start of $n$ year is $1,846,800 - 1,296,800(1.026)^{n-1}$.</div></li><li data-list-item-id="item-1761911484686-k1asd1e5l" data-list-item-path="item-1761911484686-k1asd1e5l"><div class="list-item-paragraph">Determine the earliest month and year he will fully repay his loan and the amount of his final instalment.</div></li><li data-list-item-id="item-1761911485924-xim92m9os" data-list-item-path="item-1761911485924-xim92m9os"><div class="list-item-paragraph">Calculate the total interest he will have paid upon completing the loan repayment.</div></li></ol>

At the beginning of January 2025, James took a housing loan of $\$550\;{000}$ from National Housing Board (NHB) that charges $2.60\%$ at the end of every year. James pays back $\$3900$ in the middle of every month, starting from January 2025.

Financial Arithmetic Q1

<p>At the start of January 2014, Leon took a home loan of $\$500\,\,000$ from a bank that offers a fixed monthly interest rate of $0.3\%$, to buy an apartment unit in a condominium. The interest was added to the amount owed on the ${{15}^{\mathrm{th}}}$ of every month, with the first interest amount added on 15 January 2014. Leon made a monthly repayment of $\$x$ on the 20$^{\mathrm{th}}$ of every month, with the first repayment on 20 January 2014.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the amount (in dollars) that Leon owed at the end of $n$ months was</div><div class="list-item-paragraph" style="text-align: center">$500000{{\left( 1.003 \right)}^{n}}-\frac{1000x}{3}\left( {{1.003}^{n}}-1 \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Leon planned to repay the loan fully in $360$ monthly repayments.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the value of $x$, giving your answer to the nearest cents.<span class="inline-marks">[2]</span></div></li></ol>
<p>The estate management collects a monthly maintenance fee from each apartment unit within the condominium at the start of every month. The monthly maintenance fee on 1 January 2014 was $\$400$. To cope with the rising cost of maintenance over time, the estate management increased the monthly maintenance fee by $\$k$ every $2$ years such that the monthly maintenance fee increased to $\$\left(400+k\right)$ on 1 January 2016 and to $\$\left(400+2k\right)$ on 1 January 2018 and so on.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Show that the total maintenance fee (in dollars) that Leon paid at the end of $10$ years was of the form $s+tk$, where&nbsp;$s$ and $t$ are positive constants to be determined.<span class="inline-marks">[3]</span></div></li></ol>
<p>The value of Leon’s apartment unit increased to $\$750\,\,000$ at the end of December 2023. It was also known that&nbsp;$k=50$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">By considering the amount Leon owed, along with the repayment and maintenance fee that he paid over the $10$ years, explain, with justification, whether Leon should sell his apartment unit at the end of December 2023 if he received an offer of $\$750\,\,000$ from an interested buyer.<span class="inline-marks">[4]</span></div></li></ol>

At the start of January 2014, Leon took a home loan of $\$500\,\,000$ from a bank that offers a fixed monthly interest rate of $0.3\%$, to buy an apartment unit in a condominium. The interest was added to the amount owed on the ${{15}^{\mathrm{th}}}$ of every month, with the first interest amount added on 15 January 2014. Leon made a monthly repayment of $\$x$ on the 20$^{\mathrm{th}}$ of every month, with the first repayment on 20 January 2014.

Leon planned to repay the loan fully in $360$ monthly repayments.

The estate management collects a monthly maintenance fee from each apartment unit within the condominium at the start of every month. The monthly maintenance fee on 1 January 2014 was $\$400$. To cope with the rising cost of maintenance over time, the estate management increased the monthly maintenance fee by $\$k$ every $2$ years such that the monthly maintenance fee increased to $\$\left(400+k\right)$ on 1 January 2016 and to $\$\left(400+2k\right)$ on 1 January 2018 and so on.

The value of Leon’s apartment unit increased to $\$750\,\,000$ at the end of December 2023. It was also known that $k=50$.

2024 HCI Promo Q13

<ol class="abc-list"><li><div class="list-item-paragraph">Zack saves $\$x$ on 1 January 2024. On the first day of each subsequent month, he saves $\$10$ more than in the previous month, so that he saves $\$\left(x+10\right)$ on 1 February 2024, $\$\left(x+20\right)$ on 1 March 2024, and so on.</div><div class="list-item-paragraph">Find the value of $x$ if he saves $\$25\,000$ in total on 1 December 2026.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Dan opens a savings account. He deposits $\$200$ on 1 January 2024 and continues to deposit $\$200$ on the first day of each subsequent month. The interest rate of the savings account is $0.2\%$ per month, so that on the last day of each month the amount in the account on that day is increased by $0.2\%$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find how much the $\$200$ deposited on 1 January 2024 is worth at the end of 31 December 2024.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that, at the end of the $n$th month (where January 2024 is the first month, February 2024 is the second month, and so on), the total amount in the account is $\$100200\left({{1.002}^{n}}-1\right)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">In which month and year will the amount in Dan’s account first exceed $\$4500$?</div><div class="list-item-paragraph">Explain whether this occurs on the first or last day of the month.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

Compound Interest

2024 YIJC Promo Q11

<p>A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound interest of $5\%$ per annum at the end of the year, based on the amount in the account at the beginning of each year.</p>
<p>John and Sarah are both interested in this scheme.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">John invests $\$x$ at the start of the first year and a further $\$x$ on the first day of each subsequent year. He chooses to leave the money in his account for the interest to accumulate.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down the amount in John’s account, including the interest, at the end of the first year.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that John will have a total of $21x\left( {{1.05}^{n}}-1 \right)$ in his account at the end of $n$ years.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">If John invests $\$10000$ at the start of every year, find the number of years for the total in his account to first exceed $\$500\,\,000$. Determine if this happens at the start or at the end of that year.<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">Sarah invests $\$6000$ at the start of the first year. On the first day of each subsequent year, she invests $\$400$ more than the amount invested at the start of the previous year.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why the amount in Sarah’s account at the end of $n$ years can be given by $\sum\limits_{r=1}^{n}{\left( a+b\left( r-1 \right) \right)}{{\left( 1.05 \right)}^{n-\left( r-1 \right)}}$, and determine the values of the constants $a$ and $b$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence determine the number of complete years for the total amount in her account to first exceed $\$500\,\,000$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound interest of $5\%$ per annum at the end of the year, based on the amount in the account at the beginning of each year.

John and Sarah are both interested in this scheme.

Sigma Notation

2024 RI Promo Q11

<p>On his ${{45}^{\mathrm{th}}}$ birthday on $1$ July $2024$, Mr Eu purchases an annuity plan called EuRetire for a principal sum of $\$x$. The plan works like this:</p>
<p><strong>Phase 1:</strong> The plan pays compound interest at a rate of $3\%$ per annum on the last day of June each year, until Mr Eu turns $65$. The total sum at the end of Phase 1 is called the “accumulated sum”.</p>
<p><strong>Phase 2:</strong> Upon turning $65$, Mr Eu receives a monthly payout, $\$y$, on the ${{1}^{\mathrm{st}}}$ of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at $0.2\%$ per month on the last day of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-1760753493990-nobajnq7h" data-list-item-path="item-1760753493990-nobajnq7h"><div class="list-item-paragraph">Write down an expression for the accumulated sum at the end of Phase 1, giving your answer to $5$ significant figures.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760753494538-0yukkoe6j" data-list-item-path="item-1760753494538-0yukkoe6j"><div class="list-item-paragraph">Show that the amount remaining in the plan at the end of the ${{n}^{\mathrm{th}}}$ month of Phase 2, after interest has been credited, is</div><div class="list-item-paragraph" style="text-align: center">$\$1.8061\left({{1.002}^{n}}\right)x-501\left({{1.002}^{n}}-1\right)y$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760753495850-omow3pwib" data-list-item-path="item-1760753495850-omow3pwib"><div class="list-item-paragraph">Mr Eu wishes to receive $\$1000$ a month in Phase 2 for at least $10$ years. What is the minimum principal sum, correct to the nearest dollar, he would need?<span class="inline-marks">[2]</span></div></li></ol>

On his ${{45}^{\mathrm{th}}}$ birthday on $1$ July $2024$, Mr Eu purchases an annuity plan called EuRetire for a principal sum of $\$x$. The plan works like this:

**Phase 1:** The plan pays compound interest at a rate of $3\%$ per annum on the last day of June each year, until Mr Eu turns $65$. The total sum at the end of Phase 1 is called the “accumulated sum”.

**Phase 2:** Upon turning $65$, Mr Eu receives a monthly payout, $\$y$, on the ${{1}^{\mathrm{st}}}$ of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at $0.2\%$ per month on the last day of each month.

2024 EJC P1 Q11

<p>Customers who open a savings account with SRF bank will earn an interest rate of $0.2\%$ per month based on the amount in the account on the last day of each month. The interest is added to the account at the end of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-1760689764559-ualjpv8ke" data-list-item-path="item-1760689764559-ualjpv8ke"><div class="list-item-paragraph">Peter has a savings account with the SRF bank. He puts $\$2000$ into his savings account at the beginning of each month starting from ${{1}^{\mathrm{st}}}$ January 2023. Find the amount he has in his savings account on ${{31}^{\mathrm{st}}}$ December 2027 after the interest has been added if no withdrawal is made. Show your workings clearly.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760689765331-qpxbpom5w" data-list-item-path="item-1760689765331-qpxbpom5w"><div class="list-item-paragraph"><p>Sandy, another customer of SRF bank, has retired from work starting ${{1}^{\mathrm{st}}}$ January $2023$. Since then, she will no longer do any deposit of money into her own savings account with the bank. Instead, she will be using the money saved in that account for her monthly spending. She has&nbsp;$\$750\,000$ in her SRF bank savings account on ${{1}^{\mathrm{st}}}$ January $2023$.</p></div><ol class="i-list"><li data-list-item-id="subitem-1760689790367-d4jrkulvn" data-list-item-path="item-1760689765331-qpxbpom5w/subitem-1760689790367-d4jrkulvn"><div class="list-item-paragraph"><p>As she still has some money with her for her daily spending in January $2023$, she will only commence her monthly withdrawal of $\$k$ from her savings account in the middle of each month, starting from February $2023$ onwards. Show that the amount of money in her savings account, at the end of $n$ complete months from ${{1}^{\mathrm{st}}}$ January $2023$, after the interest has been added, is given by ${{1.002}^{n}}\left( \$750\,\,000\right)-\$501k\left({{1.002}^{n-1}}\right)+\$501k$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760689791260-a0sj3x2o6" data-list-item-path="item-1760689765331-qpxbpom5w/subitem-1760689791260-a0sj3x2o6"><div class="list-item-paragraph"><p>Taking $k$ to be $3000$, find the amount of money left in her savings account immediately after her last complete withdrawal of $\$ 3000$.</p><span class="inline-marks">[4]</span></div></li></ol></li></ol>

Customers who open a savings account with SRF bank will earn an interest rate of $0.2\%$ per month based on the amount in the account on the last day of each month. The interest is added to the account at the end of each month.

2023 ASRJC Promo Q10

<ol class="abc-list"><li data-list-item-id="item-1760448834847-2mymvviyx" data-list-item-path="item-1760448834847-2mymvviyx"><div class="list-item-paragraph">Given that the sum of the first $n$ terms of a series is ${{S}_{n}}$, where&nbsp;${{S}_{n}}=3{{n}^{2}}-n$, state the value of its first term and write down an expression for&nbsp;${{S}_{n-1}}$ for&nbsp;$n\ge 2$. Hence show that the terms in the series form an arithmetic progression.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760448835380-rw8oarkrn" data-list-item-path="item-1760448835380-rw8oarkrn"><div class="list-item-paragraph">The first term of an infinite geometric progression is $a$ and the common ratio is $r$, where&nbsp;$\left| r \right|&lt;1$. Given that the sum of all the terms after the $n$th term is equal to the $n$th term of the progression, show that&nbsp;$r=\frac{1}{2}$ and hence find the sum to infinity of the progression in terms of $a$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760448836093-vqo4pqusk" data-list-item-path="item-1760448836093-vqo4pqusk"><div class="list-item-paragraph">A bank has an account for investors. Compound interest is added to the account at the end of each year at a fixed rate of $3\%$ of the amount in the account at the beginning of that year.</div><div class="list-item-paragraph">Mr Ang decides to invest $\$2\mathrm{} \, 000$ on 1 January 2023 and a further $\$2\mathrm{} \, 000$ on 1 January of each subsequent year.</div><ol class="i-list"><li data-list-item-id="subitem-1760448859665-oa1uj8pcz" data-list-item-path="item-1760448836093-vqo4pqusk/subitem-1760448859665-oa1uj8pcz"><div class="list-item-paragraph">How much compound interest has his <strong>original</strong> $\$2\mathrm{} \, 000$ earned at the end of 31 December 2030?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760448860528-gii4uhvv6" data-list-item-path="item-1760448836093-vqo4pqusk/subitem-1760448860528-gii4uhvv6"><div class="list-item-paragraph">By considering the total amount in the account at the end of the $n$ th year, find the year in which the total in the account will first exceed $\$40\mathrm{} \, 000$. Explain whether this occurs at the beginning or the end of the year.<span class="inline-marks">[6]</span></div></li></ol></li></ol>

Proving the progression

Sum to infinity

$u_n = S_n - S_{n-1} $

2023 YIJC Promo Q12

<ol class="abc-list"><li><div class="list-item-paragraph">James puts $\$500$ into an empty safe on 1 January 2023. On the first day of each subsequent month he puts $\$10$ more than in the previous month, so that he puts $\$510$ on 1 February 2023, $\$520$ on 1 March 2023, and so on. On what date will he first have over $\$10,000$ in total in his safe?<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Nora puts $\$x$ on 1 January 2023 into a bank account which pays a compound interest at a rate of $0.5\%$per month on the last day of each month. She puts a further $\$x$ into the account on the first day of each subsequent month. Given that she does not withdraw any money from her account, find the least integer value of $x$ for which her account will accumulate at least $\$50,000$ on 31 December 2027.<span class="inline-marks">[5]</span></div></li></ol>

2023 TJC P1 Q5

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Mr Tan set aside $\$80\,000$ for his two sons. On the first day of the year that his sons turned $7$ and $17$ years old, he deposited $\$x$ into the younger son’s bank account and the remaining sum of money into the elder son's bank account. Mr Tan adds a further $\$1000$ into the younger son’s account on the first day of each subsequent year. The bank pays a compound interest at a rate of $2\%$ per annum on the last day of each year.</div><div class="list-item-paragraph">Each son will withdraw the full sum of money from his account (after interest had been added) on the last day of the year that he turns $21$ years old.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the amount of money the elder son will withdraw in terms of $x$. Leave your answer in exact form.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the amount of money, in terms of $n$ and $x$, in the younger son’s bank account at the end of the $n$ th year, where&nbsp;$n\ge 2$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using answer from <strong>(ii)</strong>, show if Mr Tan had initially deposited $\$20\,000$ into the younger son’s account, the younger son would withdraw $\$43,\,210.78$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the value of $x$ if Mr Tan wanted both sons to receive the same withdrawal amount, giving your answer to the nearest integer.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2015 TJC P2 Q2 [Modified]

<p>Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1st October 2022, he charged $\$7500$ to the credit card. The credit card charges a $2\%$ monthly interest rate that is applied on the last day of every month. From 1st November 2022, Antonio started paying off his credit card debt by paying $\$x$ at the start of each month, where $x$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the amount of Antonio’s credit card debt left after his first repayment, giving your answer in terms of $x$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Show that the amount of Antonio’s credit card debt after his $n$th repayment is</div><div class="list-item-paragraph" style="text-align: center">$\left( 7500-50x \right){{\left( 1.02 \right)}^{n}}+50x$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the minimum amount that Antonio must consistently pay each monthly repayment for him to fully pay off his credit card debt with his $60$th repayment. Find also the amount of credit card interest he would have paid in this case.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Suppose that up to and including 1st October 2023, Antonio had faithfully repaid $\$500$ a month to service his credit card debt. As a promotional reward, the credit card company decided to lower Antonio’s monthly interest rate to $1\%$ with immediate effect from October 2023, provided he maintains his $\$500$ monthly repayments.</div><div class="list-item-paragraph">Find the date of Antonio’s last repayment to the credit card company with this new interest rate, and the amount of his last repayment.</div><p style="text-align: right;">[4]</p></li></ol>
<p>Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at $\$1500$ for each table of $10$ guests. At the Grande Hotel, the catering is priced as follows:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760348977028-1pg2z6ifwaj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzNDg5NzcwMjgtMXBnMno2aWZ3YWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcxMzc2NSwiZXhwIjoxNzkyNzE0MDY1fQ.YYExzr9166ZTXJbK4gCreMTuKfpLzOFsJkhHliZ8a5w" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">How many tables will the couple need at least, for it to be cheaper to hold their wedding banquet at the Grande Hotel?</div><p style="text-align: right;">[3]</p></li></ol>

Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1st October 2022, he charged $\$7500$ to the credit card. The credit card charges a $2\%$ monthly interest rate that is applied on the last day of every month. From 1st November 2022, Antonio started paying off his credit card debt by paying $\$x$ at the start of each month, where $x$ is a constant.

Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at $\$1500$ for each table of $10$ guests. At the Grande Hotel, the catering is priced as follows:

2022 ACJC Promo Q11

<p>Mrs Tan plans to start a business which requires a start-up capital of $\$700,000$. She decided to first save $\$200,000$ by depositing money every month into a savings plan. For the remaining $\$500,000$, she intends to take a loan from a finance company.</p>
<p>She deposited $\$3000$ into the savings plan in the first month and on the first day of each subsequent month, she deposited $\$100$ more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $\$200,000$. It is given that this savings plan pays no interest.</p>
<ol class="i-list"><li data-list-item-id="item-1760261886772-61fdl9r6i" data-list-item-path="item-1760261886772-61fdl9r6i"><div class="list-item-paragraph">Find the month in which Mrs Tan’s monthly deposit will exceed $\$6550$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760261887531-o7awzqx80" data-list-item-path="item-1760261887531-o7awzqx80"><div class="list-item-paragraph">Find the number of months that it will take for Mrs Tan to save $\$200,000$ and hence find the amount that she would have deposited in the last month.<span class="inline-marks">[4]</span></div></li></ol>
<p>After Mrs Tan has saved $\$200,000$, she took a loan of $\$500,000$ from a finance company. To repay the loan from the finance company, Mrs Tan would have to pay a monthly payment of $\$x$ at the beginning of each month, starting from the first month. An interest of $0.3\%$ per month will be charged on the outstanding loan amount at the end of the month.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760261910344-wkmf4r3uo" data-list-item-path="item-1760261910344-wkmf4r3uo"><div class="list-item-paragraph">Show that the outstanding amount at the end of ${{n}^{\mathrm{th}}}$ month, after the interest has been charged, is $A\left( {{1.003}^{n}} \right)-Bx\left( {{1.003}^{n}}-1 \right)$, where&nbsp;$A$ and $B$ are exact constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760261911987-teh2q0rrs" data-list-item-path="item-1760261911987-teh2q0rrs"><div class="list-item-paragraph">Find the amount of $\$x$, to $2$ decimal places, if Mrs Tan wants to fully repay her loan in $8$ years.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760261912572-h3okmetny" data-list-item-path="item-1760261912572-h3okmetny"><div class="list-item-paragraph">Using the value of $x$ found in part <strong>(iv)</strong>, calculate the total interest that the finance company will earn from Mrs Tan at the end of $8$ years.<span class="inline-marks">[2]</span></div></li></ol>

Mrs Tan plans to start a business which requires a start-up capital of $\$700,000$. She decided to first save $\$200,000$ by depositing money every month into a savings plan. For the remaining $\$500,000$, she intends to take a loan from a finance company.

She deposited $\$3000$ into the savings plan in the first month and on the first day of each subsequent month, she deposited $\$100$ more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $\$200,000$. It is given that this savings plan pays no interest.

After Mrs Tan has saved $\$200,000$, she took a loan of $\$500,000$ from a finance company. To repay the loan from the finance company, Mrs Tan would have to pay a monthly payment of $\$x$ at the beginning of each month, starting from the first month. An interest of $0.3\%$ per month will be charged on the outstanding loan amount at the end of the month.

2021 ACJC Promo Q12

<p>Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for $6$ years which charges an interest rate of $3.00\%$ per annum, compounded monthly. Interest is chargeable immediately when the loan sum is drawn out. The monthly payment, $\$m$, is fixed throughout the loan tenure.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the calculated loan balance at the end of the ${{n}^{\mathrm{th}}}$ loan month, after the monthly repayment is made, is given by</div><div class="list-item-paragraph" style="text-align: center">$40000{{\left( \frac{401}{400} \right)}^{n}}-400m\left[ {{\left( \frac{401}{400} \right)}^{n}}-1 \right]$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By legislation, banks can approve a car loan only if the monthly repayment does not exceed $15\%$ of an applicant’s monthly income. Prove that Mary will not be able to apply for the car loan.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">If the interest rate for all car loans by the banks is compounded monthly, find the range of interest rates chargeable which will enable Mary to apply for the car loan successfully. Give your answer in the form of $r\%$ per annum, correct to $1$ decimal place.</div><p style="text-align: right;">[3]</p></li></ol>

Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for $6$ years which charges an interest rate of $3.00\%$ per annum, compounded monthly. Interest is chargeable immediately when the loan sum is drawn out. The monthly payment, $\$m$, is fixed throughout the loan tenure.

2013 AJC Promo Q12

<p>A bank offers both an ordinary account and a savings account on 1st January 2020 Perlin puts $\$100$ into an ordinary account. On the first day of each subsequent month she saves $\$5$ more than in the previous month, so that she saves $\$105$ on 1st February 2020, $\$110$ on 1st March 2020, and so on. This account pays no interest.</p>
<ol class="i-list"><li><div class="list-item-paragraph">On what date will the amount in Perlin's account first exceed $\$5000$?</div><p style="text-align: right;">[5]</p></li></ol>
<p>On 1st January 2020, Pauline puts $\$100$ into a savings account, and on the first day of each subsequent month she puts another $\$x$ into the account. The interest rate is $0.5\%$ per month, so that on the last day of each month the amount in the account on that day is increased by $0.5\%$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Show that the expression for the amount in Pauline’s account on the last day of the ${{n}^{\mathrm{th}}}$ month (where January 2020 is the 1st month, February 2020 is the 2nd month, and so on) is given by</div><div class="list-item-paragraph" style="text-align: center">${{1.005}^{n}}\left( 200x+100 \right)-201x$.</div><p style="text-align: right;">[3]</p><ol class="abc-list"><li><div class="list-item-paragraph">If $x=150$, find the number of complete months for the total amount in Pauline's account to first exceed $\$5,000$</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Pauline wishes for the amount in her account to be at least $\$5,000$ on 31st October 2021 in order to purchase a first-class flight to Japan for her year-end vacation. What is the smallest integer value of $x$ would achieve this?</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A bank offers both an ordinary account and a savings account on 1st January 2020 Perlin puts $\$100$ into an ordinary account. On the first day of each subsequent month she saves $\$5$ more than in the previous month, so that she saves $\$105$ on 1st February 2020, $\$110$ on 1st March 2020, and so on. This account pays no interest.

On 1st January 2020, Pauline puts $\$100$ into a savings account, and on the first day of each subsequent month she puts another $\$x$ into the account. The interest rate is $0.5\%$ per month, so that on the last day of each month the amount in the account on that day is increased by $0.5\%$.

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