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<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">A train is travelling at a constant speed of $x$ km/h for $40$ km. Then it slows down and travels $\left(25 + \frac{x}{5}\right)$ km at a constant speed of $15$ km/h slower. Given that it takes $12$ minutes shorter to travel the first $40$ km, form an equation and show that it reduces down to $\frac{200 + x}{5x} = \frac{125 + x}{5x - 75}$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Solve the equation in <strong>(i)</strong>. Hence, or otherwise, find the average speed of the train for the whole journey.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">After the train slows down, it takes $72$ seconds to pass a lamp post. Find the length of the train.</span><span class="inline-marks">[2]</span></div></li></ol>

A train is travelling at a constant speed of $x$ km/h for $40$ km. Then it slows down and travels $\left(25 + \frac{x}{5}\right)$ km at a constant speed of $15$ km/h slower. Given that it takes $12$ minutes shorter to travel the first $40$ km, form an equation and show that it reduces down to $\frac{200 + x}{5x} = \frac{125 + x}{5x - 75}$.

After the train slows down, it takes $72$ seconds to pass a lamp post. Find the length of the train.

Applications of Linear Equations in Real Life

Speed, Distance and Time

Average Speed over a Journey

IP1 Class Test (May 2025) Q13

Linear Equations and Inequalities

Ratio, Rate and Speed

<p>A book at bookstore A is marked at $\$30$ but sold at $\$24$. Another bookstore B sells this kind of book at $\$28$ after a percentage discount of $30\%$. And bookstore B offers a further percentage discount of $20\%$ on each book after the first 100 when more than 100 books are purchased.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the percentage discounts of the book at bookstore A.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the marked price of the book at bookstore B.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the selling price and the overall percentage discount of the book at bookstore B for the part of purchasing beyond $100$.</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">A school is planning to buy this kind of book for the students. Find which bookstore, A or B, has a cheaper total price if the school wants to buy $400$ such books.</div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">Find the minimum number of books one will find it cheaper to buy from bookstore B than bookstore A.</div></li></ol>

A book at bookstore A is marked at $\$30$ but sold at $\$24$. Another bookstore B sells this kind of book at $\$28$ after a percentage discount of $30\%$. And bookstore B offers a further percentage discount of $20\%$ on each book after the first 100 when more than 100 books are purchased.

Find the percentage discounts of the book at bookstore A.

Find the marked price of the book at bookstore B.

Find the selling price and the overall percentage discount of the book at bookstore B for the part of purchasing beyond $100$.

A school is planning to buy this kind of book for the students. Find which bookstore, A or B, has a cheaper total price if the school wants to buy $400$ such books.

Find the minimum number of books one will find it cheaper to buy from bookstore B than bookstore A.

Percentages involving Finances: Discount

Successive Percentage Changes

Percentages

<p>A storeowner buys a certain number of goods from supplier A at $\$6$ as a unit price and sell at $\$12$ as a unit price. The storeowner bears all other costs at $\$4,000$ in total. Suppose the owner manages to sell all the goods and earns a profit of $20\%$ on the total cost.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of goods the owner buys from supplier A.</span></div></li></ol>
<p>Another supplier B offers a lower unit price at $\$4$ to the storeowner. The storeowner buys a new number of such goods from supplier B and sell at the unit price $\$12$. But this time the storeowner only sells $45\%$ of the goods and made a loss of $10\%$ on the total cost. Suppose all other costs are also $\$4,000$ in total.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the new number of goods the owner buys from supplier B.</span></div></li></ol>

A storeowner buys a certain number of goods from supplier A at $\$6$ as a unit price and sell at $\$12$ as a unit price. The storeowner bears all other costs at $\$4,000$ in total. Suppose the owner manages to sell all the goods and earns a profit of $20\%$ on the total cost.

Find the number of goods the owner buys from supplier A.

Another supplier B offers a lower unit price at $\$4$ to the storeowner. The storeowner buys a new number of such goods from supplier B and sell at the unit price $\$12$. But this time the storeowner only sells $45\%$ of the goods and made a loss of $10\%$ on the total cost. Suppose all other costs are also $\$4,000$ in total.

Find the new number of goods the owner buys from supplier B.

Percentages involving Finances: Profit and Loss

<p>In a $2$-digit positive number, the digit in the ones place is half of the digit in the tens place. The sum of this number and the number when the digits are reversed is $99$. Find the number.</p>

In a $2$-digit positive number, the digit in the ones place is half of the digit in the tens place. The sum of this number and the number when the digits are reversed is $99$. Find the number.

Solving Linear Equations

<p>In a class of $35$ students, there are $17$ boys and $18$ girls. Some girls left the class. Find the number of girls who left the class such that the fraction of the remaining girls in the class becomes $\frac{14}{31}$.</p>

In a class of $35$ students, there are $17$ boys and $18$ girls. Some girls left the class. Find the number of girls who left the class such that the fraction of the remaining girls in the class becomes $\frac{14}{31}$.

<p>The diagram below shows a rectangular plot of land with a circular pond. It is known that the length is $3$ metres longer than twice of the width. The width of the land is $x$ m.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600843964-s9ivokqew7i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA4NDM5NjQtczlpdm9rcWV3N2kucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MTc0NzM5NCwiZXhwIjoxNzkxNzQ3Njk0fQ.FVSNSbe0ikQgPKUZ3VTVjvqX-CSWksg3e9_11tLpWTs" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the length of the land (in metres).</span></div></li><li><div class="list-item-paragraph">The perimeter of the land is $48$ meters and the radius of the pond is $3$ metres. Find the value of $x$.</div></li><li><div class="list-item-paragraph">Find the area of the land which is not covered by the pond, correct to $3$ significant figures.</div></li></ol>

The diagram below shows a rectangular plot of land with a circular pond. It is known that the length is $3$ metres longer than twice of the width. The width of the land is $x$ m.

Write down an expression, in terms of $x$, for the length of the land (in metres).

Perimeter and Area of Composite Figures

Forming Algebraic Expressions and Formulae

Perimeter and Area of Rectangles and Squares

Algebraic Expressions, Identities and Formulae

Perimeter and Area of Plane Figures

<p>A string which forms the rectangle $ABCD$ is used to form a square $PQRS$. The rectangle has length and width of $(2x + 7)$ cm and $(x + 5)$ cm respectively and the square has sides of length $(3x - 11)$ cm.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600692494-ldvnf79p4d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA2OTI0OTQtbGR2bmY3OXA0ZC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkxNzQ3Mzk0LCJleHAiOjE3OTE3NDc2OTR9.qLcfOk9N1wXA9QrM6TBg_wqXMACoEiPmbbddotX74Rw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down an equation in $x$ and find the value of $x$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the square $PQRS$.</span></div></li></ol>

A string which forms the rectangle $ABCD$ is used to form a square $PQRS$. The rectangle has length and width of $(2x + 7)$ cm and $(x + 5)$ cm respectively and the square has sides of length $(3x - 11)$ cm.

Write down an equation in $x$ and find the value of $x$.

 Applications of Linear Equations in Real Life

<p>A man drives from town $A$ to town $B$ at an average speed of $67$ km/h. Using the same route, he travels back from town $B$ to town $A$ at an average speed of $72$ km/h. If the total travelling time to complete the journey is $85$ minutes, find the distance travelled by the man from town $A$ to town $B$.</p>

A man drives from town $A$ to town $B$ at an average speed of $67$ km/h. Using the same route, he travels back from town $B$ to town $A$ at an average speed of $72$ km/h. If the total travelling time to complete the journey is $85$ minutes, find the distance travelled by the man from town $A$ to town $B$.

<p>The sum of two consecutive even numbers is $178$. Find the product of the two numbers.</p>

The sum of two consecutive even numbers is $178$. Find the product of the two numbers.

 Solving Linear Equations

<p>The numerator of a fraction is $8$ less than its denominator. If $2$ is added to the numerator and $3$ is subtracted from the denominator, the new fraction is $\frac{5}{8}$. Find the fraction in its lowest terms.</p>

The numerator of a fraction is $8$ less than its denominator. If $2$ is added to the numerator and $3$ is subtracted from the denominator, the new fraction is $\frac{5}{8}$. Find the fraction in its lowest terms.

<p>The age of Alan’s father is $5$ years older than $3$ times that of Alan. Let $x$ be Alan’s age (in years).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express the father’s age (in years) in terms of $x$.</span></div></li><li><div class="list-item-paragraph">After $35$ years, Alan will be as old as his father now. Write down an equation in $x$ and solve it to find Alan’s age.</div></li></ol>

The age of Alan’s father is $5$ years older than $3$ times that of Alan. Let $x$ be Alan’s age (in years).

Express the father’s age (in years) in terms of $x$.

<p>Alice has $3x$ marbles. Beatrice has $6$ more marbles than Alice. Candice has half as many marbles as Beatrice.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the number of marbles that Candice has.</span></div></li><li><div class="list-item-paragraph">The total number of marbles is $114$. Find the number of marbles that Beatrice has.</div></li></ol>

Alice has $3x$ marbles. Beatrice has $6$ more marbles than Alice. Candice has half as many marbles as Beatrice.

Write down an expression, in terms of $x$, for the number of marbles that Candice has.

<p>Two consecutive odd numbers are such that the sum of twice the smaller number and thrice the larger number is $61$. Find the numbers.</p>

Two consecutive odd numbers are such that the sum of twice the smaller number and thrice the larger number is $61$. Find the numbers.

[EPQ Vol 1A] Question 25

<p>The denominator of a fraction is $10$ more than its numerator. If $7$ is subtracted from the numerator and $5$ is added to the denominator, the new fraction becomes $\frac{5}{16}$. Find the fraction in its lowest terms.</p>

The denominator of a fraction is $10$ more than its numerator. If $7$ is subtracted from the numerator and $5$ is added to the denominator, the new fraction becomes $\frac{5}{16}$. Find the fraction in its lowest terms.

[EPQ Vol 1A] Question 24

<p>The diagram shows a rectangle $ABCD$ with a square $PBQR$ of side $x$ cm removed. It is given that&nbsp;$AP=3$ cm and $QC=4$ cm.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1777701404992-afflzdye0"><div class="list-item-paragraph">Find, in terms of $x$, an expression for the area of the rectangle $RQCS$.</div></li><li data-list-item-id="item-1777701405629-y2ax2d65d"><div class="list-item-paragraph">Find, in terms of $x$, an expression for the area of the rectangle $APSD$.</div></li></ol><p>The area of the rectangle $APSD$ is twice the area of the rectangle $RQCS$.</p><ol class="abc-list" start="3"><li data-list-item-id="item-1777701413409-roiwtmnt0"><div class="list-item-paragraph">Form an equation in $x$ and solve it.</div></li><li data-list-item-id="item-1777701415009-cx1ib9ofc"><div class="list-item-paragraph">Hence find the area of the rectangle $APSD$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764334567877-7yp8q1yl3ll.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzMzQ1Njc4NzctN3lwOHExeWwzbGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MTc0NzM5NCwiZXhwIjoxNzkxNzQ3Njk0fQ.YCn1nwsx5YUuzkFnUnx_TK64KBz0ybSFaKQe0uiJUuE" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows a rectangle $ABCD$ with a square $PBQR$ of side $x$ cm removed. It is given that $AP=3$ cm and $QC=4$ cm.

Algebraic Areas in Composite Rectangles

<p>Angie is $x$ years old.  Her sister is $12$ years older than her while their mother is twice as old as her sister.  Lily is $3$ years younger than Angie while Lily’s mother is $4$ times as old as Lily.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express the age of Angie’s mother in terms of $x$.</div></li><li><div class="list-item-paragraph">Express the age of Lily’s mother in terms of $x$.</div></li><li><div class="list-item-paragraph">If Angie’s mother is $6$ years older than Lily’s mother, find Angie’s age.</div></li></ol>

Angie is $x$ years old.  Her sister is $12$ years older than her while their mother is twice as old as her sister.  Lily is $3$ years younger than Angie while Lily’s mother is $4$ times as old as Lily.

Word Problem: Ages and Relationships Between Family Members

<p>Amin is $x$ years old and his father is $4$ times as old as him.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down, in its simplest form, an algebraic expression in terms of $x$,</div><ol class="i-list"><li><div class="list-item-paragraph">for Amin’s father’s age now.</div></li><li><div class="list-item-paragraph">for Amin’s age in $5$ years’ time.</div></li></ol></li><li><div class="list-item-paragraph">Given that the sum of their ages in $5$ years’ time is $55$, form an equation in $x$ and find Amin’s age now.</div></li></ol>

Amin is $x$ years old and his father is $4$ times as old as him.

Word Problem: Ages (Amin and His Father)

<p>Amy has $47$ sweets and Benny has $65$ sweets.  If Amy gives some sweets to Benny, the ratio of the sweets of Amy to that of Benny becomes $3:5$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">If the number of sweets that Amy gives to Benny is $x$, write down an expression, in terms of $x$, for the remaining sweets that Amy has.</div></li><li><div class="list-item-paragraph">Write down an expression in terms of $x$, for the number of sweets that Benny has after receiving $x$ number of sweets from Amy.</div></li><li><div class="list-item-paragraph">Hence, form an equation and solve for $x$.</div></li></ol>

Amy has $47$ sweets and Benny has $65$ sweets.  If Amy gives some sweets to Benny, the ratio of the sweets of Amy to that of Benny becomes $3:5$.

Word Problem: Ratio and Transfer of Quantities

<p>There are $3$ consecutive odd integers.  Let $n-1$ be the smallest integer.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down expressions for the next two bigger consecutive odd integers after $n-1$.</div></li><li><div class="list-item-paragraph">Find, in its simplest form, the expression for the sum of these three consecutive odd integers.</div></li><li><div class="list-item-paragraph">If the sum is $159$, find the value of $n$.</div></li></ol>

There are $3$ consecutive odd integers.  Let $n-1$ be the smallest integer.

Word Problem: Consecutive Odd Integers

<p>David is $x$ years old.  David’s father is thrice the age of David.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the father’s age.</div></li></ol>
<p>In $5$ years’ time the sum of their ages is equal to $70$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the father’s age in $5$ years’ time.</div></li><li><div class="list-item-paragraph">Form an equation in terms of $x$.</div></li><li><div class="list-item-paragraph">Solve the equation and find the father’s present age.</div></li></ol>

David is $x$ years old.  David’s father is thrice the age of David.

In $5$ years’ time the sum of their ages is equal to $70$.

Math Question Bank