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Rate and Ratio

1. Rate and Ratio. Case Study. 1.1 Rate. 1.2 Ratio. 1.3 Applications of Ratios. Chapter Summary. Case Study. Mandy is planning to study abroad next year. She wants to compare the school fees among 3 different countries.

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Rate and Ratio

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  1. 1 Rate and Ratio Case Study 1.1 Rate 1.2 Ratio 1.3 Applications of Ratios Chapter Summary

  2. Case Study Mandy is planning to study abroad next year. She wants to compare the school fees among 3 different countries. The table below shows the school fees in different currencies and their corresponding exchange rates. The exchange rate is the amount of Hong Kong dollars needed to exchange for one unit of a foreign currency. We can use the exchange rate to convert the foreign school fees into Hong Kong dollars first, and then compare them.

  3. ‘/’ means per. 1.1 Rate Rate is the comparison of 2 quantities of different kinds. In the figure, the price of milk per carton is found by comparing 2 different kinds of quantities: ‘the total price’ and ‘the number of cartons’. When expressing the relationship in rate, we express the amount of one quantity as per unit of the other quantity, by the symbol ‘/’. For example, the price of each brand of milk can be expressed as ‘$8/carton’ and ‘$7/carton’ respectively.

  4. Example 1.1T 1.1 Rate Andy works as a programmer in a computer company. He earns a total of $51 000 in half a year. Find his income in the following units: (a) $/month (b) $/year Solution:

  5.  15 minutes  720 m  5.4 km/h 1.1 Rate Example 1.2T Betty runs at a speed of 1.5 m/s. (a) Express her speed in the unit km/h. (b) How long does she take to run 1350 m? (Give the answer in minutes.) (c) How far can she run in 8 minutes? (Give the answer in m.) Solution: (a) 1.5 m  (1.5  1000) km  0.0015 km (b) Time required  (1350  1.5) s  900 s  (900  60) minutes (c) Distance run  1.5  (8  60) m

  6.  9.94 HKD/EU  $8450.45 For higher accuracy, we use 9.9417 as the exchange rate in the calculation in part (b). 1.1 Rate Example 1.3T Mrs. Wong gets 240 Euros (EU) for HKD 2386. (a) Find the exchange rate in the unit HKD/EU. (b) How much Hong Kong dollars can she get with 850 Euros? (Give the answers correct to 2 decimal places.) Solution: (a) Exchange rate  HKD 2386  EU 240  9.9417 HKD/EU (cor. to 2 d. p.) (b) Amount of Hong Kong dollars she can get  $(850  9.9417) (cor. to 2 d. p.)

  7. 1.2 Ratio Ratio is the comparison of quantities of the same kind. The ratio of a to b is usually expressed as a : b or (where a 0 and b 0). We say that the ratio of the volume of orange juice to that of the soft drink is 2 : 1. This can be also written in the form . A. Basic Concepts of Ratio In the figure, a fruit punch is mixed by adding a cup of soft drink into 2 cups of orange juice. That means, the volume of orange juice in the fruit punch is always twice that of the soft drink. We compare 2 quantities by division: ‘the volume of orange juice’ and ‘the volume of soft drink’ and these quantities are of the same kind.

  8. For example, 0.75 m : 40 cm  75 cm : 40 cm 1.2 Ratio A. Basic Concepts of Ratio A ratio is usually expressed in its simplest form, e.g. 75 : 40  15 : 8. A ratio can be written as a fraction and we know that the value of the fraction remains unchanged when we multiply (or divide) both the numerator and the denominator by the same non-zero number.  15 : 8

  9. m 3 1.2 Ratio A. Basic Concepts of Ratio Example 1.4T If m : 4  (m 6) : 12, find the value of m. Solution: m : 4  (m 6) : 12 12m 4(m 6) 3mm 6 2m 6

  10. Note that a : bb : a.  3 : 4  2 : 3 1.2 Ratio A. Basic Concepts of Ratio Example 1.5T Peter has 18 coins. Nancy has 6 more coins than Peter, and she has twice as many as Stella. Find the ratio of (a) Peter’s coins to Nancy’s coins, (b) Stella’s coins to Peter’s coins. Solution: Number of coins that Nancy has 18  6  24 Number of coins that Stella has  24  2  12 (a) Required ratio  18 : 24 (b) Required ratio  12 : 18

  11. 1.2 Ratio A. Basic Concepts of Ratio Example 1.6T Emily bought a bottle of apple juice of volume 650 mL. She pours the juice into 2 cups such that the volumes of juice in these cups are in the ratio 6 : 7. Find the volume of juice in these 2 cups. Solution: Since the volumes of juice are in the ratio 6 : 7, we can imagine that the bottle of apple juice is divided into (6  7)  13 equal parts. Volume of the cup with less juice Volume of the cup with more juice

  12. We can compare the ratios directly, without finding the exact value of each small part. 1.2 Ratio A. Basic Concepts of Ratio Example 1.6T Emily bought a bottle of apple juice of volume 650 mL. She pours the juice into 2 cups such that the volumes of juice in these cups are in the ratio 6 : 7. Find the volume of juice in these 2 cups. Alternative Solution: If we divide the juice into 13 parts, then 6 parts belong to the cup with less juice and the other 7 parts belong to the cup with more juice. Volume of the cup with less juice Volume of the cup with more juice

  13. 1.2 Ratio A. Basic Concepts of Ratio Example 1.7T Educational Secondary School has a total of 57 teachers, of which 27 of them are male teachers. (a) Find the ratio of the number of male teachers to the number of female teachers. Solution: (a) Number of male teachers : Number of female teachers  27 : (57  27)  27 : 30

  14. 1.2 Ratio A. Basic Concepts of Ratio Example 1.7T Educational Secondary School has a total of 57 teachers, of which 27 of them are male teachers. (b) The principal has just hired 8 new teachers. The ratio of male teachers to female teachers now becomes 6 : 7. How many female teachers has the principal hired? Solution: (b) Let x be the number of female teachers hired. Number of male teachers hired  8 x. [27  (8  x)] : (30  x)  6 : 7 6(30  x)  7(35  x) 180  6x  245  7x 13x  65 x  5 6(30  x)  7(35  x) ∴ 5 female teachers has been hired.

  15. Continued ratios can only be expressed in the form a : b : c, but not in a fraction. 1.2 Ratio B. Continued Ratio We can also use ratio to compare 3 or more quantities of the same kind. For example, the expression a : b : c 4 : 5 : 9 compares the 3 quantities a, b and c, with a : b 4 : 5, b : c 5 : 9 and a : c 4 : 9. Such an expression is called a continued ratio. For 3 quantities given, if we only know the ratio between individual quantities, we can rewrite the ratios into a continued ratio.

  16. 1. First, find the ratios a : b and b : c. 2. Then, make the common terms equal in both ratios. ∴ and  5  4 : 3  4  4  3 : 5  3 1.2 Ratio B. Continued Ratio Example 1.8T If 3a 5b 4c, find the ratio a : b : c. Solution: Since 3a 5b 4c, we have 3a 5b and 5b 4c. ∴ a : b  5 : 3 and b : c  4 : 5 a: b 5: 3 b:c  4:5  20: 12  12 : 15

  17. Required ratio  24 : 2 : 1 1.2 Ratio B. Continued Ratio Example 1.9T There are 540 seats in a plane. The number of economy class seats and business class seats are in the ratio 12 : 1. The number of business class seats and first class seats are in the ratio 2 : 1. (a) Find the ratio of the number of economy class seats : the number of business class seats : the number of first class seats. (b) Find the number of first class seats. Solution: (a) Economy: Business  12 : 1 Business:First  2 : 1  24: 2  2 : 1

  18. 1.2 Ratio B. Continued Ratio Example 1.9T There are 540 seats in a plane. The number of economy class seats and business class seats are in the ratio 12 : 1. The number of business class seats and first class seats are in the ratio 2 : 1. (a) Find the ratio of the number of economy class seats : the number of business class seats : the number of first class seats. (b) Find the number of first class seats. Solution: (b) We can imagine that the total number of seats can be divided into (24  2  1)  27 equal parts. Number of first class seats

  19. 1.3 Applications of Ratios A. Similar Figures If 2 figures have the same shape but their sizes are not the same, then the 2 figures are said to be similar. If we compare the lengths of the corresponding sides in the 2 photos, we will have: In general, similar figures have the following property: For 2 similar figures, the ratios of the corresponding sides are always the same.

  20. When finding the side lengths of similar figures, we should identify which of them are the corresponding sides. x  2.5 (m) y  16 (m) 1.3 Applications of Ratios A. Similar Figures Example 1.10T In the figure, the 2 parallelograms are similar to each other. Find x and y. Solution: 6x  15 3y  48

  21. 1.3 Applications of Ratios A. Similar Figures Example 1.11T In the figure, a boy with a height of 1.8 m stands in front of the tree. Assume that DABC and DDEF are similar triangles. What is the length of his shadow? Solution: Let y m be the length of his shadow, i.e., EF y m. ∴ The length of his shadow is 0.45 m.

  22. Note that 1 : nn : 1. 1.3 Applications of Ratios B. Scaling If we want to draw something which is very large or small in size, such as a country or an insect, we need to reduce or enlarge it according to a specified ratio in a diagram. This kind of drawing is called scale drawing. When using scale drawing, we need to specify the ratio in which the object is enlarged or reduced in the picture. This ratio is called the scale of the drawing, and is usually represented in the form 1 : n or n : 1.

  23. We can also express the scale in the form 1 cm : 15 km. 1.3 Applications of Ratios B. Scaling For example, the map of Hong Kong Island shown has a scale of 1 : 1 500 000. This means that a length of 1 cm on the map represents an actual length of 1 500 000 cm. In the figure, a length of 1 cm on the figure represents an actual length of 0.2 cm. Thus the scale is 1 : 0.2, i.e., 5 : 1.

  24. 1.3 Applications of Ratios B. Scaling Example 1.12T The picture on the right shows the top view of a tennis court of actual length 36 m. If the length of the picture is 4.8 cm, find the scale of the picture. Solution: Scale of the picture  4.8 cm : 36 m  4.8 cm : 3600 cm

  25.  0.64 km 1.3 Applications of Ratios B. Scaling Example 1.13T Consider a map of a city with a scale of 1 : 20 000. If the distance between 2 buildings is 3.2 cm, find the actual distance between them. Give the answer in the unit of km. Solution: Actual distance  (3.2  20 000) cm  64 000 cm  640 m

  26.  25 : 12 1.3 Applications of Ratios B. Scaling Example 1.14T According to the floor plan, find the ratio of the actual area of the master bedroom to the actual area of the kitchen. (Hint: Assume the scale of the floor plan to be 1 cm : n m.) Solution: Actual side length of the master bedroom  (2.5 n) m  2.5n m Similarly, the actual length and the actual width of the kitchen are 2n m and 1.5n m respectively. ∴ The required ratio  (2.5n  2.5n) m2 : (2n 1.5n) m2  6.25 : 3

  27. Chapter Summary 1.1 Rate Rate is the comparison of 2 quantities of different kinds.

  28. Chapter Summary • Ratio is the comparison of quantities of the same kind. The ratio of • a to b is usually expressed as a : b or (where a 0 and b 0). 1.2 Ratio • If the ratios a : b and b : c are given, we can find the continued ratio a : b : c by finding the L.C.M. of the values corresponding to the common term b.

  29. Chapter Summary 1.3 Applications of Ratios 1. Similar figures For 2 similar figures, the ratios of the corresponding sides are always the same. 2. Scale drawing If we reduce or enlarge the drawing of the real object by a certain scale, the drawing is similar to the original object.

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