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GCSE Ratio, Scale and Similar Triangles Dr J Frost Last modified: 22 nd February 2014

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RECAP: Ratio Most ratio problems come in three flavours: Total given.1 The ratio of cats to dogs in Battersea Dogs Home is 3:5. There are 240 animals in total. How many cats are there? 8 parts= part= 30 3 parts= 90 One quantity given2 The ratio of boys to girls in a class is 5:6. There are 15 boys. How many girls are there? 5 parts= 15 1 part= 3 6 parts= 18 Difference given3 Some pocket money is shared between Alice and Bob in the ratio 3:7. Bob gets £16 more than Alice. What was the total amount of pocket money shared? 4 parts= £16 1 part= £4 10 parts= £40 ? ? Alternately use a fractional approach. 3/8 of the animals are cats. 3/8 of 240 = 90 ?

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[Edexcel Nov 2012] Talil is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight. Talil wants to make 180 kg of concrete mix. Talil has 15 kg of cement 85 kg of sand 100 kg of gravel Does Talil have enough cement, sand and gravel to make the concrete mix? [Edexcel Nov 2012] Talil is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight. Talil wants to make 180 kg of concrete mix. Talil has 15 kg of cement 85 kg of sand 100 kg of gravel Does Talil have enough cement, sand and gravel to make the concrete mix? Test Your Understanding 9 parts = 180kg 1 part (cement) = 20kg 3 parts (sand) = 60kg 5 parts (gravel) = 100kg Insufficient cement. 1 ? I need to make 300ml of squash, which consists of 1 part concentrate to 5 parts water. How much water do I use? 2 6 parts = 300ml 1 part = 50ml 5 parts = 250ml The ratio of red to blue marbles in a bag is 7:4. If there are 49 red marbles, how many blue marbles are there? 3 7 parts = 49 1 part = 7 4 parts = 28 Frostonia has a population 10,000 more than Yusufland. If the ratio of people in Frostonia to Yusufland is 11:7, what is the population of Yusufland? 4 4 parts = 10,000 1 part = 2,500 7 parts = 17,500 ? ? ?

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RECAP: Map Scale Edexcel IGCSE Jan 2014 Method 1: 14cm is 14/4 = 3.5 times bigger. 1km x 3.5 = 3.5km Method 2: 4cm : 1km so 1cm : 0.25km So 14cm : 3.5km ? 1km = 1000 x 100 = 100,000cm 100,000 / 4 = 25,000 So ratio is 1 : 25,000 ?

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Test Your Understanding A map scale is 1 : Two towns are 14cm apart on the map. What actual distance does this represent in km? 1.4km A map scale is 1 : Two towns are 14cm apart on the map. What actual distance does this represent in km? 1.4km 5cm on a map represents 18km in real life. a) What does 8cm represent? 28.8km b) What is the map scale in the form 1:n? 1: cm on a map represents 18km in real life. a) What does 8cm represent? 28.8km b) What is the map scale in the form 1:n? 1: cm on a map represents 12km in real life. a) What does 20cm represent? 15km b) What is the map scale in the form 1:n? 1: cm on a map represents 12km in real life. a) What does 20cm represent? 15km b) What is the map scale in the form 1:n? 1: A map scale is 1 : If a supermarket is 3.2km away in real life, how close will it appear on the map? 6.4cm A map scale is 1 : If a supermarket is 3.2km away in real life, how close will it appear on the map? 6.4cm ? ? ? ? ? ?

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Similarity vs Congruence Two shapes are congruent if: They are the same shape and size (flipping is allowed) Two shapes are similar if: They are the same shape (flipping is again allowed) a a a b b b ? ?

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Similarity These two triangles are similar. What is the missing length, and why? ? There’s two ways we could solve this:

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Quickfire Examples Given that the shapes are similar, find the missing side (the first 3 can be done in your head) ? ? ? ?

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(Vote with your diaries) What is the length x? x 1 4 8

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(Vote with your diaries) What is the length x? x 4 8 9

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(Vote with your diaries) What is the length x? x

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Harder Problems In the diagram BCD is similar to triangle ACE. Work out the length of BD. ? 1 2 The diagram shows a square inside a triangle. DEF is a straight line. What is length EF? (Hint: you’ll need to use Pythag at some point) (both these questions were on past Year 9 landmarks) ?

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Harder Problems 3 4 A square is inscribed in a right- angled triangle as shown. What is the side-length of the square? ? ?

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Exercises ? ? ? ? ?

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A4/A3/A2 paper A4 A3 “A” sizes of paper (A4, A3, etc.) have the special property that what two sheets of one size paper are put together, the combined sheet is mathematically similar to each individual sheet. What therefore is the ratio of length to width? ?

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Scaling areas and volumes 3cm 2cm 6cm 9cm A Savvy-Triangle is enlarged by a scale factor of 3. Area = 3cm 2 Area = 27cm 2 Length increased by a factor of 3 Area increased by a factor of 9 ? ? ? ? ? ?

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Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example: A shape X is enlarged by a scale factor of 5 to produce a shape Y. The area of shape X is 3m 2. What is the area of shape Y? Bro Tip: This is my own way of working out questions like this. You really can’t go wrong with this method! Shape XShape Y Length: Area: 3m 2 75m 2 ? ? Example: Shape A is enlarged to form shape B. The surface area of shape A is 30cm 2 and the surface area of B is 120cm 2. If shape A has length 5cm, what length does shape B have? Shape AShape B Length: Area: 30cm 2 120cm 2 ? 5cm 10cm ? ?

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2160cm 3 Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example 3: Shape A is enlarged to form shape B. The surface area of shape A is 30cm 2 and the surface area of B is 270cm 2. If the volume of shape A is 80cm 3, what is the volume of shape B? Shape AShape B Length: Area: Volume: 30cm 2 270cm 2 ? ? ? 80cm 3 ?

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Test Your Understanding A B These 3D shapes are mathematically similar. If the surface area of solid A is 20cm 2. What is the surface area of solid B? Volume = 10cm 3 Volume = 640cm 3 640cm 3 Solid ASolid B Length: Area: Volume: 20cm 2 320cm 2 10cm 3 Answer = 320cm 2 ?

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Exercises 1 Copy the table and determine the missing values. Shape A Shape B Length: Area: Volume: 3cm 5cm 2 10cm 3 6cm 20cm 2 80cm 3 ? ? ? ? 2 Determine the missing values. Shape A Shape B Length: Area: Volume: 5m 8m 2 12m 3 15m 72m 2 324m 3 ? ? ? ? 3 Determine the missing values. Shape A Shape B Length: Area: Volume: 1cm 4cm 2 3cm 3 5cm 100cm 2 375cm 3 ? ? 4 Determine the missing values. Shape A Shape B Length: Area: Volume: 6m 8m 2 10cm 3 9m 18m cm 3 ? ? ? ? ? ? ? ? ? ? ? ? ?

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Test Your Understanding Bro Hint: Scaling mass is the same as scaling what? Volume ? ?

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