Ratios and similar figures Century Year 11/05_ratios...SIMPLIFYING RATIOS The parts of a ratio are called its terms. A simplified ratio is one where the terms are whole numbers with

RATIOS AND SIMILAR FIGURES

163

55

Ratios andsimilar figures

MEASUREMENT

The invention of the personal computer in the early 1980s was made possible by the development of the microchip, illustrated below. The electrical circuitry needed to operate a computer system could now be housed on a tiny square of silicon measuring 3 mm by 3 mm. Hardware engineers use a scale diagram when designing a microchip to ensure that the many circuits are positioned in the correct places.

While most scale diagrams, like maps and house plans, show a reduction of the real object, the scale diagram of a microchip shows an enlargement. The photo enlargement below shows a microchip 100 times its real size. We say that the ratio of the enlargement to the actual chip is 100 : 1 and that the enlargement (or

image

) and the actual chip (

object

) are similar figures.

In this chapter you will learn how to:

�

simplify ratios

�

use the unitary method to solve ratio problems

�

divide a quantity in a given ratio

�

establish properties of similar figures and find their scale factors

�

enlarge and reduce figures using measurement or a centre of enlargement

�

calculate unknown sides and angles in similar figures

�

use a shadow stick to find the height of tall objects

�

use scales and develop scale drawings of objects and images

�

obtain measurements from, and interpret symbols on, house and building plans

�

calculate lengths and areas from floor plans

�

transfer measurements between floor plans and elevations.

164

NEW CENTURY MATHS GENERAL: PRELIMINARY

SIMPLIFYING RATIOS

The parts of a ratio are called its

terms

.

A

simplified ratio

is one where the terms are whole numbers with no common factor.

To simplify a ratio, we divide or multiply each of its terms by the same number.

Example 1

Express these ratios in simplest form.

(a) 8 : 12 : 28 (b) :

Solution

(a) 8 : 12 : 28

=

2 : 3 : 7 Divide each term by 4.

(b) :

=

×

20 :

×

20 Multiply each term by 20.

=

15 : 8

Example 2

Express 2 hours to 45 minutes as a ratio in simplest form.

Solution

2 hours : 45 minutes

=

150 min : 45 min Change to the same units.

=

150 : 45 Omit the units.

=

10 : 3 Reduce to simplest form.

Example 3

Express the ratio 12 : 5 in the ratio

m

: 1.

Solution

12 : 5

=

: Make the second term equal to 1.

=

: 1 or 2.4 : 1

1.

Express these ratios in simplest form.(a) 5 : 15 (b) 4 : 2 : 10 (c) 14 : 14

(d) (e) (f) 12 : 36 : 9

(g) 0.04 : 0.004 (h) : 1 (i) :

2.

Simplify these ratios.

(a) 250 mL to 2 L (b) 3 days to 4 weeks

(c) 2 thousand dollars : 1 million dollars (d) 45 cm to 3 m

(e) $4.20 : 95c (f) 2 kg : 800 g

(g) cup to 2 cups (h) 1 fortnight : 3 weeks : 1 year

(i) 1 decade to 1 century (j) 2 ha : 100 m

2

34--- 2

5---

34--- 2

5--- 3

4--- 2

5---

12---

12---

125

------ 55---

125

------

Exercise 5-01: Simplifying ratios

10575

--------- 1768------

12--- 4

5--- 2

3---

12--- 1

2---


165

3.

A square has side 5 cm. (a) What is the ratio of the side to the perimeter?(b) Why is it inappropriate to find the ratio of the side to the area?

4.

Joyce’s Discount Store has a midyear sale and offers a 10% discount on the marked price of all goods in the store. What is the ratio of:(a) discount to marked price?(b) discount to sale price?(c) marked price to sale price?

5.

Concrete is made by mixing sand, cement and lime in the ratio 2.5 : 0.5 : 0.25. Express this ratio in its simplest form.

6.

Express these ratios in the form 1 :

m

.(a) 25 : 1000 (b) 150 : 90(c) $2.50 : $15.00 (d) 20 laps to 60 laps

7.

Express these ratios in the form

m

: 1.(a) 48 : 36 (b) 50c to $2.00(c) 1 h to 15 min (d) 81 mm to 27 cm

8.

Half of Robert’s class walk to school, a third ride a bike and the rest travel by train. What is the ratio of:(a) bike riders to walkers?(b) train travellers to the whole class?(c) walkers to train travellers?

9.

Among male smokers, 20 in 90 are likely to develop lung cancer, as are 15 in 130 female smokers. Express these ratios in the form 1 :

n

.

10.

Which of these fertilisers has nutrients in the ratio 4 : 2 : 3?Fertiliser A: 18 : 8 : 2 Fertiliser B: 12 : 3 : 10Fertiliser C: 16 : 8 : 3 Fertiliser D: 8 : 4 : 6

THE UNITARY METHOD

As we saw in Chapter 2, the unitary method considers the value of 1 part and uses this to calculate other parts. The unitary method can be used to solve problems involving ratios.

Example 4

A 2 kg bag of fertiliser contains nitrogen, phosphorus and potassium in the ratio 2 : 3 : 5. How much phosphorus does the fertiliser contain?

Solution

Total parts

=

2

+

3

+

5

=

1010 parts

=

2000 2 kg

=

2000 g.

1 part

=

Divide by 10

=

200 .3 parts

=

200

×

3 Multiply by 3.

=

600

The fertiliser contains 600 g phosphorus.Omit units in the working out and put them in at the end.

200010

------------

166


Example 5

Jeremy and Fiona invested money in a townhouse in the ratio 4 : 7. If Jeremy invested $50 000, how much did the townhouse cost?

Solution

Jeremy’s investment

=

4 parts4 parts

=

50 000

1 part

=

Divide by 4.

11 parts

=

×

11 Multiply by 11.

=

137 500

The townhouse cost $137 500.

1.

Julie cuts a length of ribbon in 3 pieces in the ratio 1 : 3 : 5. If the largest piece is 75 cm long:(a) how long are the other 2 pieces?(b) how long was the length of ribbon before cutting?

2.

Picnic Point High School had an enrolment last year of 1150 pupils. The ratio of boys to girls was 23 : 27. (a) How many boys were enrolled?(b) How many girls were enrolled?

3.

(a) If 3 L of juice are made up of concentrate and water in the ratio 2 : 13, how much concentrate is needed to make 3 L of juice?

(b) If 4.5 L of juice are made with the same ratio of concentrate and water, how much water is needed?

4.

Lynne made new curtains and cut lengths of material in the ratio 2 : 3 : 4. If the smallest length was 2.5 m, how much material did she use altogether?

5.

Sugar is made up of carbon, hydrogen and oxygen in the ratio 12 : 22 : 11. How much carbon, hydrogen and oxygen are contained in a 2 kg bag of sugar? (Answer to the nearest gram.)

6.

The ratio of teachers to students in a TAFE college is 1 : 15. In a college of 1200 students, how many teachers are there?

7.

Heath and Joy own a business with shares in the ratio of 5 : 3. How should they share a profit of $53 600?

8.

Lawnmower fuel is made up of 3 parts petrol to 2 parts oil. How much of each is required to mix 7.5 L of fuel?

9.

Ewes and rams are mated in the ratio 35 : 1. How many rams are needed for a flock of 455 ewes?

10.

Thelma and Louise share a company in the ratio 7 : 8. If Louise invests $120 000 in the company, how much does Thelma invest?

11.

Trent and Keiran won first prize of $2400 in a raffle and divided it between them in the ratio 5 : 3. How much did each receive?

50 0004

----------------

50 0004

----------------

Exercise 5-02: Unitary method


167

12.

Jason’s soccer team had a win : loss ratio of 8 : 5. How many losses did they have over 39 games?

13.

The ratio of fiction to non-fiction books in the library is 2 : 3. If there are 1388 fiction books, how many books are in the library?

14.

In a certain region, 10 000 lobsters were tagged and released. Later, in a catch of 2500 lobsters in the region, 20 were found to be tagged.(a) What was the ratio of tagged lobsters caught to lobsters caught?(b) Using this ratio, estimate the number of lobsters in the region.

15.

Trish made 2 kg of chocolate crackles containing copha, coconut, rice bubbles and cocoa in the ratio 2 : 2 : 5 : 1.(a) How much of each ingredient did she use?(b) If she agreed to make 10 kg of crackles for the school fete, how much of each

ingredient would she need?

16.

A concrete mix contains cement, sand and gravel in the ratio 1 : 4 : 6. (a) How many shovels of sand are needed in a mix containing 3 shovels of cement?(b) If 2 t of gravel are used to make concrete mix, how many kilograms of sand are used?

17.

Elsie wants to make a larger quantity of coconut ice than in her recipe. The recipe requires 250 g copha, 500 g coconut and 400 g sugar and makes 3 trays of coconut ice.(a) What is the ratio of the ingredients in simplest form?(b) How much (to the nearest gram) of each ingredient does Elsie need to make 7 trays?

18.

This information is on a packet of macaronicheese. (a) What is the ratio of protein to calcium?(b) What is the ratio of fat to carbohydrate

to sugar?

19.

The population of a town is 68 850. Of these, 30 600 are female.(a) How many males are in the town?(b) What is the ratio of males to females?(c) If 2000 males come to live in the town, how many extra females are needed to

maintain the same ratio?

20.

In Monique’s veterinary clinic there are 2 budgies, 6 cats and 10 dogs waiting.(a) What is the ratio of budgies to cats to

dogs?(b) Two dogs and a cat are treated and

leave. What is the ratio of animals waiting now?

(c) After 10 minutes, another cat is treated and leaves. What is the ratio now?

(d) Three dogs arrive. How many animals are waiting now?

Nutritional information Per 100 g

Protein 3.3 g

Fat 5.3 g

Carbohydrate 16.8 g

Sugar 2.4 g

Calcium 42 mg

168


DIVIDING A QUANTITY IN A GIVEN RATIO

When finding the value of the parts of a quantity divided in a given ratio, we can use the unitary method or consider each part as a fraction of the whole quantity.

Example 6Ron’s estate of $350 000 was divided between his daughters Rhonda and Carolyn in the ratio 3 : 2. How much did each daughter receive?

SolutionUsing unitary methodTotal estate is 3 + 2 = 5 parts.

5 parts = 350 000

1 part = = 70 000

3 parts = 70 000 × 3 = 210 0002 parts = 70 000 × 2 = 140 000

Rhonda received $210 000 and Carolyn received $140 000.

Using fraction of a wholeTotal estate is 5 parts.

5 parts = 350 000

Rhonda’s share = × 350 000

= 210 000

Carolyn’s share = × 350 000

= 140 000Rhonda received $210 000 and Carolyn received $140 000.

1. Divide each quantity in the given ratio.(a) $5000 in the ratio 5 : 3 (b) 375 kg in the ratio 13 : 2(c) 3 × 106 L in the ratio 2 : 3 (d) 10 ha in the ratio 4 : 5 (nearest m2)

(e) 25 t in the ratio 7 : 6 (nearest kg) (f) 2 cups in the ratio 1 : 4

2. Glenys pays 30% tax from an income of $36 000. The remaining income is divided into savings and living expenses in the ratio 2 : 3. (a) How much does she save?(b) How much is used for living expenses?

3. Adolf won $200 000 in the lottery and divided it between his 4 grandchildren in the ratio 3 : 2 : 2 : 1. How much money did each receive?

4. Two-stroke fuel for an outboard motor is made up of oil and petrol in the ratio 1 : 49. How much oil and petrol are needed to make 20 L of fuel?

5. 550 kg of concrete are to be made by mixing cement, sand and aggregate in the ratio 1 : 4 : 6. How many kilograms of each component are needed?

6. Punch is made up of fruit juice, lemonade and ginger ale in the ratio 2 : 2 : 1. (a) How many millilitres of each ingredient do you need to make 6 L of punch?(b) What size bowl (in cubic centimetres) is needed to hold 6 L of punch?

350 0005

-------------------

Hint: Don’t use units in your working out. Put them in at the end.

35---

25---

Exercise 5-03: Dividing a quantity in a given ratio

12---

RATIOS AND SIMILAR FIGURES 169

7. Rosemary divides her lotto winnings of $75 000 between her daughter, son and grandson in the ratio 3 : 2 : 1. How much does her daughter receive?

8. Baby formula is made up of 3 tablespoons of powdered formula to 6 tablespoons of water. How much water (in millilitres) is needed to make up 300 mL of formula?

9. Nancy’s daily diet consists of protein, fat and carbohydrate in the ratio 3 : 0.5 : 9.5. (a) Simplify this ratio.(b) How much protein does Nancy have in one day if she consumes 3.5 kg of food?(c) How much would she eat in a day if her diet contained 20 g of fat?(d) How much carbohydrate is in her 500 g bowl of pasta if it contains the same ratio of

components?

10. A weed spray is made up of wetting agent and a herbicide in the ratio 1 : 100. To make 85 L of weed spray, how much wetting agent (to the nearest millilitre) do you need?

11. The ratio of boys to girls at a local high school is 4 : 5. How many girls are in the school if the total enrolment is 846 students?

12. An alloy contains copper and nickel in the ratio 7 : 4. How much of each metal is in 500 kg of the alloy (correct to the nearest kg)?

13. Monica’s inheritance was divided between her daughter and two sons. They received shares of 32%, 26% and 42% respectively.(a) Express the shares as a ratio in simplest form.(b) If her daughter received $6500, how much did each of her sons receive?(c) How much was Monica’s inheritance?

14. Osmo fertiliser is made up of nitrogen (N), potassium (K) and phosphorus (P) in the ratio 4 : 4 : 1. A customer requires exactly 1 t of fertiliser.(a) How much of each nutrient is required to make 1 t of Osmo fertiliser?(b) If the manufacturer only has scales marked in kilograms, how can he make sure that

he supplies exactly 1 t of fertiliser to a customer?

A cook often needs to vary the quantities in a recipe to feed more or less people than stated in the recipe. The new quantities aren’t always exact, so a few assumptions may need to be made. For example, if the new quantity was calculated to be 1 eggs, you would need to decide whether to use 2 small eggs or 1 large egg; or if the new quantity was calculated to be 18.75 g of butter, you might choose to use 20 g.

Let’s look at the problem of varying this family favourite recipe to serve 20 people.

Investigation: Varying recipes

12---

Sticky toffee pudding with butterscotch sauce

Pudding: 1 cup water cup castor sugar

200 g fresh dates 1 teaspoon vanilla essence1 teaspoon bicarbonate of soda 2 eggs

60 g butter 1 cups self-raising flour

Sauce: 300 mL cream 1 cups firmly packed brown sugar

125 g butter, chopped Serves: 6–8

34---

12---

12---

170 NEW CENTURY MATHS GENERAL: PRELIMINARY

Assume that the recipe serves 8 people. (We could assume that it serves 6 or 7.) Then:

New quantity = × old quantity = 2.5 × old quantity

A spreadsheet is a useful tool when varying quantities. The first four ingredients are shown here along with the relevant formulas.

1. Complete the ingredients in columns B and C to row 14.

2. Put the formula (as shown) in D4, then fill down to D14 to get the new quantities in columnD.

Other investigations1. Find the quantities for 20 people, assuming that the original recipe serves 6.

2. Vary the recipe to serve 50 people, assuming that the original recipe serves 7.

3. Use your favourite recipe and vary the ingredients to serve more or less people.

4. Comment on any assumptions you need to make or problems you encounter.

A B C D

1 Recipe for Sticky Toffee Pudding with Butterscotch Sauce

2 No. of people 8 20

3

4 1 cup water =B4/$B$2*$D$2

5 200 g dates =B5/$B$2*$D$2

6 1 tsp bicarb soda =B6/$B$2*$D$2

7 60 g butter =B7/$B$2*$D$2

8

208

------

Old quantityPudding

1 cup water

200 g fresh dates

1 tsp bicarb. soda

60 g butter

cup castor sugar

1 tsp vanilla essence

2 eggs

1 cups self-raising flour

Sauce300 mL cream

1 cups brown sugar

125 g butter, chopped

34---

12---

12---

New quantityPudding

2 cups water

500 g fresh dates

2 tsp bicarb. soda

150 g butter

1.875 cups castor sugar

2 tsp vanilla essence

5 eggs

3 cups self-raising flour

Sauce750 mL cream

3 cups brown sugar

312 g butter, chopped

12---

12---

12---

34---

34---

12---

× 2.5→

Technology: Using a spreadsheet

You need the $ since cells B2 and D2 are constant.


SCALE FACTORS AND CENTRE OF ENLARGEMENTScale factors and similar figuresSimilar figures are the same shape but not necessarily the same size.� All figures similar to a given figure will be an enlargement or reduction of that figure.� The scale factor shows by how much a figure is enlarged or reduced. � The original figure is called the object and the enlarged figure is called the image.� The matching angles in each figure are equal to preserve the same shape.

Example 7

The figure DAVE is a parallelogram. It has been enlarged by a scale factor of 1.5 togive figure D′A′V′E′.The figures are similar and the sides are in the ratio 1.5 : 1 or 3 : 2.

= = = =

Example 8

ΔTOM has been reduced by a scale factor of to give ΔT′O′M′.The figures are similar and the sides are in the ratio 1 : 4.

= = =

D

E

A

V2 cm

3 cmObject

D´

E´

A´

V´3 cm

4.5 cmImage

Scale factor = 1.5

We put the image first in the ratio.D′A′

DA------------ A′V′

AV----------- V′E′

VE----------- E′D′

ED------------ 3

2---

T

O

M

Object

T´

O´

M´

Image

Scale factor = 14--

14---

T′O′TO

----------- O′M′OM

------------- M′T′MT

------------ 14---

Similar figures are the same shape but not necessarily the same size.The matching angles in similar figures are equal to preserve the same shape.Two similar figures that have a scale factor of 1 are said to be congruent.


On a photocopier you have the opportunity to enlarge and reduce figures.

1. Draw a shape and enlarge it by selecting 120%. The resulting image is a similar shape with a scale factor of 1.2.

2. Use your original shape and reduce it by selecting 70%. The resulting image is a similar shape with a scale factor of 0.7.

3. Investigate enlarging and reducing other shapes.

Centre of enlargement and similar figuresA centre of enlargement (or reduction) will help you to draw similar figures.

Example 9Use a point C as the centre of enlargement to:(a) enlarge ΔABD using a scale factor of 2

(b) reduce ΔABD using a scale factor of

Solution

Step 1 Draw lines from C through A, B and D.Step 2 Measure CA and mark a point A′ so that CA′ = 2 × CA.Step 3 Repeat for B′ and D′.ΔABD has been enlarged by a scale factor of 2 to give ΔA′B′D′. The sides of the two similar triangles are in the ratio 2 : 1.

= = =

Step 1 Draw lines from C through A, B and D.Step 2 Measure CA and mark a point A″ so that CA″ = × CA.Step 3 Repeat for B″ and D″.ΔABD has been reduced by a scale factor of to give ΔA″B″D″.The sides of the two similar triangles are in the ratio 1: 3.

= = =

Investigation: Photocopiers

13---

(a)

D´

A´

B´A

B

C

D

A′B′AB

----------- B′D′BD

------------ D′A′DA

------------ 21---

(b) A

B

C

D

A˝

D˝

B˝

13---

13---

A″B″AB

------------- B″D″BD

-------------- D″A″DA

-------------- 13---


Notation: The symbol ||| means ‘is similar to’. In Example 9, ΔABD ||| ΔA′B′D′ and ΔABD ||| ΔA″B″D″.

1. Copy each figure onto graph paper, then draw a similar figure using the scale factor.(a) scale factor = 2 (b) scale factor = 0.5 (c) scale factor = 4

2. Write down the scale factor of each enlargement or reduction (B′ denotes the enlargement or reduction).

3. Copy each figure onto graph paper, then enlarge or reduce it using O as the centre of enlargement and the given scale factor.

(a) scale factor of (b) scale factor of 1.5 (c) scale factor of 2.5

1. A triangle of side 1 is enlarged to give a triangle of side 2, and the length and area scale factors are shown. Write down the length and area scale factors when:(a) a triangle of side 1 is enlarged to give a

triangle of side 3(b) a triangle of side 2 is enlarged to give

a triangle of side 3

Exercise 5-04: Scale factors and centre of enlargment

(a) (b) (c)

B´

B B´

B

B

B´

12---

O

O

O

Investigation: Scale factor and area

Length scale factor = 2Area scale factor = 4


2. Write down length and area scale factors when:(a) a square of side 3 is reduced to give a

square of side 1(b) a square of side 3 is reduced to give a

square of side 2

PROPERTIES OF SIMILAR FIGURESWe have looked at how to draw similar figures using scale factors and a centre of enlargement (or reduction). Now let us summarise the properties of similar figures:

Example 10Are these two figures similar? Give reasons.

SolutionThe matching angles are equal:

∠B = ∠B′, ∠L = ∠L′, ∠U = ∠U′, ∠E = ∠E′The corresponding sides are in the same ratio:

= = = =

BLUE ||| B′L′U′E′ because the matching angles are equal and the matching sides are in the same ratio.

Example 11

(a) Which two triangles are similar and why?(b) What is the scale factor here?

Length scale factor =

Area scale factor =

12---

14---

� Similar figures have all matching angles equal.� Similar figures have matching sides in the same ratio.

L

U

E

B

L´

U´

E´

B´

3.5

4.5

3

4

6

8

7

9

The scale factor is .12---

B′L′BL

----------- L′U′LU

----------- U′E′UE

------------ E′B′EB

----------- 12---

9

15

10

18

12

7B

A

6

9 3.5C


Solution(a) Two angles in ΔA are equal to two

angles in ΔC, so the third angles must be equal. In ΔA and ΔC, the ratios of the matching sides are equal,

since = = = .

ΔA ||| ΔC because their matchingangles are equal and their matchingsides are in the same ratio.

(b) We can say that ΔA is an enlargement of ΔC with a scale factor of 2 or that ΔC is a reduction of ΔA with a scale factor of .

Example 12(a) Find the scale factor for these two similar figures.(b) Find the values of the pronumerals.

Solution(a) The second figure is a reduction of the first figure and the scale factor is = .

(b) = and =

2y = 8 × 5 8x = 2 × 12y = 20 x = 3

1. Choose the similar shapes and write down the scale factor.

6

9 3.5C

18

12

7A

It often helps to redraw the figures with the same orientation.

126

------ 73.5------- 18

9------ 2

1---

12---

y

8

12x

2 5

28--- 1

4---

5y--- 2

8--- x

12------ 2

8--- Take care to write down the ratios of the

matching sides in the correct order.

Exercise 5-05: Properties of similar figures

(a)

1012

6

E6

3

5

F6 9

7.5

G

(b)

B

9

7

9.8

12.6AC

14

4.5


2. Find: (i) the scale factors for these similar shapes(ii) the values of the pronumerals

(c) 6

8S

3

6

TR

2

4

(d)

64

4.8

5

T

8

A

10 B

(e)

6.5

7.5J

2.5

6.5

HG

513

(a) (b)

5.2

3.4

4

x

10

6

x

12

7

6

y

z

(c) (d)5

4

3

x

yz

58

25

15

7 x

(e)

30°

α°y

x

12

9

θ° z120°

6


3. (a) Measure the sides of each triangle and state the scale factor.

(b) Are the triangles similar? Give reasons for your answer.

(c) What is the ratio of the areas?

When you take a photograph of an object, the image on film is smaller, upside down and back to front. The object and the image are similar figures.

1. A photograph of Jeff shows him to be 3.5 cm tall. If the photograph has a scale factor of 1 : 50, what is Jeff’s real height?

2. If Felicity is 168 cm tall, what is her height on a photograph with a scale factor of 1 : 50?

USING SHADOWS AND SIMILAR TRIANGLESOn a sunny day, a stick of known length, such as a metre rule, and a long tape can be used to determine the heights of trees, flagpoles or buildings in your neighbourhood with the aid of similar triangles. The stick is often referred to as a shadow stick.

Example 13The metre rule throws a shadow of 1.6 m while the tree throws a shadow of 48 m. What is the height of the tree?

A

B

C

A′

B′

C′

Investigation: Photographic images

Lens

A

B

C

D

A′B′

C′

D′

ImageObject

1.6 m48 m

h

1 mTo use similar triangles, shadows must be measured at the same time of day.


SolutionUsing matching sides

=

=

h = 30

Height of the tree is 30 m.

1. Robert is 2 m tall and casts a shadow 1.5 m long. At the same time, a tower casts a shadow 14 m long. How high is the tower?

2. A 5.2 m flagpole casts a shadow of 8.1 m while Arlene casts a shadow of 2.9 m. How tall is Arlene?

3. A metre rule casts a shadow of 2.3 m and at the same time a nearby hill casts a shadow of 42 m. How high is the hill?

4. A 2 m stick casts a shadow of 3.5 m while a flagpole casts a shadow of 8.5 m. How high is the flagpole?

5. A school building throws a shadow of 26 m at the same time as a 3 m high tree throws a shadow of 5 m. What is the height of the building?

6. A mountain of height 1500 m casts a long shadow at dusk of length 6 km while a TV tower throws a shadow of 15 m. (a) Why are shadows longer at dusk?(b) How tall is the tower?

7. A 1.5 m high fence has a shadow of length 1.8 m while at the same time a shadow is thrown by a building of height 4.8 m. How long is the building’s shadow?

Using the scale factor

Scale factor =

= 30Height of tree = 30 × height of stick

= 30 m

481.6-------

Height of treeHeight of stick----------------------------------- shadow of tree

shadow of stick-------------------------------------

h1--- 48

1.6-------

Exercise 5-06: Shadows and similar triangles

1.5 m14 m

h

8.1 m

5.2 m

2.9 m

2.3 m

42 m

1 m

h


8. Jaani holds a 30 cm ruler and measures its shadow to be 24 cm. At the same time Costa measures the shadow of a lamp post to be 10.5 m long. How tall is the lamp post?

9. A church spire is 9.4 m high and casts a shadow of 18.6 m. At the same time, young Cassius casts a shadow 2.3 m long. How tall is Cassius?

10. Anthony is on one side of a canal and takes measurements of 15 m and 24 m as shown. He then stands at point A so that he is in line with the pier and the tree on the other side of the canal. The distance of the tree from the jetty is known to be 150 m. Use two similar triangles to find the width of the canal.

11. A fireman’s ladder reaches 12.5 m up a building. The ladder is propped up by a support placed 5.4 m from the building and 3.2 m from the base of the ladder. What is the length (correct to 1 decimal place) of the support?

12. Galila is standing 32 m from a flagpole. She holds up a 30 cm ruler and moves it until it appears the same height as the flagpole. If this occurs when the ruler is held 70 cm from her eye, how tall is the flagpole (correct to 1 decimal place)?

6

A

Tree

150 m

Jetty

15 m

24 m

Pier

Anthony

5.4 m

12.5 m

3.2 m

32 m70 cm

30 cm

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Study tips


SCALE DRAWINGSA scale drawing is usually a reduction of a real object, such as a building, but can be an enlargement of a very small object, such as a computer chip. The scale factor used in a scale drawing is called the scale. Some common scale drawings are house plans and maps.

Here are some ways of representing scales:

�

� 10 mm to 1 m� 1 : 100� 25 : 1� 1 cm = 2 m�

Example 14Here is a part of an architect’s working drawings for a new shopping centre.(a) What is the real length of a shop if the length on the

drawing is 15 mm?(b) Each shop is to be 4.5 m wide. What distance represents

the shop width on the drawings?

SolutionA scale of 1 : 1000 means 1 unit on the plan represents 1000 units of actual length.(a) Real length of shop = 1000 × 15 mm

= 15 000 mm or 15 m

(b) Scaled width of shop = × 4.5 m

= × 4500 mm

= 4.5 mm

Example 15A surveyor makes a scale drawing of a block of land and uses a scale of 10 mm to 5 m.(a) Write the scale as a simplified ratio.(b) One side of the block is 120 m long. What length represents this side on the scale

drawing?(a) If another side is 8 mm long on the drawing, what is its actual length?

Solution(a) Scale is 10 mm : 5 m = 10 mm : 5000 mm

= 10 : 5000= 1 : 500

(b) Scaled length = × 120 m

= × 120 000 mm

= 240 mmThe side is 240 mm long on the scale drawing.

(c) Actual length of side = 500 × 8 mm= 4000 mm or 4 m

0 5 10 15 20 km

0 10 20 30 40 50 60 m

Shop

1

Shop

6

Scale 1 : 1000

mm and m are the standard units of measurement in the construction industry.

is the scale factor.11000------------1

1000------------

11000------------

The scale factor is .1

500---------1

500---------

1500---------


Example 16Mario and Stella hired an architect to do a scale drawing of the proposed extension to their house. They wanted a guest suite (bedroom and bathroom) and a parents’ retreat. The floor plan is shown here.

By measurement and calculation find:(a) the actual dimensions of the new guest suite(b) the length of the window in the main bedroom(c) the area of the floor to be tiled in the ensuite

shower

SolutionScale is 1 : 200.

(a) For the guest suite:Scaled length = 35 mmActual length = 200 × 35 mm

= 7000 mm or 7 mScaled width = 14 mmActual width = 200 × 14 mm

= 2800 mm or 2.8 mActual dimensions of the guest suite are 7 m by 2.8 m.

(b) For the window in the main bedroom:Scaled length = 10 mmActual length = 200 × 10 mm

= 2000 mm or 2 m

(c) For the ensuite shower:Scaled length = 6 mm Actual length = 200 × 6 mm

= 1200 mm or 1.2 mScaled width = 4 mmActual width = 200 × 4 mm

= 800 mm or 0.8 m

Area to be tiled is 1.2 × 0.8 = 0.96 m2.

1. Write down each scale as a ratio in simplest form:

(a)

(b) 2 mm : 1 cm

(c)

(d) 10 cm to 1 m(e) 5 m : 2 cm

New extension

Guest bedroom

Dre

ssin

g

Bath Ensuite

Main bedroom

Parents’retreat

Scale 1 : 200

Porch

Take inside measurements.

Exercise 5-07: Scale drawings

0 1 2 3 4 5 6 km

0 10 20 30 40 50 60 m


2. The map ‘Routes out of Sydney’ is a scale drawing. Find:(a) the scale factor(b) the distance (in a straight line)

from Sydney to Katoomba(c) the distance from Newcastle to

Nowra(d) the distance from Goulburn to

Gosford(e) the distance from Sydney to

Cessnock(f) the town that is 92 km from

Sydney(g) the towns that are exactly 40 km

apart

3. By measurement and calculation, find the real lengths of these objects.(a) (b)

(c)

(d)

4. A house is drawn to scale as shown.(a) What does the scale of 1 : 100 mean?(b) If the height of the door in the

drawing is 18 mm, what is its actual height?

(c) Find the actual width of:(i) the house

(ii) a window(d) Find the actual height above

ground level of:(i) the ceiling

(ii) the peak of the roof

Routes out of Sydney

Newcastle

Nowra

Goulburn

Cessnock

Gosford

SYDNEY

PenrithKatoomba

Mittagong Wollongong

0 30 60 90 120 km

Robocar 1 : 100Tennis racquet 1 : 12

Ship 1 : 3000

Ant 4 : 1

Scale 1 : 100


5. What is the distance (in a direct line)between:(a) Liverpool and Campbelltown?(b) Hoxton Park and Ingleburn?(c) Campbelltown and Hoxton Park?

6. Here is a scale drawing of a hockey field. (a) Copy the figure carefully, then calculate and write the actual dimensions on your

drawing.(b) What is the actual width of the hockey field?(c) What is the actual length of the field?

7. This is a scale drawing of an indoor cricket venue. (a) Copy the figure, then calculate and mark the actual dimensions on your drawing.(b) What is the actual length of the venue?(c) What is the actual width of the cricket pitch?(d) What is the actual length of the pitch?

8. A map has a scale of 1 : 50 000. What real distance is represented by a scaled length of:(a) 1 mm? (b) 2 cm? (c) 25 mm?

9. The distance from Sydney to Penrith is 55 km. What scaled distance would this be on a map with a scale of 1 : 2 000 000?

Liverpool

Ingleburn

Hoxton Park

Campbelltown

Scale: 1 cm to 4 km

Scale: 1 cm = 10 m

Scale: 1 cm to 3 m


10.

Here is a plan (not to scale) of one wall and the floor of an office. The office has no windows and only one door. (a) Make a scale drawing of the floor, using a scale of 1 to 50.(b) Make a scale drawing of the wall with the door, using the same scale.(c) Find the area (in square metres) of carpet needed to carpet the office.(d) Find how many litres of paint are needed to paint the four walls (not including the

door), if 4 L of paint cover 10 m2. Answer correct to the nearest litre.

FLOOR PLANS AND ELEVATIONSWhen an architect is drawing up two-dimensional plans for a new home, the view from the top looking directly down is called the floor plan or plan, and the views from the front, back and sides are called elevations. The plan is a scale drawing of the floor of the house.

Example 17Here is a site plan for a new factory. The block of land is 50 m wide and 90 m long.(a) What scale is used here?(b) What is the area of the block of land, in square metres?(c) What percentage of the land does the factory occupy (correct to 1 decimal place)?

Solution(a) Scaled width of the block = 50 mm By measurement of plan

Actual width of the block = 50 mScale = 50 mm : 50 m

= 50 mm : 5000 mm= 1 : 1000

(b) Area of block = 90 m × 50 m= 4500 m2

Office wall

2.9 m

2.75 m

0.8 m

2.1 m

Office floor

2.9 m

2.9 m

Factory


(c) Area of factory = area A + area B + area C= (40 × 25 + 18 × 10 + 25 × 15) m2

= 1555 m2

% occupied by factory = × 100%

≈ 34.6%

Example 18A view of a single garage is shown here. Sketch the floor plan and the front and side elevations of the garage.

Solution

The plan and front and side elevations of the garage are shown above. The diagram also enables you to see the relationship between the plan and elevations. The plan and elevations are part of the working drawings and are not always drawn to scale.

To draw the diagram, follow these steps:Step 1 Draw the plan (not necessarily to scale) and mark dotted lines down and across as

shown. Step 2 Draw the front elevation using the dotted lines as a guide, then draw dotted lines

across as shown.Step 3 Draw line ABC (about 45º but it does not need to be exact).Step 4 Draw dotted lines down from B and C.Step 5 Draw the side elevation as shown and mark in the door and window.

40

15

25

10

18

A

BC 25

15554500------------

Front elevation Side elevation

Plan

Roof

Wall

B

A

C


1. The following prisms represent roof types. Draw the plan and front and side elevations showing the appropriate measurements on your diagrams.

(a) Hip roof

(b) Gabled roof

(c) Skillion roof

2. (a) Sketch the fishing shack with this plan (P), front elevation (F) and side elevation (S).

(b) What type of roof does the shack have?

(c) What is the height of the back wall?(d) What is the width of the shack?(e) What is the floor area?

3. (a) Sketch the barn with this plan (P), front elevation (F) and side elevation (S).

(b) What type of roof does the barn have?(c) What is the height of the highest point

of the roof from the ground?(d) What is the length of the barn?(e) What is the floor area?

Exercise 5-08: Floor plans and elevations

2.5 m

8 m

10 m

5.3 m5 m

12 m

5 m4 m

6 m

20 m

10 m

3 m

1 m

P

Scale 1 cm : 2.5 m

F S

Scale 1 : 400

P

SF


4. (a) Sketch the plan, front and side elevations of this hayshed.

(b) What is the floor area of the shed?(c) What volume of hay does the shed hold

when full to the roof?

5. The plan and front elevation of a house are drawn to scale.

(a) By measuring and calculating, find the actual value of x (roof height above ceiling).(b) Find the actual value of y (ceiling height).(c) Calculate the floor area of the house (to the nearest square metre).(d) Sketch a side elevation of the house.

10 m

8 m6 m

6 m9 m

4 m

x

y

Plan

Front elevation

9.4 m

5.4 m10.4 m

3.4 m

10 m

6 m

8.6 m

Scale 1 : 200


SYMBOLS AND CALCULATIONS FROM PLANS AND ELEVATIONS

These plans (not to scale) are of a four-bedroom house. Use the plan to work through the following excamples. Measurements are all in mm.

SymbolsExample 19(a) Here are some symbols used on house plans. Can you locate them on this house plan?

(b) How many windows are in the house? 16(c) How many hinged doors are on the plan (including wardrobes)? 19(d) In the kitchen, what indicates that there are wall cupboards? Dotted lines

Rumpus3700 × 4000

Bed 42700 × 3150

Bed 32700 × 3150

Family5140 × 3770

Meals2880 × 2660

RobeRobe

Bed 23180 × 2970

Dining2900 × 3000

Robe

Kitchen

Laundry

Line

nPtry Ref.

Lin.

Living4830 × 3990

Bath

Ens

Garage5630 × 5500

Bed 13700 × 3800

Porch

Entry

W.I.R.

N

WM

WM

Hinged door

Toilet

Kitchen sink

Vanity

Shower

Linen cupboard

Laundry tub

Stove

Bath

Washing machine


CalculationsExample 20 The dimensions shown on the house plan on page 188 refer to inside measurements.(a) Find the area of the garage to the nearest square metre.(b) The rumpus room is to be tiled with cork tiles. The tiles are 30 cm square and come in a

pack of 10. How many packs are needed?(c) The four bedrooms are to be carpeted. Carpet is 4.2 m wide and costs $120 per metre

laid. How much carpet (to nearest half metre) is needed and how much will it cost?

Solution(a) Area of garage = 5630 mm × 5500 mm

= 5.63 m × 5.50 m≈ 31 m2

(b) For rumpus room:Length of rumpus room = 4000 mm

No. of tiles needed for one length = ≈ 14

Width of rumpus room = 3700 mm

No. of tiles needed for one width = ≈ 13

No. of tiles needed for rumpus room = 14 × 13 = 182Hence, 19 packs of tiles are needed.Note: There will always be some wastage for cutting when laying tiles.

(c) Amount of carpet for Bed 1 = 3700 mm = 3.70 mAmount of carpet for Bed 2 = 2970 mm = 2.97 mAmount of carpet for Bed 3 = 3150 mm = 3.15 mAmount of carpet for Bed 4 = 3150 mm = 3.15 m

Total amount of carpet = 12.97 m ≈ 13 m Total cost of carpet = $120 × 13

= $1560

Imperial units are still used in the construction industry today and they do not convert exactly to metric units. Some commonly used imperial units are:

1 inch (1″) ≈ 25.4 mm1 foot (1 ft or 1′) ≈ 0.305 m

1 yard (1 yd) ≈ 0.915 mAlso 12″ = 1′and 3′ = 1 yard

1. A piece of ‘4 by 2’ is a piece of timber with a cross-section of 4 inches by 2 inches.What is the approximate size in metric units?What is the approximate size in metric units of a piece of ‘2 by 1’?

2. Houses are usually measured in ‘squares’. A square is equal to an area 10 ft by 10 ft.What is the metric equivalent of a square?How many square metres of floor area in a 20-square home?

4000 mm = 14 tiles

3700 mm

= 13 tiles

4000300------------

3700300------------

Throughout a house, carpet is laid in one direction only. Here it is laid along the length of the house. (You could also lay it the other way.)

Investigation: Imperial units

4″

2″


1. These working drawings have been reduced in size, so no dimensions are shown. The site plan, floor plan and elevations are of a three-bedroom holiday house.

(a) What does the abbreviation W.C. refer to? (b) How many windows in the house?(c) What are the abbreviations for the ground level, floor level and ceiling level?(d) What is a footing? (e) Which direction does the front of the house face?(f) What type of roof does the house have?(g) What indicates the roof shape on the floor plan?(h) What is section A–A?

2. This basic hut is one of several in a National Park.

Exercise 5-09: Symbols and calculations from plans and elevations

Laundry

Kitchen Bed 3

Bed 2

Bed 1

Living room

W.C

.

Bath

Porch

A

A

Floor plan

South elevation

North elevation East elevation

West elevation

Section A–A

Footing detail

Site plan

Bed 1Bed 2Bed 3

G.L.

C.L.

F.L.

G.L.

C.L.

F.L.

G.L.

N

Plan

Living

Bathroom

Bedroom

Scale 1 : 200Porch

9.2 m

7.8

m


(a) What is the total area inside the hut? (b) What type of roof is on the hut?(c) Measure the pitch angle of the roof.(d) Will the pitch angle change if you enlarge the diagram? Why?(e) What is the area of the porch?(f) What are the internal dimensions of the living room?(g) How wide are the doors onto the porch?(h) What is the ceiling height?(i) What is the height (in metres) of the highest point of the roof?

3. These working drawings are of a small ranger’s hut.

Front elevation Side elevation

C.L.

F.L.G.L.

1400

2400

Ranger’s hut

6000

850

3700

1025

3500

1300 2250 575

100

A

A

450

450

100

3500

450

710 10

010

0

100

100

Plan

Scale 1 : 100

6200

Section A–A

Ranger’s hut

F.L.

G.L.

2100 2150

Scale 1 : 50


(a) What type of roof does the hut have?(b) Measure the pitch angle of the roof.(c) Which way does the door face?(d) What is the thickness of the walls?(e) What is the ceiling height at the front of the hut?(f) How wide is the window?(g) What is the internal floor area?(h) In which corner of the hut is the downpipe situated?(i) How many brick piers support the hut?(j) Each pier sits on a concrete footing measuring 450 mm × 450 mm × 300 mm. How

much concrete was purchased for the footings, if concrete can only be purchased in increments of 0.2 m3?

4. This is the ground floor of a tri-level home.

C.L.

F.L.

G.L.

21002150

West elevation North elevationScale 1 : 100

WO

Rumpus5.2 × 4.9

Kitchen

Family7.3 × 4.1

Meals

Study3.7 × 3.0

Dining4.2 × 3.3

Living6.5 × 4.0

Double garage5.6 × 5.5Porch

8.4 × 1.8

Entry

Ref

M/O

Line

n

N

HWS

Deck


(a) How many steps do you walk up from the family room to the middle level?(b) Draw the symbols used on the plan to represent:

(i) hot water service (ii) laundry tub(iii) hotplates (iv) wall oven(v) linen press (vi) microwave oven

(vii) front door (viii) sliding door(c) The porch is to be tiled with terracotta tiles. How many square metres of tiles are

needed (rounded to the nearest 0.5 m2)?(d) If the tiles are 30 cm by 30 cm each, how many whole tiles are needed?(e) The dining and living areas are to be carpeted. How much will it cost to carpet the

rooms if the carpet is 4.2 m wide and costs $155 per metre laid?

5. Here are the floor plan, the south elevation and a cross-section of a home built on a sloping block of land.

A

APorch

EntryLivingDining Bed 3

Linen

Laundry

FamilyKitchen

Bed 2 Bed 1

W.I.R.

Scale 1 : 100

Plan

Bath

Ens

x

N

South elevation

C.L.

F.L.

2400


(a) What scale is used in the plan shown on the previous page?(b) What is the actual length of the house (denoted by x)?(c) How many windows in the house?(d) How many external doors are there?(e) What type of roof does the house have?(f) Sketch the north elevation of the house.(g) Measure the pitch angle of the roof.(h) Use the plan to write down the dimensions of bedroom 1 plus wardrobe and ensuite.(i) What is the area of the porch in square metres? Answer to the nearest m2.(j) What is the ceiling height?(k) Describe the symbol on the plan that indicates the posts supporting the roof at the

front.(l) Bedrooms 2 and 3 are to be recarpeted. How much will it cost to carpet the two

rooms with carpet tiles, each 40 cm by 40 cm, if a box of 10 costs $85 plus a charge of $10 per square metre of tiles to lay?

(m) The family room and kitchen are to have slate tiles. These cost $75 per square metre laid. How much will it cost to have these two rooms tiled? Answer to the nearest $100.

(n) At what angle does the land slope?(o) How many floor supports are shown in section A–A?

6. The plan shown at the top of the next page is of a three-bedroom home. Dimensions are in millimetres.(a) What is the internal floor area (to the nearest square metre)?(b) What are the internal dimensions of bedroom 1?(c) A wardrobe, 600 mm wide, is to be built along the eastern wall of bedroom 1.

(i) What is the maximum length it can be?(ii) What floor area will be left to carpet?

(d) What is the area of the deck (to the nearest square metre)?(e) How much will it cost to build this house (excluding the deck) if the current building

rate is $540 per square metre? (f) The bathroom floor is to be tiled with 20 cm square tiles. Find:

(i) the number of tiles that will fit along the wall adjacent to the hallway(ii) the number of tiles that will fit along the wall adjacent to the laundry

(iii) the number of tiles required

C.L.

F.L.

Section A–A

Bed 1 W.I.R. Ens


7. The following extensions to the existing house are proposed:

Build new double garage.Extend the deck to 3 m wide.Pave the shaded area.

(a) Use the plan on the next page to find the width of the garage door.

(b) In the front elevation, the door height is 2.8 m. What is the height of the apex of the roof from the top of the garage door?

(c) Sketch the side elevation with the door.(d) Draw a 3D sketch of the garage.(e) If the concrete slab for the garage floor

is to be 125 mm thick, find the amount of concrete needed for the slab (rounded up to the nearest 0.2 m3).

(f) What is the area of the new deck?(g) If timber decking is to be in a north–

south direction, how many boards are needed if they measure 3200 × 75 mm?

(h) How much will the decking cost at $2.30 per linear metre (offcuts disgarded)?

(i) What area is to be paved?(j) What percentage of the block is not

built on?

Kitchen/meals

FoyerLiving

Bedroom 1 Bedroom 3

Bedroom 2 Bathroom Laundry

3600 2700 2100 6000300300

15 000

17002700 2700 3700 3600300300

Deck

2700

1200

300

900

3600

300

3600

300

3600

300

7800

N

Existing house

Proposedgarage

Proposed deck

Scale 1 : 400

Site plan

N

10 m

3 m

10 m

6 m

10 m

40 m

1 m

7.5 m1 m

20 m

4 m


1. Collect floor plans from newspapers, magazines or the Internet. Discuss features you like or dislike in the houses. What would you change? What would you keep?

2. Draw a plan and elevation of your house or unit.

3. Draw a scale drawing showing all features of your kitchen, bathroom, classroom.

Proposed garage

Front elevation

Automatic door

6670

250

250

7500

220

110

6000

5280250 110 110 250

Plan

C.L.

G.L.

Scale 1 : 100

N

Modelling activity: House plans

BEFORE AN EXAM� Make a study plan early—don’t leave it until the last minute.� Read and memorise your summaries.� Work on your weak areas—learn from your mistakes.� Don’t spend too much time studying work you know already. � Use revision exercises, past exams and assignments for practice. � Vary the way you study so that you don’t become bored. For example, write a

summary, have someone quiz you, record your summary on tape, explain the work to someone, design a poster, use a mind map.

� Anticipate the exam:Which topics are being tested?How many questions?What type of questions (e.g. multiple choice, short answer, long answer,investigation)?How long is the exam?How many marks for each part?How much time will I spend on each question?

Study tips


Chapter review

Ratios and similar figures1. Simplifying ratios2. The unitary method3. Dividing a quantity in a given ratio4. Scale factors and centre of enlargement5. Properties of similar figures6. Using shadows and similar triangles7. Scale drawings8. Floor plans and elevations9. Symbols and calculations from plans and elevations

This chapter, Ratios and similar figures, looked at ratios and problems involving ratios. In it you investigated similarity in the real world, drew scale diagrams, and investigated the use of scale, common terms and symbols in house and building plans.

Make a summary of this topic. Use the chapter outline above as a guide. An incomplete mind map has also been started below. Use your own words, symbols, diagrams, boxes and reminders. Use the questions in Your say below to think about your understanding of the topic. Gain a ‘whole picture’ view of the topic and identify any weak areas.

Topic summary

Ratios and similar figures

Scale drawings

Shadow sticks

Floor plans and elevations

Similarfigures

Enlargements and reductions

Ratio problems


� Have you satisfied the outcomes listed at the front of this chapter?� What was the most important thing that you learned?� How did you feel about the topic? Did you enjoy it?� What was new?� What are your weaknesses? What will you need to study more?� How will you revise and summarise this topic?

1. Express in the simplest form:(a) 500 g : 3.5 kg (b) 1 century : 1 millennium

(c) 32 : 12 : 96 (d) 2 : 1

2. Adam, Beryl and Colleen shared a $500 prize in the ratio of their ages. Adam is 8 years old and received $100 and Colleen is 12 years old.(a) How old is Beryl?(b) How much did Colleen receive?

3. John made enough formula to feed Joshua for 1 day. The formula is made up of 2 parts concentrate to 7 parts water. Joshua has 6 feeds per day and takes 120 mL at each feed.(a) How much concentrate is needed to make up a day’s formula?(b) How much water is used to make a day’s formula?

4. Shannon and Sylvia set up a small business. Shannon invested $12 000 and Sylvia invested $16 000.(a) What was the ratio of their investments?(b) What amount of the $14 000 profit should each receive?

5. A photograph 8 cm by 6 cm is enlarged by a scale factor of 2.5. What is the size of the enlargement?

6. Catriona measured the width of a river by taking measurements as shown. What was the width of the river?

7. Here is a recipe for Strawberry Souffle, to serve 8 persons:400 g strawberries150 g castor sugar12 egg whitesicing sugar for dusting

(a) Vary the souffle recipe to serve 10 people.(b) Vary the recipe to serve 3 people.(c) Comment on any problems you could encounter by varying this recipe.

Your say: Reflecting about the topic ● ● ● ●

Chapter assignment

14--- 1

3---

Tree

6.9 m12.8 m

5.4 m

Catriona


8. A large family room in the shape of a rectangle has length 6.4 m and width 5.2 m.(a) How many lengths of specially made skirting board (only 3 m lengths are

available) are needed for the room? (b) Draw a neat scale diagram using a scale of 1 : 100. (c) Find the ratio of the scaled area to the actual area.

9. This is a scale diagram of a squash court.(a) What is the scale used?(b) What is the area of the front of court

section (to the nearest 0.5 m2)?(c) What is the area of each service square?(d) How many square metres of timber

flooring are needed for a squash court (to the nearest m2)?

10. These shapes are similar. Find the values of the pronumerals.

11. An alloy contains lead and tin in the ratio 5 : 2.(a) How much lead is contained in 154 kg of the alloy?(b) An amount of the alloy was analysed and found to contain 29 g of tin. How much

lead did it contain?

9.75 m

6.4 m

3.2 m 3.2 m

1.6 m

1.6 m 1.6 m

1.6 m

5.45 m

4.3 m

Front

34

9

y38°

67° θ°38°

α°

β°

3

6

x

26

y

7

15

y

25

16 x

1.55

x6

(a) (b)

(c) (d)

(e)


12. Petra left $16 000 to be divided between three friends in the ratio 2 : 4 :3. How much did each receive?

13. An object is 20 cm from a camera lens, and the image is projected onto a screen that is 3 m from the lens. If the object is 20 mm tall:(a) what is the scale factor?(b) how tall is the image?

14. Huynh is 140 cm tall and casts a shadow of length 2 m while at the same time a 9 m pole casts a shadow. What is the length of the pole’s shadow?

15. Copy this figure onto graph paper, then enlarge it bya scale factor of 2.

16. Dagmar’s rent was increased from $135 per week to $150 per week. Find:(a) the ratio of the old rent to the new rent(a) the ratio of the increase in rent to the old rent(b) by what percentage the rent increased (correct to 1 decimal place)

17. To make up some milk, Keir used cup of powdered milk in cup of water. How many cups of powdered milk would be needed in 3 L of water (given that 1 cup = 250 mL)?

18. The plan of a house is drawn to a scale of 1 : 250. If a bedroom measures 18 mm by 15 mm on the plan, what is the actual area of the bedroom (to the nearest m2)?

O

14--- 3

4---

Documents

Ratios and similar figures Century Year 11/05_ratios...SIMPLIFYING RATIOS The parts of a ratio are called its terms. A simplified ratio is one where the terms are whole numbers with