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<p>Three companies, A, B, and C, cooperate with a technological university to develop a project. The investment ratio of companies A, B, and C is $a : b : c$ in the simplest form. The university does not invest money but will share $25%$ of the profits. The rest of the profits will be shared by companies A, B and C according to the ratio of their investments.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The pie chart below shows the investment of the three companies. Find the values of $a$, $b$ and $c$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-22-i-diagram-1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yMi1pLWRpYWdyYW0tMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNzA5LCJleHAiOjE3OTI2NjMwMDl9.GxLPkXvNFVsI0OxCpsXNBQxidtEhDb9anLJj0E2rPl0" alt="List item image" class="aligncenter custom-width" style="max-width: min(250px, 100%); height: auto;" /></div></li></ol>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Construct a pie chart using the figure below to illustrate the profit distribution among companies A, B, C, and the university, stating clearly the degree of each sector, profit distribution in percentage and the name of each entity.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-22-ii-diagram-1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yMi1paS1kaWFncmFtLTEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjcwOSwiZXhwIjoxNzkyNjYzMDA5fQ.uZNwB-kBhzCBPOVPQ_I-XqhmC8HVrRxmiME05ljcB3E" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">Given that the project yields a total profit of $\$2\ 800\ 000$, find the amount of profit that companies A, B, C and the university receive, respectively.</div></li><li><div class="list-item-paragraph">Given that the profit rate of companies A, B and C is $40%$, find the investment of companies A, B and C, respectively.</div></li><li><div class="list-item-content"><span class="list-item-text">State one advantage of representing the investment of the companies and profit distribution in pie charts.</span></div></li></ol>

Three companies, A, B, and C, cooperate with a technological university to develop a project. The investment ratio of companies A, B, and C is $a : b : c$ in the simplest form. The university does not invest money but will share $25%$ of the profits. The rest of the profits will be shared by companies A, B and C according to the ratio of their investments.

The pie chart below shows the investment of the three companies. Find the values of $a$, $b$ and $c$.

Construct a pie chart using the figure below to illustrate the profit distribution among companies A, B, C, and the university, stating clearly the degree of each sector, profit distribution in percentage and the name of each entity.

State one advantage of representing the investment of the companies and profit distribution in pie charts.

Dividing Quantities in a Given Ratio

Percentages involving Finances: Profit and Loss

Interpreting and Evaluating Statistical Representations

Pie Charts

Simplifying Ratios

Statistics

Percentages

Ratio, Rate and Speed

<p>The pie chart below shows the composition of types of products that a factory produces, where the ratio of the number of Type A to the number of Type B to the number of Type E produced is $2:3:6$.</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/1767697388127-ghr4a24gwdo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTczODgxMjctZ2hyNGEyNGd3ZG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjcwOSwiZXhwIjoxNzkyNjYzMDA5fQ.6YGCRZK6syHJjvlCPeeew7JDqqLkd7ABO_iTq-uW8hA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the ratio of the number of Type C to the number of Type D produced.</span></div></li><li><div class="list-item-paragraph">Given that the factory produces $500$ Type D products, find the total number of all $5$ types of products that the factory produces.</div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the number of Types A, B, C and E products that the factory produces, respectively.</span></div></li></ol>

The pie chart below shows the composition of types of products that a factory produces, where the ratio of the number of Type A to the number of Type B to the number of Type E produced is $2:3:6$.

Find the ratio of the number of Type C to the number of Type D produced.

Hence find the number of Types A, B, C and E products that the factory produces, respectively.

<p>In this question, the average tangential speed along the circular path is being calculated by the following formula:\[\text{Average tangential speed} = \frac{\text{Total circular distance travelled}}{\text{Total time taken}}\]The Singapore Flyer is one of the world’s most famous Ferris wheels, which has a radius of $75$ m.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Given that it takes $30$ minutes for the Singapore Flyer to finish one full rotation, find the average tangential speed of the Singapore Flyer in one full rotation in m/s.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the average speed of ascending in vertical direction, for a capsule of the Singapore Flyer to travel from the lowest point to the highest point.</div></li></ol>
<p>The London Eye is another famous Ferris wheel in the world, which also finishes one full rotation in $30$ minutes.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Given that the ratio of the radius of the Singapore Flyer to the radius of the London Eye is $5 : 4$, find the average tangential speed of the London Eye in one full rotation in m/s.</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Using your answer in (iii), find the average speed of descending in vertical direction, for a capsule of the London Eye to travel from the highest point to the lowest point.</div></li></ol>
<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/1767674389740-uhh3znrfqv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2NzQzODk3NDAtdWhoM3pucmZxdi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNzA5LCJleHAiOjE3OTI2NjMwMDl9.SeIsB779PObzO7C4rMGPT_AddtEgtY069howI0BhFBE" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Use the stated dimensions as a mathematical model and assume both wheels rotate uniformly. Then a half-turn takes half the full-turn time.</p>

In this question, the average tangential speed along the circular path is being calculated by the following formula:\[\text{Average tangential speed} = \frac{\text{Total circular distance travelled}}{\text{Total time taken}}\]The Singapore Flyer is one of the world’s most famous Ferris wheels, which has a radius of $75$ m.

Given that it takes $30$ minutes for the Singapore Flyer to finish one full rotation, find the average tangential speed of the Singapore Flyer in one full rotation in m/s.

Find the average speed of ascending in vertical direction, for a capsule of the Singapore Flyer to travel from the lowest point to the highest point.

The London Eye is another famous Ferris wheel in the world, which also finishes one full rotation in $30$ minutes.

Given that the ratio of the radius of the Singapore Flyer to the radius of the London Eye is $5 : 4$, find the average tangential speed of the London Eye in one full rotation in m/s.

Using your answer in (iii), find the average speed of descending in vertical direction, for a capsule of the London Eye to travel from the highest point to the lowest point.

Use the stated dimensions as a mathematical model and assume both wheels rotate uniformly. Then a half-turn takes half the full-turn time.

Speed, Distance and Time

Circumference and Area of Circles

Speed

Perimeter and Area of Plane Figures

<p>Ali, David and Priya agree to invest $\$315\,000$, $\$420\,000$ and $\$285\,000$ respectively into a project, and they cooperate with a technological university to develop their project. The university does not invest money into the project but will share $20\%$ of the profit of this project. The rest of the profit will be shared by Ali, David and Priya according to the ratio of their investments.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the ratio of Ali, David and Priya’s investments in the simplest form.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the fraction of total profit that Ali, David and Priya share, respectively.</span></div></li></ol>
<p>Suppose the project yields a total profit of $\$2\,550\,000$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the amount of profit that Ali, David, Priya and the university received, respectively.</span></div></li></ol>

Ali, David and Priya agree to invest $\$315\,000$, $\$420\,000$ and $\$285\,000$ respectively into a project, and they cooperate with a technological university to develop their project. The university does not invest money into the project but will share $20\%$ of the profit of this project. The rest of the profit will be shared by Ali, David and Priya according to the ratio of their investments.

Find the ratio of Ali, David and Priya’s investments in the simplest form.

Find the fraction of total profit that Ali, David and Priya share, respectively.

Suppose the project yields a total profit of $\$2\,550\,000$.

Find the amount of profit that Ali, David, Priya and the university received, respectively.

Percentage of a Quantity

Ratio

<p>There are $600$ girls in a school, and the ratio of the number of girls to the number of boys is $1 : 0.96$. Find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the number of boys in this school,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the ratio of the number of girls to the total number of students in this school.</span></div></li></ol>

There are $600$ girls in a school, and the ratio of the number of girls to the number of boys is $1 : 0.96$. Find

the ratio of the number of girls to the total number of students in this school.

<p>A bottle of beverage is mixed by syrup, apple juice and water in the ratio $1 : 2 : 5$. Given that the net volume of one bottle of such beverage is $500\text{ml}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the volume of syrup, apple juice and water that one bottle of such beverage contains, respectively.</span></div></li><li><div class="list-item-content"><span class="list-item-text">A bottle of such beverage is a half-full. If it is re-filled fully with pure water, find the new ratio of the volume of syrup, apple juice and water.</span></div></li></ol>

A bottle of beverage is mixed by syrup, apple juice and water in the ratio $1 : 2 : 5$. Given that the net volume of one bottle of such beverage is $500\text{ml}$.

Find the volume of syrup, apple juice and water that one bottle of such beverage contains, respectively.

A bottle of such beverage is a half-full. If it is re-filled fully with pure water, find the new ratio of the volume of syrup, apple juice and water.

Changing Ratios

<p>Danny and Eric started an online business. Danny invested $\$60\,000$ and Eric invested $\$24\,000$.  They agreed that all the profits should be divided in the same ratio as the amounts of money they have invested. The profit was $\$14\,700$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write the ratio $60\,000:24\,000$ in its simplest form.</div></li><li><div class="list-item-paragraph">Calculate Eric’s share of the profit.</div></li><li><div class="list-item-paragraph">How much more profit does Danny get?</div></li></ol>

Danny and Eric started an online business. Danny invested $\$60\,000$ and Eric invested $\$24\,000$.  They agreed that all the profits should be divided in the same ratio as the amounts of money they have invested. The profit was $\$14\,700$.

Ratio Word Problem: Sharing Profits

<p>A cake was made using the ingredients - eggs, butter and flour in the ratio $2:3:5$.<br>$900$ grams of butter was used in the making of the cake.<br>Calculate the</p>
<ol class="abc-list"><li><div class="list-item-paragraph">total weight of the ingredients in grams.</div></li><li><div class="list-item-paragraph">weight of flour used in grams.</div></li><li><div class="list-item-paragraph">new ratio of eggs, butter and flour if 100 more grams of flour was being used to make the cake.</div></li></ol>

A cake was made using the ingredients - eggs, butter and flour in the ratio $2:3:5$.
$900$ grams of butter was used in the making of the cake.
Calculate the

Ratio Word Problem: Ingredients in a Recipe

<p>The number of marbles Alex, Ben and Charles have is in the ratio $3:x:5$.</p>
<p>Charles has&nbsp;$32$ marbles more than Alex, while Ben had $16$ marbles less than Charles.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the number of marble Charles has.</div></li><li><div class="list-item-paragraph">Calculate the value of $x$.</div></li></ol>

The number of marbles Alex, Ben and Charles have is in the ratio $3:x:5$.

Charles has $32$ marbles more than Alex, while Ben had $16$ marbles less than Charles.

Ratio Word Problem: Using Ratios and Differences

<p>In a class, $\frac{1}{4}$ of the boys and $\frac{1}{3}$ of the girls take Spanish as a third language.</p>
<p>The ratio of the number of boys to the number of girls is $2:3$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">What is the ratio of the number of boys who do not take Spanish as a third language to the number of girls who do not take Spanish as a third language?</div></li><li><div class="list-item-paragraph">Given that there are $28$ students in the class who do not take Spanish as a third language, find the number of students in the class.</div></li></ol>

In a class, $\frac{1}{4}$ of the boys and $\frac{1}{3}$ of the girls take Spanish as a third language.

The ratio of the number of boys to the number of girls is $2:3$.

Applications of Ratio

<p>A liquid $X$ consists of ingredients $A$ and $B$ in the ratio $4:3$ by volume. Another liquid $Y$ consists of ingredients $C$ and $B$ in the ratio $11:10$ by volume.</p>
<ol class="abc-list"><li data-list-item-id="item-1760834287180-5cbgk042h" data-list-item-path="item-1760834287180-5cbgk042h"><div class="list-item-paragraph">Find the volume of ingredient $A$ required in order to have&nbsp;$420$ ml of liquid $X$.</div></li><li data-list-item-id="item-1760834288267-lfhagbm83" data-list-item-path="item-1760834288267-lfhagbm83"><div class="list-item-paragraph">The volumes of ingredient $B$ in liquids $X$ and $Y$ are equal. Find the ratio of the volume of $A$ in $X$, this common volume of $B$, and the volume of $C$ in $Y$. The liquids are not combined.</div></li></ol>

A liquid $X$ consists of ingredients $A$ and $B$ in the ratio $4:3$ by volume. Another liquid $Y$ consists of ingredients $C$ and $B$ in the ratio $11:10$ by volume.

Combining Ratios

IP1U9 HOT Q1

<p>Ali, David and Priya agree to invest $\$315\,000$, $\$420\,000$ and $\$285\,000$ respectively into a project, and they cooperate with a technological university to develop their project. The university does not invest money into the project but will share $20\%$ of the profit of this project. The rest of the profit will be shared by Ali, David and Priya according to the ratio of their investments.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the ratio of Ali, David and Priya’s investments in the simplest form.</div></li><li><div class="list-item-paragraph">Find the fraction of total profit that Ali, David and Priya share, respectively.</div></li></ol>
<p>Suppose the project yields a total profit of $\$2\,550\,000$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the amount of profit that Ali, David, Priya and the university received, respectively.</div></li></ol>

IP1U9 Exercise Q2

<p>Lynn left her house at ${8.20\ \mathrm{am}}$ and drove from Town ${A}$ to ${B}$.</p>
<p>Along the way, she stopped at a café for ${15}$ minutes to buy coffee.</p>
<p>She drove off to Town ${B}$ from the café at a speed of ${60\ \mathrm{km/h}}$.</p>
<p>The ratio of the times she spent on the first part of her journey, buying coffee and the final part of the journey was ${2:1:3}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Convert ${60\ \mathrm{km/h}}$ into ${\mathrm{m/s}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">What time did Lynn reach Town ${B}$?</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">If the distance between Town ${A}$ and the café is ${34\ \mathrm{km}}$, find the</div><div class="list-item-paragraph">average speed of the whole journey in kilometres per hour.</div><p style="text-align: right;">[3]</p></li></ol>

Lynn left her house at ${8.20\ \mathrm{am}}$ and drove from Town ${A}$ to ${B}$.

Along the way, she stopped at a café for ${15}$ minutes to buy coffee.

She drove off to Town ${B}$ from the café at a speed of ${60\ \mathrm{km/h}}$.

The ratio of the times she spent on the first part of her journey, buying coffee and the final part of the journey was ${2:1:3}$.

Converting Units of Rates and Speed

Average Speed over a Journey

2023 AISS EM S1 EOY P1 Q14

<p>The ratio of ${x:y}$ is ${1\frac{1}{2} : 2\frac{2}{5}}$ and ${x:z}$ is ${3:4}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express the ratio of ${x:y}$ in the simplest form.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the ratio of ${x:y:z}$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Alan, Ben and Caius shared a sum of money in the ratio found in part <strong>(b)</strong> respectively. If Caius has $\$60$ more than Alan, find the total amount of money the three of them have.</div><p style="text-align: right;">[2]</p></li></ol>

The ratio of ${x:y}$ is ${1\frac{1}{2} : 2\frac{2}{5}}$ and ${x:z}$ is ${3:4}$.

Math Question Bank