CHAPTER 5 RATIOS, RATES AND PROPORTIONS

CHAPTER 5 RATIOS, RATES AND PROPORTIONS

5.1 Ratio of two quantities

5.1a Comparison two quantities in the form of a : b or

(i). The ratio of two quantities is the comparison between two quantities of the same kind or same unit.(ii) There is no unit of measurement given when expressed in rations.(iii) Ratio of two quantities is always in the form of a : b or

which is read as ratio of a to b 1. Write the ratios in the form a : b or Ratio

a : b

(a) 4 to 7

(b) 12 : 36

(c) 12 cm to 5 cm

(d) 42 g to 29 g

(e) 60 sen to RM 5

(f) 1 kg to 800 g(g) 17 sec to 1 min(h) 100 cm : 0.9 m(i) 5 days : 2 weeks

(j) 450 ml : 2 liters5.1b Simplifying ratios to the lowest terms

(i) Ratios must be stated in their lowest terms.2. Simplify each ratio to the lowest terms.(a) 4:12 (b) 36 : 27

(c) 121:396 (d) 1.5 : 2.5(e) 0.004 : 0.100 (f) 0.25 : 40(g) : (h) :

(i) : (j) : 5.1c Determining whether given rations are equivalent ratios(i) To determine whether or not ratios are equivalent, multiply or divide the two quantities in the given ratio by the same factor.(ii) The ratio of two quantities does not change if both the quantities are multiplied or divided by the same factor.

3. Determine whether the following pairs of ratios is equivalent ratios.(a) 3 : 12 and 6 : 24(b) 6 : 5 and 20 : 24

(c) 100 : 44 and 25 : 11

(d) : and 4 : 12

(e) : and :

(f) and

5.1d To state ratios related to a given ratio4. Complete the table below

m : n

n : m

m : n + m

n : m + n

m : m n

n : m n

5 : 2

8 : 3

3.5 : 2.5

:

:

:

5. Refer to the figure below and write down the ratios.

(a) The ratio of the perimeter of rectangle ABCD to the perimeter of triangle ADE

(b) The ratio of the area of rectangle ABCD to the area of triangle ADE

6. Solve the following problems(a) The ratio of male students to female students in a class is 4 : 5. If the difference between male students and female students is 7, what is the total number of students in the class?(b) A hawker sold x apples and y oranges. If the ratio of apples to oranges is 4 : 7 and the total number of apples and oranges is 77, calculate the number of oranges sold.5.2 Concept of ProportionA proportion is a relationship between two ratios whereby the first ratio is equal to second.

5.2a To state whether two pairs of quantities are a proportion

(i) When the ratios of two pairs of quantities are the same, the quantities are in proportion.

(ii) For example c : d = m : n is read as c is to d equals m is to n 7. State whether the following pairs of quantities are proportional.(a) 2 : 5 ; 18 : 45

(b) : (c) : ; 15m : 0.0004 km

(d) kg : 700g

(e) day : 14 hours

5.2b To determine if a quantity is proportional to another quantity given two values of each quantity.

(i) Two quantities are proportionate if the ratios of both pairs of quantities are the same.

8. Determine whether each of the following pairs

(a)

(b) The price of 4 shirts is RM64 and the price of 6 shirts is RM102.

(c) 2 kg of rambutan costs RM5 and 10 kg of rambutan costs RM25.

(d) 100 g of chocolate costs RM5. Another 150 g chocolate costs RM6.

(e) A rectangular of width 3 cm and length 4 cm. Another rectangle of width 12 cm and length 16 cm

5.2c To find the value of a quantity when the ratio of the two quantities and the value of another quantity.

Two methods that can be used.

(a) Unitary method

(b) Proportion method ( or Cross multiplication method)

9. Evaluate each of the following using unitary method(a) Ratio of x : y is 3 : 5 . If the value of x = 24 kg, find the value of y.Ans. x ( 3 parts = 24 kg

1 part = 24 3 kg

= 8 kgThus, y = 5 parts = 5 x 8 kg

= 40 kg

(b) Given u : v = 4 : 3 and the value of u = 20 cm. what is the value of v?(c) Given that h : k = = 39 : 12, find the value of h if k = 4.

(d) Ratio of a : b is 2 : 9 . If the value of b is RM45, what is the value of a(e) Given that p : q = 6 : 5, find the value of p if q = 40

10. Solve the following using the proportion method / cross multiplication method(a) x : 5 = 32 : 40

40x = 160 x = 4

(b) m : 5 = 40 : 25

(c) 12 : 7 = 96 : x

(d) Given that p : q = 7 : 10, find the value of q if p = 42

(e) Given that x : y = 2 : 5 4, find the value of y if x = 8

(f) Given 5 : 4 = R : 28. Find the value of R

5.2d Finding the value of a quantity given the ratio and the sum of the two quantities.11. (a) Given the a : b = 7 : 9 and a + b = 112, find the value of a and b(b) Given that x : y = 4 : 7 and x + y = 99, find the value of y(c) Given that f : g = 25 : 7 and f + g = 128, find the value of f and g.

(d) Given that j : k = 11 : 5 and j + k = 1328, find the value of j and k.(e) Given that x : y = 4 : 3 and x + y = 28, find the value of x and y

5.2e Find the sum of two quantities given the ratio of the quantities and the difference between the quantities.

12. (a) Given e : f = 5 : 6 and f e = 40, find the value of e + f(b) Given p : q = 5 : 3 and p q = 56, find the value of p + q

(c) Given that w : v = 23 : 11 and w v = 48, find the value of w + v.

(d) Given that s : t = 15 : 6 and s t = 81, find the value of s + t(e) Given x : y = 5 : 8 and y x = 39, find the value of y + x.

13. My father gave RM68 to John and me based on the ratio 3 : 1. Calculate amount of money received by John.

14. Three similar ribbons have a total length of 105 cm. Find the total length of 11 such ribbons.

15. Each of the 3 blades of a particular fan makes 14 revolutions per second. How many revolutions does each of the blades make per minute?

16. RM76 was divided into 2 portions in the ratio 4 :12. Find the value of the smaller portion17. School A and school B raised funds fir an orphanage. The ratio of funds raised by school A to the raised by school B is 7 : 5. If the difference between the funds raised by both schools is RM400, calculate the total funds raised by both schools

18. The ratio of Samys age to his brothers age is 5 : 4. If the sum of their ages is 27 years. What is his brother age?

19. Melanie had RM65. She spent it on books and clothes in the ratio of 1 : 4. How much did she spend on clothes?

20. A sum of RM390 is to be shared by Alan and his brother in the ratio of 17 :13. Find the amount received by Alans brother.

21. Tasta was given 2057 stamps by her uncle. She shared the stamps with her sister in the ratio of 8 : 3. If her sister got the bigger share, how much more did her get?

22. For rectangle, the ratio of its width to its length is 2 : 7. Given that the width is 30 cm less than its length, calculate the perimeter of the rectangle.

23. The number of sweets in 2 containers is in the ratio 8 : 5. If the difference between the numbers of sweets in the 2 containers is 120, calculate the total number of sweets in the 2 containers.24. A wire is cut into two pieces in the ratio of 5 : 8 . Given that the difference between the two pieces is 21 cm, find the length of the shorter piece of wire.

25. Encik Murad and his wifes monthly income are in the ratio of 5 : 3 . Given that Encik Murads income is RM478 more than his wifes, calculate his wifes income.

26. Lily and her sister, Mimi, received their ang paus in the ratio of 7 : 5. If the difference between the amounts they received is RM150, calculate the total amount that the siblings received.27. A dough is to be prepared by combining wheat flour and rice flour in the ratio of 3 : 2. Given that the difference in weight between these two types of flour used is 150 g, find the total weight of these two types of flour used to prepare the dough.

5.3 Concept of ratio of three quantities Three quantities of same kind can be compared as ratio

The order of the quantities is important

The units of measurement must be the same.

5.3a Comparing three quantities in the form a : b : c 1. Complete the table below

a

bca : b : c4

6

7 3 m6 m9 mRM4RM6RM730 cm1 m40 cm500 g1.5 kg100 g18 sec0.25 min4 sec5.3b Simplifying the ratio of three quantities to the lowest terms. - The ratio of three quantities can be simplified to its lowest term.2. Simplify each of the following ratios to its lowest terms.(a) 9 : 27 : 18 (b) 1.6 : 4 : 2.4

(c) 1 2: 2 : 1 3 (d) 3 4: 1 16: 1 2

5.3c Determining whether given ratios are equivalent ratios.3. Determine whether the given ratios are equivalent

(a) 3 : 5 : 7 and 15 : 25 : 35

(b) 32 : 24 : 40 and 4 : 3 : 5 (c) 1.2 : 2 : 2.8 and 6 : 5 : 7

(d) 62 5: 9.6 : 16 and 2 : 3 : 5(e) 11 3: 2 6: 1 12 and 16 : 2 : 1

4. Complete the table below

a : b : cabc2 : 1 : 5

6

0.4 : 0.8 : 0.016

1 2: 1 4: 3 84

5.3d Stating the ratio of any two quantities given the ratio of the three quantities.- The ratio of any two quantities can be identified if the ratio of all three quantities is given.5. Complete the following table.

a : b : ca : b a : c b : ca : a + b + ca + b : a c 3 : 1 : 96 : 5 : 846 : 72 : 150.6 : 0.2 : 041 5: 4 25: 3 20

5.3e Identifying and calculating the ratio of a : b : c given the ratio a : b and a : cExampleGiven that a : b = 4 : 5 and b : c = 6 : 7 , calculate a : b : c

a : b : c .

4x 6 5x6 6x 5 7x5 24 : 30 : 35

a : b : c = 24 : 30 : 35

6. Find the ratio of a : b : c, given the ratio of two quantities(a) a : b = 5 : 8 and b : c = 8 : 12

(b) a : b = 3 : 7 and a : c = 3 : 12

(c) a : c = 4 : 6 and b : c = 5 : 6

(d) a : b = 3 : 5 and b : c = 7 : 4

(e) a : b = 4 : 13 and a : c = 7 : 8

(f) a : c = 6 : 11 and b : c = 5 : 2(g) a : c = 12 : 7 and b : c = 11 : 17

5.3f The value of the other quantity, given the ratio of three quantities and the value of one of the three quantities- When the value of one of the three quantities and the ratio of the three quantities are given, the other values can be calculated.7. Solve each of following(a) Given that a : b : c = 3 : 5 : 8 and a = 15, find the value of b and c

(b) Given that p : q : r = 36 : 54 : 18 and q = 15, find the value of p and r

(c) Given that x : y : z = 15 : 45 : 17 and z = 85, find the value of x and y

(d) Given that k : m : n = 8 : 13 : 17 and k = 48, find the value of m and n

(e) Given that u : v : w = 54 : 108 : 81 and w = 3, find the value of u and v

5.3g The value of each quantities given

(a) the ratio and the sum of three quantities.(b) the ratio and the difference between two of the three quantities.8. Solve each of the following(a) Given that p : q : r = 3 : 5 : 7 and p + q + r = 120 kg, find the value of p, q and r

(b) Given that k : m : n = 12 : 3 : 8 and k + m + n = 69, find the value of k, m and n

(c) Given that a : b : c = 3 : 4 : 6 and c b = RM32, find the value of a, b and c

(d) Given that x : y : z = 3 : 8 : 15 and y + z = 69, find the value of x, y and z(e) A wire measuring 60 cm is cut into 3 pieces in the ratio 3 : 4 : 5 . Calculate the length of the shortest piece of wire.

5.3h The sum of three quantities given the ratio and the difference between two of the three quantities.9. Solve each of the following

(a) Given that x : y : z = 2 : 5 : 9 and z x = 35, find the value of x + y + z

(b) Given that a : b : c = 2 : 8 : 3 and b a = 42, find the value of a + b + c

(c) Given that p : q : r = 12 : 5 : 3 and p r = 70, find the value of p + q + r

(d) Given that u : v : w = 3 : 4 : 8 and w u = 6, find the value of u + v + w

(e) Given that k : m : n = 3 : 5 : 9 and n m = 28, find the value of k + m + n

Exercise 11. A rope is cut into three pieces in the ratio 4 : 5 : 10 . If the difference between the longest and the shortest piece is 42 m, find the total length of the rope.

2. The ages of Aishah, Salina and Ester are in the ratio 4 : 7 : 10. If the difference in age between Salina and Ester is 9 years, find Aishahs age

3. The triangle ABC has three angles a, b and c. The ratio a : b : c = 3 : 5 : 4. What is the value of angles a, b and c respectively?

4. In a mathematics test, the marks obtained by Aznil, Kim Lee and Joseph are in the ratio 6 : 5 : 4. Given that the difference between the highest mark and the lowest marks is 40, calculate their total marks.

5. The distance from town A to town B is 32 km. If the ratio of the distance of town A to town B and the distance of town B to town C is 4 : 5, calculate the distance from town B to town C : 3

6. A sum of money is shared between Azhar, Bakar and Amin in the ratio 5 : 17 : 18. If the largest share is RM162, find the difference between the largest and the smallest shares.11. Chong, Hanif and Muthu invested in a stall for the school Canteen Day in the ratio 2 : 3 : 4. The total amount invested by Chong and Hanif was RM415, How much did muthu invest in?

7. A Sum of money is divided among Sathi, Suresh and Kamal in the ratio 6 : 5 : k. If Sathi received RM54 and Kamal receives RM 36, find the value of k 12. A photography competition offers RMy as prize money. Given that the prize money is divided the first, second and third prize winners in the ratio 9 : 5 : 2 and the third prize winner gets RM520 lesser than second prize winner, find the sum received by the first prize winner.

8. The ratio of Jamess Salary to Mohans salary is 3 : 5 and the ratio of Mohans salary to Peters salary is 4: 7 Find the ratio of Jamess salary to Peters salary.13. Three baskets, P, Q and R, contain rambutans in the ratio 5 : 4 : 3. Given that the total number of rambutans in Basket p and Q is 42 more than the number of rambutans in basket R, calculate the total rambutans in the three baskets.

9. The ratio of the number of books in the box P and box Q is in the ratio 3 : 7. The ratio of the number of books in box Q and box R is in the ratio of 2 : 1. Given that there are 12 books in box P, find the total number of books in boxes P, Q and R14. The triangle ABC has three sides. Given that the length of sides AB, BC and AC is in the ratio of 3 : 4 : 5 and the total length of sides AB and BC is 14 cm, find the perimeter of the triangle.

10. Ahmad and Bakar each has RM210 and RM150 respectively. The ratio of Chias money to Bakars money is 5 : 6. Find the ratio of Ahmads money to Bakars money to Chias money.15. The price of a pair of shoes, a shirt and a storybook is in the ratio of 12 : 35 : w. Given that the difference the price of the pair of shoes and the shirt is RM26 and the total cost of the three items is RM90, find the value of w

A

B

C

E 3 cm D

5 cm

= 4 cm

=

4 cm

|

|

P

25

27

q

25

45

=

x 5 32 40

Or

LCM of 5 and 6

2 5, 6

5, 3

5, 1

1, 1

LCM = 2 x 3 x 5

= 30

PAGE 7K C KHOO F2 Chapter 5 Ratios, Rates and Proportions 07

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