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PSLE Foundation Mathematics

<p>$p$ is inversely proportional to the square of $q$, where&nbsp;$p$, $q$ are positive. It is known that&nbsp;$p={{p}_{0}}$ when&nbsp;$q={{q}_{0}}$.  Find $q$ in terms of ${{q}_{0}}$, when&nbsp;$2p=3{{p}_{0}}$.	<span class="inline-marks">[3]</span></p>

$p$ is inversely proportional to the square of $q$, where $p$, $q$ are positive. It is known that $p={{p}_{0}}$ when $q={{q}_{0}}$.  Find $q$ in terms of ${{q}_{0}}$, when $2p=3{{p}_{0}}$.	

Inverse Proportion

IP2 Class Test (May 2025) Q8

Direct and Inverse Proportion

<p>The illumination, $I$ units, of a bulb varies inversely as the square of the distance, $d$ metres.</p>
<p>Given that the illumination is $9$ units when the distance is $2$ m,</p>
<ol class="abc-list"><li data-list-item-id="item-1764484479368-7669o5ydr" data-list-item-path="item-1764484479368-7669o5ydr"><div class="list-item-paragraph">express $I$ in terms of $d$,</div></li><li data-list-item-id="item-1764484480283-hxkq8v238" data-list-item-path="item-1764484480283-hxkq8v238"><div class="list-item-paragraph">find the percentage change in illumination when the distance from the bulb is reduced to $\frac{1}{5}$ of its original value.</div></li></ol>

The illumination, $I$ units, of a bulb varies inversely as the square of the distance, $d$ metres.

Given that the illumination is $9$ units when the distance is $2$ m,

Inverse Proportion: Illumination and Distance Squared

<p>The brightness of an object, $b$ $\mathrm{W}/\mathrm{m}^2$ with respect to a light source is inversely proportional to the square of the distance, $d$ metres, from the light source.</p>
<ol class="abc-list"><li data-list-item-id="item-1764484158746-svp89rc0u" data-list-item-path="item-1764484158746-svp89rc0u"><div class="list-item-paragraph">Write down a formula connecting $b,d$ and a constant $k$.</div></li><li data-list-item-id="item-1764484159766-l2gqb9f7v" data-list-item-path="item-1764484159766-l2gqb9f7v"><div class="list-item-paragraph">When the object is $5$ km from the light source, the brightness is $2$ W/m$^{2}$. Find the brightness when the object is $10$ km from the light source.</div></li><li data-list-item-id="item-1764484160523-cqnnflvu8" data-list-item-path="item-1764484160523-cqnnflvu8"><div class="list-item-paragraph">When the object is a certain distance from the light source, the brightness is $x$ W/m$^{2}$.  What is the brightness in terms of $x$ when this distance is halved?</div></li></ol>

The brightness of an object, $b$ $\mathrm{W}/\mathrm{m}^2$ with respect to a light source is inversely proportional to the square of the distance, $d$ metres, from the light source.

Inverse Proportion: Brightness and Distance Squared

<p>The length, $l$, of a rectangle is inversely proportional to its breadth, $b$.  The breadth is increased by $100\%$.  Find the percentage decrease in the length.</p>

The length, $l$, of a rectangle is inversely proportional to its breadth, $b$.  The breadth is increased by $100\%$.  Find the percentage decrease in the length.

Inverse Proportion: Rectangle Length and Breadth

<p>Given that&nbsp;$y$ is inversely proportional to $\sqrt{x-1}$ and that the difference in the values of $y$ is $\frac{1}{4}$ when the values of $x$ are $5$ and $10$, express $y$ in terms of $x$.</p>
<p>Source check: an unsigned difference of $1/4$ permits either sign of the proportionality constant. The positive answer assumes $y(5)-y(10)=1/4$.</p>

Given that $y$ is inversely proportional to $\sqrt{x-1}$ and that the difference in the values of $y$ is $\frac{1}{4}$ when the values of $x$ are $5$ and $10$, express $y$ in terms of $x$.

Source check: an unsigned difference of $1/4$ permits either sign of the proportionality constant. The positive answer assumes $y(5)-y(10)=1/4$.

Inverse Proportion Involving a Square Root Expression

<p>A construction firm pays its worker $\$100$ each per day and each worker works at the same rate. If&nbsp;$15$ workers are able to complete a project in $30$ days,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">calculate the total labour cost of the project,</div></li><li><div class="list-item-paragraph">find how many workers needed to be hired if the project must be completed in $10$ days,</div></li><li><div class="list-item-paragraph">determine whether the total labour cost will be affected as a result of hiring more workers in part <strong>(b)</strong>. Show your working clearly.</div></li></ol>

A construction firm pays its worker $\$100$ each per day and each worker works at the same rate. If $15$ workers are able to complete a project in $30$ days,

Labour Cost and Work Rate Problem

<p>$p$ is inversely proportional to the cube of $q$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that when $p=5,q=2$, express $p$ in an equation in terms of $q$.</div></li><li><div class="list-item-paragraph">It is known that $p=\theta $ for a particular value of $q$.  Find the value of $p$ in terms of $\theta $, when this value of $q$ is halved.</div></li></ol>

$p$ is inversely proportional to the cube of $q$.

Inverse Proportion Involving the Cube of $𝑞$

<p>Two quantities, $\left( x+1 \right)$ and $(y-2)$ are in inverse proportion.  Values of $x$ and $y$ are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$x$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$b$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$y$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$a$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation connecting $\left( x+1 \right)$ and $(y-2)$.</div></li><li><div class="list-item-paragraph">Hence, find the values of $a$ and $b$.</div></li></ol>

Two quantities, $\left( x+1 \right)$ and $(y-2)$ are in inverse proportion.  Values of $x$ and $y$ are shown in the table below.

Inverse Variation Involving Linear Expressions

<p>$y$ is inversely proportional to ${{x}^{2}}+1$.  It is given that&nbsp;$y=5$ when&nbsp;$x=4$.  Find the values of $x$ when&nbsp;$y=4$.</p>

$y$ is inversely proportional to ${{x}^{2}}+1$.  It is given that $y=5$ when $x=4$.  Find the values of $x$ when $y=4$.

Inverse Proportion Problem with Squared Terms

<p>The value of a car, $\$C$, is inversely proportional to its age, $Y$ years.  The value of a car after 3 years is $\$54\,000$.</p>
<ol class="abc-list"><li data-list-item-id="item-1764481999981-u1r059j2x" data-list-item-path="item-1764481999981-u1r059j2x"><div class="list-item-paragraph">Write down an equation connecting $C$ and $Y$.</div></li><li data-list-item-id="item-1764482001316-l2ev645ma" data-list-item-path="item-1764482001316-l2ev645ma"><div class="list-item-paragraph">Find the value of the car after 5 years.</div></li><li data-list-item-id="item-1764482001834-ubwj6o21e" data-list-item-path="item-1764482001834-ubwj6o21e"><div class="list-item-paragraph">In how many years should the owner sell the car before the car is worth less than 	$\$20\,000$.</div></li></ol>
<p>For part (c), distinguish the age of the car from additional years from now. The model gives an age limit of $8.1$ years; a current age is needed to calculate an additional waiting time.</p>

The value of a car, $\$C$, is inversely proportional to its age, $Y$ years.  The value of a car after 3 years is $\$54\,000$.

For part (c), distinguish the age of the car from additional years from now. The model gives an age limit of $8.1$ years; a current age is needed to calculate an additional waiting time.

Inverse Proportion: Car Value and Age

<p>The table below shows the corresponding value of $P$ and $Q$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$P$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0.3$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$\frac{1}{2}$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$b$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$Q$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$P$ and $Q$ are in inverse proportion.</div></li><li><div class="list-item-paragraph">Find the value of $b$.</div></li></ol>

The table below shows the corresponding value of $P$ and $Q$.

Inverse Proportion Between $𝑃$ and $𝑄$

<p>$12$ men can build a house in $32$ days. Find the time taken for&nbsp;$6$ men to build the same house. State the assumption made.</p>

$12$ men can build a house in $32$ days. Find the time taken for $6$ men to build the same house. State the assumption made.

Inverse Proportion: Workers and Time

<p>$x$ is inversely proportional to the cube of $y$.  It is known that&nbsp;$x=9$ for a particular value of $y$.  When the value of $y$ is increased by $100\%$, find</p>
<ol class="abc-list"><li data-list-item-id="item-1764481118980-h1lz333sh" data-list-item-path="item-1764481118980-h1lz333sh"><div class="list-item-paragraph">the value of $x$.</div></li><li data-list-item-id="item-1764481119535-7aavc3awg" data-list-item-path="item-1764481119535-7aavc3awg"><div class="list-item-paragraph">the percentage decrease in the value of $x$.</div></li></ol>

$x$ is inversely proportional to the cube of $y$.  It is known that $x=9$ for a particular value of $y$.  When the value of $y$ is increased by $100\%$, find

Application of Inverse Proportion

<p>Given that&nbsp;$y$ is inversely proportional to $x+3$ and $y=4$ when&nbsp;$x=3$, find the value of $y$ when&nbsp;$x=8$.</p>

Given that $y$ is inversely proportional to $x+3$ and $y=4$ when $x=3$, find the value of $y$ when $x=8$.

Finding Equation of Another Form of Inverse Proportion

<p>Step $1$:</p>
<p>For inverse proportion, equation will be in the form of $\Box = \frac{k}{\Box}$</p>
<p>Step $2$:</p>
<p>Substitute the given values of $A$ and $B$ to find $k$.</p>
<p>Step $3$:</p>
<p>Substitute the value of $k$ into step $1$ equation to form the equation connecting $A$ and $B$.</p>
<p><strong>Example</strong></p>
<p>$A$ is inversely proportional to $B$. Given that&nbsp;$A = 8$ and $B = 36$, find an equation connecting $A$ and $B$.</p>

For inverse proportion, equation will be in the form of $\Box = \frac{k}{\Box}$

Substitute the given values of $A$ and $B$ to find $k$.

Substitute the value of $k$ into step $1$ equation to form the equation connecting $A$ and $B$.

$A$ is inversely proportional to $B$. Given that $A = 8$ and $B = 36$, find an equation connecting $A$ and $B$.

Essential Steps – Equation of Inverse Proportion

<p>The table shows some corresponding values of $x$ and $y$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$x$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$20$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$100$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$y$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1.2$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$x$ and $y$ are in inverse proportion.</div></li><li><div class="list-item-paragraph">Sketch the graph of $y$ against $x$.</div></li></ol>

The table shows some corresponding values of $x$ and $y$.

Proving Two Variables in Inverse Proportion

<p>$6$ identical pipes can fill a pool in $2$ hours. Calculate the time taken for&nbsp;$8$ such pipes to fill up the same pool.</p>

$6$ identical pipes can fill a pool in $2$ hours. Calculate the time taken for $8$ such pipes to fill up the same pool.

Inverse Proportion: Number of Pipes and Time Taken

<p>It takes $4$ workers to paint a house in $1\frac{1}{2}$ days. Assuming the workers paint at the same rate,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">how many days will $3$ workers take to paint the same house?</div></li><li><div class="list-item-paragraph">how many more workers are needed if the house is to be painted in half a day?</div></li><li><div class="list-item-paragraph">how many days will $6$ workers take to paint $5$ such houses?</div></li></ol>

It takes $4$ workers to paint a house in $1\frac{1}{2}$ days. Assuming the workers paint at the same rate,

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