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<p>Ashlyn, Hayden and Kahlan took a multiple choice test which consists of $40$ questions.</p>
<p>Kahlan attempted all the questions.</p>
<p>The table shows the results of the test and the marks awarded.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center">Correct</th><th style="text-align: center; --cell-justify: center">Incorrect</th><th style="text-align: center; --cell-justify: center">Not attempted</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Ashlyn</td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$4$</td></tr><tr><td style="text-align: center; --cell-justify: center">Hayden</td><td style="text-align: center; --cell-justify: center">$32$</td><td style="text-align: center; --cell-justify: center">$7$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">Kahlan</td><td style="text-align: center; --cell-justify: center">$x$</td><td style="text-align: center; --cell-justify: center">$40 - x$</td><td style="text-align: center; --cell-justify: center">$0$</td></tr><tr><td style="text-align: center; --cell-justify: center">Marks</td><td style="text-align: center; --cell-justify: center">$2$</td><td style="text-align: center; --cell-justify: center">$-1$</td><td style="text-align: center; --cell-justify: center">$0$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Evaluate $\begin{pmatrix} 30 & 6 & 4 \\ 32 & 7 & 1 \\ x & 40 - x & 0 \end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$.</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Explain what your answer to <strong>(a)</strong> represents.</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">If Kahlan has scored the highest marks, using your answer in part <strong>(a)</strong>, find the least number of correct answers she has scored.</span></div></li></ol>

Ashlyn, Hayden and Kahlan took a multiple choice test which consists of $40$ questions.

The table shows the results of the test and the marks awarded.

Evaluate $\begin{pmatrix} 30 & 6 & 4 \\ 32 & 7 & 1 \\ x & 40 - x & 0 \end{pmatrix} \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$.

Explain what your answer to **(a)** represents.

If Kahlan has scored the highest marks, using your answer in part **(a)**, find the least number of correct answers she has scored.

Application of Matrices

Problem Solving Involving Inequalities

Matrices

Linear Equations and Inequalities

<p>A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.</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/1767694925729-e3sgrgfwk9f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MTc0NzE1OCwiZXhwIjoxNzkxNzQ3NDU4fQ.IzDJd77gJKTVZhcMLZl6o7JRnPgM9CucNDpDeDzCaT4" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the volume of water contained in the cuboid tank.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that the maximum number of cylinder cans that can be completely filled is $8$, find the integer value of $h$. Hence, find the volume of remaining water in the tank.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.</div></li></ol>
<p>Part (b) does not specify a unique integer height. The full condition allows $h=31,32,33$ or $34\text{ cm}$; the solution lists the corresponding remainders. A request for the greatest possible integer height would select $34\text{ cm}$.</p>

A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.

Find the volume of water contained in the cuboid tank.

Given that $h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.

Part (b) does not specify a unique integer height. The full condition allows $h=31,32,33$ or $34\text{ cm}$; the solution lists the corresponding remainders. A request for the greatest possible integer height would select $34\text{ cm}$.

Volume and Surface Area of Cylinders

Volume and Surface Area of Prisms

Conservation of Volume and Displacement

Volume and Surface Area of Cylinder

Mensuration (Pyramid, Cones and Spheres)

<p>To receive a grade B for a course, a candidate must have a mean score of at least $80$ but less than $90$ in three examination papers.  </p>
<p>The full score in each paper is $100$.</p>
<p>A student scored $83$ and $75$ in the first two papers.  </p>
<p>What range of score should this student have in the third paper in order to get a grade B for the course?</p>

To receive a grade B for a course, a candidate must have a mean score of at least $80$ but less than $90$ in three examination papers.  

A student scored $83$ and $75$ in the first two papers.  

What range of score should this student have in the third paper in order to get a grade B for the course?

Forming and Solving a Compound Inequality from a Grade Requirement

<p>A newspaper company charges $25$ cents per word and a fixed charge of $80$ cents for and advertisement.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the total cost of an advertisement containing $15$ words.</div><div class="list-item-paragraph">Leave your answer in dollars.</div></li><li><div class="list-item-paragraph">The total cost of an advertisement containing $n$ words is $C$ cents.  </div><div class="list-item-paragraph">Find a formula for&nbsp;$C$ in terms of $n$.</div></li><li><div class="list-item-paragraph">Arthur does not want to spend more than $\$5.00$ on an advertisement.</div><div class="list-item-paragraph">What is the greatest number of words he can use?</div></li></ol>

A newspaper company charges $25$ cents per word and a fixed charge of $80$ cents for and advertisement.

Forming and Solving an Inequality from a Word Count Pricing Scenario

<p>Company $X$ quotes a car rental rate of $\$65$ per day.</p>
<p>Company $Y$ quotes a rate of $\$56$ per day with an additional initial charge of $\$86.50$.  </p>
<p>Let&nbsp;$x$ be the number of days to rent a car.</p>
<p>Form an inequality in $x$ and use it to find the minimum number of complete days required such that charges of Company $Y$ will become cheaper than Company $X$.</p>

Company $X$ quotes a car rental rate of $\$65$ per day.

Company $Y$ quotes a rate of $\$56$ per day with an additional initial charge of $\$86.50$.  

Let $x$ be the number of days to rent a car.

Form an inequality in $x$ and use it to find the minimum number of complete days required such that charges of Company $Y$ will become cheaper than Company $X$.

Comparing Rental Costs Using Inequalities: Finding the Minimum Number of Days

<p>The table below shows the cost of desks and chairs from a Furniture Shop.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 300px;"><colgroup><col style="width: 200px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>Items</p></th><th style="text-align: center; --cell-justify: center"><p>Cost</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>Student Desk</p></td><td style="text-align: center; --cell-justify: center"><p>$\$30$ per item</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>Student Chair</p></td><td style="text-align: center; --cell-justify: center"><p>$\$25$ per item</p></td></tr></tbody></table></div></div>
<p>A tuition centre would like to purchase $20$ desks and as many chairs as possible given a budget of $\$1,200$.  </p>
<p>There is a delivery charge of $\$30$.</p>
<p>Using an inequality, calculate the maximum number of chairs she can purchase.</p>

The table below shows the cost of desks and chairs from a Furniture Shop.

A tuition centre would like to purchase $20$ desks and as many chairs as possible given a budget of $\$1,200$.  

Using an inequality, calculate the maximum number of chairs she can purchase.

Furniture Purchase Problem: Solving a Budget Inequality

<p>A Telco company charges $\$5$ for the first $10$ minutes of international call and $90$ cents per minute subsequently.</p>
<p>On a particular month, Kenny travelled overseas for work.  </p>
<p>The bill for his international call was not more than $\$20.25$ for the month.</p>
<p>Form an inequality to find the maximum number of minutes Kenny spent on international call for the month.</p>

A Telco company charges $\$5$ for the first $10$ minutes of international call and $90$ cents per minute subsequently.

On a particular month, Kenny travelled overseas for work.  

The bill for his international call was not more than $\$20.25$ for the month.

Form an inequality to find the maximum number of minutes Kenny spent on international call for the month.

Forming an Inequality from a Real-World Cost Scenario Involving Time and Money

<p>John scored $38$, $32$, $49$ and $x$ marks for four Mathematics tests.  </p>
<p>He must score an average of at least $50$ marks if he wants to obtain a pass in Mathematics.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Form an inequality in $x$ for the information above so that John can pass his Mathematics.</div></li><li><div class="list-item-paragraph">Solve the inequality.</div></li><li><div class="list-item-paragraph">Hence, state the minimum mark that John must score for the last Mathematics test in order to pass.</div></li></ol>

John scored $38$, $32$, $49$ and $x$ marks for four Mathematics tests.  

He must score an average of at least $50$ marks if he wants to obtain a pass in Mathematics.

Forming and Solving an Inequality from an Average Score Requirement

<p>Alicia and Betty planned for a party and they decided not to spend more than $\$150$. Alicia spent $\$25$ more than one-fourth of what Betty spent.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Let the amount that Betty spent be $\$x$, form an inequality and do not simplify.</div></li><li><div class="list-item-paragraph">Solve the inequality to find the maximum amount that Betty spent.</div></li></ol>

Alicia and Betty planned for a party and they decided not to spend more than $\$150$. Alicia spent $\$25$ more than one-fourth of what Betty spent.

Forming and Solving Inequalities from Real-World Money Problems

<p>A bowl of noodles costs $\$3.80$, By setting up an inequality, find the maximum number of bowls of noodles that can be bought with $\$50$.  </p>

A bowl of noodles costs $\$3.80$, By setting up an inequality, find the maximum number of bowls of noodles that can be bought with $\$50$.  

Inequality Application: Maximum Number of Items Purchasable

<p>A truck could carry a load $4500$kg. By setting up an inequality, find the least number of truck needed to carry $28\,000$kg of construction materials. <br><br></p>

A truck could carry a load $4500$kg. By setting up an inequality, find the least number of truck needed to carry $28\,000$kg of construction materials. 



Inequality Application: Minimum Number of Trucks Needed

<p>A test has a passing mark of $65$ marks. If number of marks allocated to each question is $2$ marks, by setting up an inequality, find the least number of questions a student must answer correctly to pass the test.  </p>

A test has a passing mark of $65$ marks. If number of marks allocated to each question is $2$ marks, by setting up an inequality, find the least number of questions a student must answer correctly to pass the test.  

Problem involving Simple Inequalities

<p>An egg tart costs $\$1.80$. By setting up an inequality, find the greatest number of egg tarts Mr. Ahmad can buy with $\$45$. </p>

An egg tart costs $\$1.80$. By setting up an inequality, find the greatest number of egg tarts Mr. Ahmad can buy with $\$45$. 

<ol class="i-list"><li><div class="list-item-paragraph">A wooden artwork is designed to take the form of a right pyramid with square base of sides $2x$m and fixed volume $\frac{4}{3}$m$^{3}$. All its exterior faces, including the base, need to be painted. The cost per square metre of painting each of the $4$ inclined faces is twice that of painting the base. Given that the cost of painting the base is $\$p$/m$^{2}$, show that the total cost of painting the artwork is</div><div class="list-item-paragraph" style="text-align: center">$\$\,4p{{x}^{2}}\left(1+2\sqrt{1+{{x}^{-\,6}}}\right)$.</div></li><li><div class="list-item-paragraph">Determine the possible heights the artwork may take if the budget given for the paint works is $\$16p$.</div></li></ol>

Graphical Method

TJC Inequalities Tutorial Q10

Inequalities

<p>The ones digit of a two-digit number is non-zero. The number is more than thrice the number formed if the digits are reversed. How many such numbers are there?</p>

The ones digit of a two-digit number is non-zero. The number is more than thrice the number formed if the digits are reversed. How many such numbers are there?

Number Reversal and Inequality Problem

<p>The Australian Mathematics Competition consists of $30$ questions.</p>
<p>$4$ marks are awarded for a correct answer while $1$ mark is deducted for a wrong answer.</p>
<p>No marks will be awarded or deducted for questions not attempted.</p>
<p>A student skipped $3$ questions and had a score of more than $44$.</p>
<p>Using an inequality, find the minimum number of correct answers obtained.</p>

The Australian Mathematics Competition consists of $30$ questions.

$4$ marks are awarded for a correct answer while $1$ mark is deducted for a wrong answer.

No marks will be awarded or deducted for questions not attempted.

A student skipped $3$ questions and had a score of more than $44$.

Using an inequality, find the minimum number of correct answers obtained.

Math Question Bank