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Direct and, verse, "portions =, a“, , , , results tn q corresponding chan, bein variation,, , There are Many situations in our daily life where the variation in one quantity brings, @ Variation in the other., , EXAMPLES, , (1) More articles will cost more., (ii) More is the money, Period., , the work,, , Direct Proportion (or Direct Variation) Two quantities, , x and y are said to be in direct Proportion, if whenever the value Of x increases (or, , decreases). then the value of y increases, x, (or decreases) in such @ way that the rane remains constant., , Thus, x and y are in direct proportion, ¥ =k, where k isa : te, Sb 2 _ Xe, ¥ U2 Us, Examples (i), (ii) and (iii) given above are the cases of direct proportion., , , , REMARK When x and y are in direct proportion, we also say that x and y have a direct, variation., , 158

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y, |, |, , i tie, SOLVED EXAMPLES, , 1. Observe the ., , aut a —— aes sven below and fin whether xand yar directly proportional:, , 315 7, , 719] 12, , — 10 | 4 | 18 | 2, mt 4 7 | 12, , 15, , Ly Tatar 36 60, , goutin i) Clearly, * 28.5 7 9 12 1, , y 6 10 ia" ig" a7 Coat), , “ Xand y are dir, (ti) We have- ectly proportional., , 4.7 21, =~ 15, 12° 21 36 “35, . 2.4712 8, y 12° 213660., , . and y are not directly proportional., , puter If x and y are directly proportional, find the values of x,, x, and y, in the table, given below,, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , i, 4, , , , x 3 x x, | 10, y | 3) 0 | 6) y, , seution Since x and y are directly proportional, we have:, , , , , , , , , , , , , , , , , , 3 4410, 36 60 96 y,, , Now, 5 = 2 = A xs -{ 500) 5., bee deel), ze? ie 12x10) = y, =120., , x, =5, x, =8 and y, =120., BAWPLEa. A car covers 432 km in 36 litres of petrol. How much distance would tt cover in, 25 litres of petrol?, Solution Let the required distance be x km. Then. we have:, Quantity of petrol {in litres) 36 25, Distance (in km) 432 x, Clearly, less is the quantity of petrol consumed, less is the distance covered., So, it is a case of direct proportion., , 9625 tS iexty= 1225) > x=300., "492 x «12 x, & required distance is 300 km., , , , , , , , , , , , , , , , , , , , —_—t T

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= Mathematics for Class 8, , chased for® 17839, {50 metres ofa cloth cosis® 3725, how much loth can be P, , , , , , , , , , , , , , , , , , Solution Let the required length of cloth be x metres. Then, we have:, [Length of cloth (in metres) | 50 | x, Its cost (in rupees) 3725 | 1788, Clearly. less is the cost, less is the length of cloth., SO. 118 eas of direct proportion. _1788x2, j 3705 “Tas a iene = (49x x) =1788%2) > X= yg, * Tequired length of cloth = 24 m., EIAMPLE:, I the weight of § sheets of thick paper is 30 grams, how many sheets Of the same, Paper would weigh ktagrams?, Solution Let the required number of sheets be x., 17 ($1000) gums = 1250 guns, Thus, we have;, Number of sheets 9 x, , , , , , , , , , , , | Weight of sheets {in grams) 30 1250, , More is the weight, more is the number of sheets., , So. it is a case of direct proportion., o « o& 3, . a ~X =|— = 375., 30 “1250 ~ 10 “i250 > * (7510), Hence, the required number of sheets is 375., , EXAMPLE 6, A train is moving at a uniform speed of 75 kmvhr., () How far will it travet in 24 minutes?, (t) In how much time will it cover 175 km?, , Solution Let the distance covered by the train in 24 minutes be x km and let it cover 175 km, in y minutes., , Then, we have:, , Distance covered (in km) 75 x | 1%, Time taken (in minutes) 60 | 24 y, , Since the speed is uniform, more distance will be covered {n more time., , So, it is a case of direct proportion., 7% _ x (17%, , , , , , , , , , , , , , , , , , , , :, 24 | = 30,, ny, , “distance covered in 24 minutes is 30 km.

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Direct and Inverse Proportions, , (tt) Algo, 23. 175, 60, , L73x4, , » yo 140., , * lime taken to cover 175 km is 140 min, Le., 2 hours 20 minutes., , 3 7, y => 7; eo, , 161, , guwrteT. A vertical pole 5 m 60 em high casts a shadow 3 m 20 cm long, Find at the same, , ie (0) the length of shadow cas by another pole 10 m 80 em high, (i he height, of a pole which casts q shadow 5 m long., , Let the required length of shadow be x cm and the required height of the pole be, , solution, , Solution, , , , , , , , ycm. Then, we have:, Height of the pole (in em) | 560 |1060| _y, (Length of its shadow (in em) | 320 | x | 500, , , , , , , , , , , , More is the height of pole, more is the length of its shadow., , So, it ts a case of direct proportion., , (i) Now, 262 _ 1050 _ 7 _ 1050, 320, , x 4 x, , => 7x=[4«1050) > x=, , .. the required length of shadow = 600 cm = 6 metres,, , (i) And ee -8- 78, 320 500 ~ 4 500, , => u (£500 =675., , “the required length of pole = 875 cm = 8 m 75 cm., , , , 41050 _, 7, , The scale of a map is 1:3x10". Two cities are 5 cm apart on the map. Find the, actual distance between them in Idlometres., , Let the required distance be x cm. Then, we have:, , , , Distance on the map (in cm), , 1, , 5, , , , , , Actual distance (in cm), , , , 3x1, , , , 0" x, , , , , , , , More is the distance on the map, more ts the actual distance., , So, it is a case of direct proportion., 1, , , , 22S (xxl)=5x3x107 > x=05x107), , , , “3x0” x, 15x10, = 115x107} em =| ———_, Required actual distance = 05 x10"} em (ies|, _{ 15x10"), 15x10" |, lio? 10" | 10° |, , = (15 «10*) km = 15 x100) km = 1500 km,, Hence. the actual distance between the two cities is 1500 km., , |

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= Mathematics for Class 8, , EXAMPLE 9, 5 men or 7 women can earn & 875 per day, how much, 5 women earn per day?, , would 10 men ang, , Solution 5 men = 7 women, , => 1 man = 2 women, , 7, => 10men =($+10) women = 14women, , = (10 men + 5 women) =(14 women + 5 women) = 19 women., , Let 10 men and 5 women would earn @ x per day. Then, we have:, , Number of women 7 19, , Earning per day (in rupees) | 875 é, , Clearly, more women will earn more per day., , So. il is a case of direct Proportion., , : we > me = (xx) =(125%19) = x= 2375., 10 men and 5 women would earn ¥ 2375 per day., , , , , , , , , , , , , , , , EXERCISE 124, 1. Observe the tables given below and in each one find whether x and y are proportional:, , m| x | 3 [5s [6 fin] 2, iy | 9 | is | 24 [33 | 7, , (fi) | x 2.5 4 7.5 | 10 14, y 10 16 | 30 | 40 | 42, , im) x | 5 | 7] 9 [15] 18] 25, Ly [15 | 21 [ 27 | 60] 72 | 75, , 2. ifxand y are directly proportional, find the values of X, Xp and y, in the table given below:, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , x 3 % | % 10, y 72 120 | 192 yy, , 8. A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover, in 20 litres of diesel?, , 4 A tax charges a fare of € 2550 for a journey of 150 km. How much would it charge for a, journey of 124 km?, , 5. A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in, 5 hours?, , 6. If 18 dolls cost % 630, how many dolls can be bought for % 455?, 7. If9 kg of sugar costs = 238.50, how much sugar can be bought for % 371?, , 8. The cost of 15 metres of a cloth 1s € 981. What length of this cloth can be purchased for, % 1308?