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PSLE

PSLE Foundation Mathematics

<p>Two cylindrical cans of milk powder of the same brand are geometrically similar to each other as shown.</p>
<p>The heights of both cans are $10$ cm and $14$ cm respectively.</p>
<p>Given that the radius of the base of can $A$ is $6$ cm, calculate the radius of the base of can $B$.</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 ratio of the volume of can $A$ to the volume of can $B$.</div></li></ol>
<p>The net weight of can $A$ is $205$ g.</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 net weight of can $B$, correct to the nearest gram.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">If the cost of cans $A$ and $B$ are $\$8.40$ and $\$19.50$ respectively, which can of milk powder is of better value? Explain your answer clearly.</div></li></ol>
<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/1770796810249-cj6kumya4y9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3OTY4MTAyNDktY2o2a3VteWE0eTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.48zGYYqTFs88Uz5rk_3ExxqqT_eTK3zXU8pq3KUrG-g" style="max-width: min(350px, 100%); height: auto; display: block;" alt="Diagram showing two similar cylindrical cans, Can A with height 10 cm and Can B with height 14 cm." class="alignright custom-width" /></figure><div class="image-caption" style="text-align: right;">Diagram showing two similar cylindrical cans, Can A with height 10 cm and Can B with height 14 cm.</div></div>

Two cylindrical cans of milk powder of the same brand are geometrically similar to each other as shown.

The heights of both cans are $10$ cm and $14$ cm respectively.

Given that the radius of the base of can $A$ is $6$ cm, calculate the radius of the base of can $B$.

Find the ratio of the volume of can $A$ to the volume of can $B$.

The net weight of can $A$ is $205$ g.

Find the net weight of can $B$, correct to the nearest gram.

If the cost of cans $A$ and $B$ are $\$8.40$ and $\$19.50$ respectively, which can of milk powder is of better value? Explain your answer clearly.

Volume Ratios of Similar Solids

Similarity

Comparing Unit Prices

IP2U8 Exercise Q11

Congruence and Similarity

Ratio, Rate and Speed

<p>The two containers $A$ and $B$ shown in the diagram are geometrically similar.</p>
<p>The height of one container is $6$ m and the other is $12$ m.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">The top of the larger container has a circumference of $22$ m, find the circumference of the top of the smaller container.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">If the surface area of the smaller container is $12.5$ m$^2$, find the surface area of the bigger container.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">The cost of the fruit juice in the larger container $B$ is $\$32$, find the cost of the fruit juice in the smaller container if both containers are completely filled with fruit juice.</div></li></ol>
<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/1770795829903-0kwn0qi6d1w.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3OTU4Mjk5MDMtMGt3bjBxaTZkMXcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.B2nQhH_Sx2bcKYkDg10C9voP1zNnnsr-ieCQWaLxrq0" style="max-width: min(250px, 100%); height: auto; display: block;" alt="Containers A and B" class="alignright custom-width" /></figure><div class="image-caption" style="text-align: right;">Containers A and B</div></div>
<p>For part (iii), assume that both containers are charged the same price per unit volume of fruit juice.</p>

The two containers $A$ and $B$ shown in the diagram are geometrically similar.

The height of one container is $6$ m and the other is $12$ m.

The top of the larger container has a circumference of $22$ m, find the circumference of the top of the smaller container.

If the surface area of the smaller container is $12.5$ m$^2$, find the surface area of the bigger container.

For part (iii), assume that both containers are charged the same price per unit volume of fruit juice.

Area Ratios of Similar Figures

IP2U8 Exercise Q10

<p><strong>QUESTION $9$</strong></p>
<p>The ratio of the areas of the bases of two geometrically similar buckets is $4 : 9$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The area of the top of the smaller bucket is $480$ cm$^2$. What is the area of the top of the larger bucket?</span></div></li><li><div class="list-item-content"><span class="list-item-text">Write down the ratio of the heights of the two buckets.</span></div></li><li><div class="list-item-paragraph">Both buckets are filled with water. The mass of the water in the larger bucket is $36$ kg. Find the mass of water in the smaller bucket.</div><div class="content-block list-item-image" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770795285717-cwsn2aoskx.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3OTUyODU3MTctY3dzbjJhb3NreC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTEwODc3LCJleHAiOjE3OTM1MTExNzd9.H2FF667TiNsD0VnycxGZvFRtVxCAQaG9zDzfW0yFhTk" alt="List item image" class="alignright custom-width" style="max-width: min(250px, 100%); height: auto;" /></div></li></ol>

The ratio of the areas of the bases of two geometrically similar buckets is $4 : 9$.

The area of the top of the smaller bucket is $480$ cm$^2$. What is the area of the top of the larger bucket?

Write down the ratio of the heights of the two buckets.

Both buckets are filled with water. The mass of the water in the larger bucket is $36$ kg. Find the mass of water in the smaller bucket.

IP2U8 Exercise Q9

<p>A container in the shape of a circular cone is filled with water to a height of $20$ cm.</p>
<p>It is given that the height of the container is $30$ cm and the diameter of the circular top is $20$ cm.</p>
<p>Show that the amount of water in the container is $\frac{8}{27} \pi$ litres.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>[$1$ litre $= 1000$ cm$^3$]</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1770794943413-5zvq0zs8poo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3OTQ5NDM0MTMtNXp2cTB6czhwb28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.no3kPoeDPZewRC57bPzq1vRG8ppSTVtPKMOHCNVrmBA" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

A container in the shape of a circular cone is filled with water to a height of $20$ cm.

It is given that the height of the container is $30$ cm and the diameter of the circular top is $20$ cm.

Show that the amount of water in the container is $\frac{8}{27} \pi$ litres.

Volume and Surface Area of Cones

IP2U8 Exercise Q8

Mensuration (Pyramid, Cones and Spheres)

<p>The two containers shown in the diagram are geometrically similar.</p>
<p>Their surface areas are $400$ $\text{cm}^2$ and $900$ $\text{cm}^2$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The height of the larger container is $24$ $\text{cm}$. Calculate the height of the smaller container.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the ratio of volume of the small container to the big container.</span></div></li></ol>
<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/1773665091793-y2wtsvzul7r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzM2NjUwOTE3OTMteTJ3dHN2enVsN3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.DpJyxnla3wtV7EcCxa2g-MvNy2E1uSjaedNnFsJ0O-8" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The two containers shown in the diagram are geometrically similar.

Their surface areas are $400$ $\text{cm}^2$ and $900$ $\text{cm}^2$ respectively.

The height of the larger container is $24$ $\text{cm}$. Calculate the height of the smaller container.

Find the ratio of volume of the small container to the big container.

IP2U8 Exercise Q7

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The lengths of sides of three squares are in the ratio of $1 : 3 : 7$. Find</span></div><ol class="abc-list"><li><div class="list-item-paragraph">the ratio of the perimeters of the three squares,</div></li><li><div class="list-item-paragraph">the ratio of the areas of the three squares.</div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The radii of three circles are in the ratio of $2 : 5 : 9$. Find</span></div><ol class="abc-list"><li><div class="list-item-paragraph">the ratio of the perimeters of the three circles,</div></li><li><div class="list-item-paragraph">the ratio of the areas of the three circles.</div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The lengths of sides of three cubes are in the ratio of $1 : 2 : 3$. Find</span></div><ol class="abc-list"><li><div class="list-item-paragraph">the ratio of the surface areas of the three cubes,</div></li><li><div class="list-item-paragraph">the ratio of the volumes of the three cubes.</div></li></ol></li></ol>

The lengths of sides of three squares are in the ratio of $1 : 3 : 7$. Find

The radii of three circles are in the ratio of $2 : 5 : 9$. Find

The lengths of sides of three cubes are in the ratio of $1 : 2 : 3$. Find

Ratio

<p>The diagram shows a solid cone that is cut into two sections, $P$ and $Q$, such that section $P$ is a cone similar to the original cone.<br>The curved surface area of cone $P$ is $90$ cm${}^2$, and the curved surface area of the original cone is $160$ cm${}^2$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">If the height of cone $P$ is $15$ cm, calculate the height, $x$ cm, of the original cone.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the ratio of the volume of the original cone to the volume of cone $P$.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Given that the mass of cone $P$ is $500$ g, find the mass of section $Q$, correcting your answer to $3$ significant figures.</div></li></ol>
<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/1767513098523-k9fpme82um.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1MTMwOTg1MjMtazlmcG1lODJ1bS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTEwODc3LCJleHAiOjE3OTM1MTExNzd9.QSS7Ycw7Ybu6skkf5So6C0lVWiO0FR2H3CdW__kwEkc" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>
<p>For the mass calculation, assume that the original cone has uniform density, so both sections have the same density.</p>

The diagram shows a solid cone that is cut into two sections, $P$ and $Q$, such that section $P$ is a cone similar to the original cone.
The curved surface area of cone $P$ is $90$ cm${}^2$, and the curved surface area of the original cone is $160$ cm${}^2$.

If the height of cone $P$ is $15$ cm, calculate the height, $x$ cm, of the original cone.

Find the ratio of the volume of the original cone to the volume of cone $P$.

Given that the mass of cone $P$ is $500$ g, find the mass of section $Q$, correcting your answer to $3$ significant figures.

For the mass calculation, assume that the original cone has uniform density, so both sections have the same density.

Geometric Similarity: Length, Area, and Volume Ratios

<p>The diagram shows two geometrically similar pots with a circular base.<br>It is given that the height of pot $A$ is $30$ cm and the base area of pot $B$ is $\frac{4}{9}$ of the base area of pot $A$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">Calculate the height of pot $B$.</div></li><li><div class="list-item-paragraph">The curved surface area of pot $B$ is $260$ cm${}^2$. <br>Find the curved surface area of pot $A$.</div></li><li><div class="list-item-paragraph">If the capacity of pot $A$ is $5.4$ litres, find the capacity of pot $B$, giving your answer in cm${}^3$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767512454856-gr1hlp05omq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1MTI0NTQ4NTYtZ3IxaGxwMDVvbXEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.DfwjjSaOJ2AnVoQuTbjYrt1e6NkxVzphc8UCGG0wg3g" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram shows two geometrically similar pots with a circular base.
It is given that the height of pot $A$ is $30$ cm and the base area of pot $B$ is $\frac{4}{9}$ of the base area of pot $A$.

Ratios of Similar Solids: From Area to Volume

<p>Two ball bearings made from the same material have masses 64 g and 729 g respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the ratio of their volumes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the ratio of their radii.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the surface area of the smaller ball bearing is 24 mm², find the surface area of the larger ball bearing.</span></div></li></ol>

Two ball bearings made from the same material have masses 64 g and 729 g respectively.

Given that the surface area of the smaller ball bearing is 24 mm², find the surface area of the larger ball bearing.

<p>Two similar cones have base areas $144\text{ mm}^2$ and $324\text{ mm}^2$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the ratio of their heights.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the ratio of their volumes.</span></div></li></ol>

Two similar cones have base areas $144\text{ mm}^2$ and $324\text{ mm}^2$ respectively.

<p>The diagram shows three geometrically similar flasks $A$, $B$ and $C$ with a circular base of diameter $4$ cm, $x$ cm and $2.5$ cm respectively.</p>
<p>The heights of flasks $A$ and $B$ are $20$ cm and $18$ cm respectively.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find</p><ol class="i-list"><li><div class="list-item-paragraph">the diameter, $x$ cm, of the base of flask $B$ ,</div></li><li><div class="list-item-paragraph">$\frac{\text{surface area of flask}\hspace{0.5em} B}{\text{surface area of flask}\hspace{0.5em} A}$ ,</div></li><li><div class="list-item-paragraph">$\frac{\text{volume of flask}\hspace{0.5em} C}{\text{volume of flask}\hspace{0.5em} A}$ .</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767506638516-nit41tbc71.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1MDY2Mzg1MTYtbml0NDF0YmM3MS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTEwODc3LCJleHAiOjE3OTM1MTExNzd9.2eTc_kVsaDldyzMBcyTd8rY8c8naf163CagSjQwtN_U" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram shows three geometrically similar flasks $A$, $B$ and $C$ with a circular base of diameter $4$ cm, $x$ cm and $2.5$ cm respectively.

The heights of flasks $A$ and $B$ are $20$ cm and $18$ cm respectively.

Problem involving Similar Solids

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Two cylindrical drums are geometrically similar.<br>The height of the smaller drum is $5$ cm.<br>The height of the larger drum is $15$ cm.</p><ol class="i-list"><li><div class="list-item-paragraph">The diameter of the base of the larger drum is $7$ cm.<br>Find the diameter of the smaller drum.</div></li><li><div class="list-item-paragraph">Find the ratio of the volume of the smaller drum to that of the larger.<br>Give your answer in the form $1:n$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767506330745-yk2ocgmva7b.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1MDYzMzA3NDUteWsyb2NnbXZhN2IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUxMDg3NywiZXhwIjoxNzkzNTExMTc3fQ.y8kD8YcbYeWqX6nrYun3gFDyX4NAACoUqNnDlMYENPk" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

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