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1. Ratios Text Book : Section Reading : ( � ) 1

2. Ratios Mohamed is making orange squash. The instructions on the bottle state that he should use one part orange to four parts water. The ratio of orange to water will be 1:4 (1 to 4). In Maths, ratios are usually written in the form a:b. 2

3. Ratio However, they are often used in a more informal way, as you will see when you work through this Chapter, which covers: Equivalent ratios and simplest form Division in a given ratio Map scales 3

4. Equivalent ratios If the instructions state that you should mix one part orange to four parts water, then the ratio of orange to water will be 1:4 (1 to 4). If we use 1 litre of orange, we will use 4 litres of water (1:4). If we use 2 litres of orange, we will use 8 litres of water (2:8). 4

5. Equivalent ratios If we use 10 litres of orange, we will use 40 litres of water (10:40). Therefore, 1:4 = 2:8 = 10:40 Both sides of the ratio can be multiplied or divided by the same number to give an equivalent ratio. 5

6. Worked example Write 40:28 in its simplest form. Solution We look for a number that will divide into 40 and 28. 2 divides into both numbers, so 40:28 can be written as 20:14.But these two numbers also divide by 2, so we can write the ratio as 10:7 6

7. Solution Nothing divides into 10 and 7, so 10:7 is the simplest form of the ratio. Note: If you noticed that the Highest Common Factor of 40 and 28 was 4, then that's even better. You'd have gone straight from 40:28 to 10:7! 7

8. Division in a given ratio Worked example 1 Salah and Ali win O.R. 500 between them. They agree to divide the money in the ratio 2:3. How much does each person receive? Solution: The ratio 2:3 tells us that for every 2 parts Salah receives, Ali will receive 3 parts. There are 5 parts in total. 8

9. Solution O.R. 500 represents 5 parts. Therefore, by dividing both sides by 5, we see that O.R. 100 represents 1 part. Salah receives 2 parts, so Salah receives 2 x O.R. 100 = O.R. 200 Ali receives 3 parts, so Ali receives 3 x O.R. 100 = O.R. 300 9

10. Division in a given ratio Remember: The order of the ratio is very important. In this example, we saw that the ratio of Salah 's money to Ali 's was 2:3. Therefore, the ratio of Ali 's money to Salah 's would be 3:2 Worked example 2 Fatima and Samia also win a sum of money, which they agree to share in the ratio 5:3. If Fatima receives O.R. 150, how much money will Samia receive? 10

11. Solution We know that Julie receives 5 parts, and that this is equivalent to O.R. 150. Therefore O.R. 150 represents 5 parts. O.R. 30 (O.R. 150 divided by 5) represents one part. Samia receives 3 parts, so Samia will receive 3 X O.R. 30 = O.R. 90. 11

12. Sample question A necklace is made using gold beads and silver beads in the ratio 3:2. If there are 80 beads in the necklace: a) How many are gold? b) How many are silver? 12

13. Solution Gold : Silver = 3:2, so there are 5 parts altogether. 80 � 5 = 16, so 1 part represents 16 beads. a)Gold = 3 x 16 = 48 beads b)Silver = 2 x 16 = 32 beads 13

14. Map scales A scale drawing is a drawing in which all dimensions have been reduced in exactly the same proportion. For example, if a model boat is made to a scale of 1:20 (1 to 20), this scale can be applied to any units, so that 1mm measured on the model is 20mm on the actual boat, 1cm measured on the model is 20cm on the actual boat, and so on... 14

15. Worked example a) If the 1:20 model boat is 15cm wide, how wide is the actual boat? b) If the boat has a mast of height 4m, how high is the mast on the model? Solution: The scale is 1:20. This means that every cm on the model is equivalent to 20cm on the real boat. 15

16. Solution a) 1cm on the model = 20cm on the actual boat, so we calculate 15cm x 20 = 300cm. 15cm on the model = 300cm (3m) on the actual boat b) 20cm on the actual boat = 1cm on the model, so we calculate �400cm� = 20cm 20 400cm (4m) on the actual boat = 20cm on the model 16

17. Sample question Now try the following question: The scale of a map is 1:50 000 a) A distance is measured as 2cm on the map. i) How many cm is this equivalent to in real life? 1 cm on the map represents 50 000cm in real life. Therefore, 2cm on the map represents 100 000cm in real life. 17

18. Sample question ii) How many m is this equivalent to? To convert from cm to m, we divide by 100. 100000cm = 1000m iii) How many km is this equivalent to? To convert from m to km we divide by 1000. 1000m = 1km. 18

19. Sample question b) What distance on the map will represent 5km in real life?(Hint: Use your answers to part a) to help you!) In part a) we saw that 1km in real life was equivalent to 2cm on the map. Therefore, 5km in real life is equivalent to 10cm on the map. 19