Tiêu đề 02-Percentage and its Applications(1).pdf ✅

PERCENTAGE AND ITS APPLICATIONS CONTENTS • Ratio • Equivalent Ratio • Proportion • Percentage • Profit and Loss • Profit and Loss Percent • Simple Interest We can compare two quantities by two methods. 1. By finding the differences of their magnitudes : When we want to see how much more or less one quantity is than the other, we find the difference of their magnitudes and such a comparison is known as comparison by difference. 2. By finding the division of their magnitudes: If we want to see how many times more (or less) one quantity is than the other, we find the ratio (or division) of their magnitudes and such a comparison is known as the comparison by division. ➢ RATIO Ratio is the comparison by division of same kind of quantities or the ratio of two quantities of same kind and in same units is a fraction that shows how many times the one quantity is of the other. The ratio a is to b is the fraction b a , and is written as a : b. We call 'a' as the first term or antecedent and 'b' the second term or consequent. Note : 1. A ratio remains unchanged if both of its terms are multiplied by the same non-zero quantity. Let k  0, then clearly, (i) b a = kb ka and therefore a : b = ka : kb (ii) b a = b / k a / k and therefore a : b =       k b : k a 2. The ratio a : b is said to be in simplest form if HCF of a and b is 1. ❖ EXAMPLES ❖ Ex.1 Express 60 : 90 in its simplest form. Sol. In order to express the given ratio in its simplest form we divide its first and second term by their HCF. We have 60 = 2 × 2 × 3 × 5 90 = 2 × 3 × 3 × 5 So, HCF of 60 and 90 is 2 × 3 × 5 i.e., 30.  60 : 90 = 90 60 = 90 30 60 30   = 3 2 = 2 : 3 Hence, the simplest form of 60 : 90 is 2 : 3.  Comparison of Ratios In order to compare two given ratios, we express each of them in simplest form and then compare these fractions by making their denominators equal. Ex.2 Compare 5 : 12 and 3 : 5 Sol. Writing, the given ratio as fractions, we have 5 : 12 = 12 5 and 3 : 5 = 5 3 LCM of 12 and 5 is 60. 2 CHAPTER

Ex.7 In a computer lab, there are 3 computer for every 6 students. How many computer will be needed for 24 students ? Sol. 6 students have = 3 computers 1 student has = 6 3 computers 24 students have = 6 3 × 24 computers = 12 computers Hence, 24 students will be needed 12 computers Ex.8 Population of Rajasthan is 570 lakh and population of UP is 1660 lakh. Area of Rajasthan is 3 lakh km2 and area of UP is 2 lakh km2 . (i) How many people are there per km2 in both these state ? (ii) Which state is less populated ? Sol.(i) Population of Rajasthan = 570 lakh Area of Rajasthan = 3 lakh km2 .  Number of people in per km2 = 3 570 = 190 and population of U.P. = 1660 lakh Area of U.P. = 2 lakh km2 .  Number of people in per km2 = 2 1660 = 830. (ii) As population of Rajasthan per km2 is less than the population of U.P. per km2 so Rajasthan state is less populated. Ex.9 The daily pocket expenses of X and Y are −j 45 and −j 90 respectively. What is the ratio of their expenses in simplest form ? Sol. HCF of 45 and 90 = 45 Required ratio = 45 : 90 = 90 45 = 2 1 90 45 45 45 =   Hence, required ratio is 1: 2. Ex.10 Are 63 42, 33, 22 in proportion ? Sol. Let a = 63, b = 42, c = 33, d = 22. As product of extremes = 63 × 22 = 1386 Product of means = 33 × 42 = 1386. So, Product of extremes = Product of means Hence, 63, 42, 33, 22 are in proportion. Ex.11 The first, second and fourth terms of a proporiton are 217, 112, 32. Find the third term. Sol. Let the third term of the proportion be x. 217 : 112 :: x : 32 We know that if numbers in proportion, then product of means = product of extremes  112 × x = 217 × 32  x = 112 217 32 ; x = 62 Hence, the third term of the given proportion is 62. Ex.12 Express the ratio (i) 24 to 48 (ii) 12 cm to 1 m in their simplest form. Sol. (i) 24 to 48 = 48 24 = 2 1 (dividing both the numbers by 24) (ii) before comparing 12 cm and 1 m they must be expressed in the same unit.  1m 12cm = 1 100cm 12cm  = 100 12 = 25 3 So 12 cm : 1 m = 3 : 25 Ex.13 Express the following ratios in their simplest form : (i) 2 : 4 3 (ii) 7 6 : 14 15 Sol. (i) 2 : 4 3 = 2 × 4 : 4 3 × 4 (Multiplying both the numbers by 4) = 8 : 3 (ii) 7 6 : 14 15 = 7 6 ÷ 14 15 = 7 6 × 15 14 = 5 4  7 6 : 14 15 = 5 4 = 4 : 5 Ex.14 Which ratio is greater, 5 : 4 or 7 : 6 ?
Sol. To compare 5 : 4 and 7 : 6 we need to compare 4 5 and 6 7 so that we may express both of them with the same denominator.  4 5 = 4 6 5 6   = 24 30 and 6 7 = 6 4 7 4   = 24 28 Clearly, 24 30 > 24 28 or 5 : 4 > 7 : 6. Ex.15 A family has 15 pets of which 6 are cats or kittens, 3 are dogs and the rest are birds. Find the ratio of the numbers of (i) birds to dogs (ii) birds to pets Sol. (i) Total no. of pets = 15 No. of cats or kittens = 6 No. of dogs = 3 No. of birds = Total no. of pets – (No. of cats + No. of dogs) = 15 – (6 + 3)  15 – 9 = 6 So, the no. of birds = 6 There are 6 birds and 3 dogs. So, the number of birds : number of dogs = 6 : 3 = 2 : 1 (ii) There are 6 birds and 15 pets So, the number of birds : number of pets = 6 : 15 = 2 : 5 Ex.16 Find the missing numbers in the following ratios : (i) : 15 = 8 : 10 (ii) 10 15 4 = Sol.(i) Let the missing number be x. therefore, x : 15 = 8 : 10  10 8 15 x =  x = 10 15 8 = 2 38 = 3 × 4 = 12 (ii) Let the missing number be x Therefore, 4 x = 10 15  x = 10 154 = 10 60 = 6 Ex.17 Two length are in the ratio 3 : 7. The second length is 42 cm. Find the first length. Sol. Let the first length be x cm. Then we write the ratio of the length as x : 42; but it must be equal to the given ratio 3 : 7  3 : 7 = x : 42  42 x = 7 3 x = 7 3 × 42 = 1 3 6 = 18 Hence, the first length is 18 cm. Ex.18 In a class of 60 pupils the ratio of the number of boys to the number of girls is 7 : 8. How many boys and girls are there ? Sol. Given that 7 are boys and 8 are girls so they are 15 together. Therefore, boys are 7 out of 15, i.e., 15 7 of 60. and girls are 8 out of 15, i.e. 15 8 of 60.  The number of boys = 15 7 of 60 = 15 7 × 60 = 7 × 4 = 28 and the number of girls = 15 8 × 60 = 8 × 4 = 32 Check : 28 + 32 = 60 Ex.19 Divide −j 2600 amongst three people so that their shares are in the ratio 4 : 5 : 4. Sol. Given ratio is 4 : 5 : 4 Now sum of the rations = 4 + 5 + 4 = 13 Therefore, the share of first person is 4 out of 13. i.e., 13 4 × −j 2600 = 4 × −j 200 = −j 800 Similarly, the share of the second person is 5 out of 13. i.e., 13 5 × −j 2600 = 5 × −j 200 = −j 1000 and the share of the third person is 4 out of 13 i.e., 13 4 × −j 2600 = 4 × −j 200 = −j 800 Check : −j 800 + −j 1000 + −j 800 = −j 2600