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17, , Proportions, , If there are 20 one-rupee coins and 4 five-rupee coins, , in Varun’s wallet., , We write the ratio of the number of five-rupee coins to, , one-tupee coins as 4:20. And it is read as 4 is to 20., , * The comparison of two quantities in terms of “how, many times” is known as ratio., , Here, the number of one rupec coins is 5 times that, of five-rupee coins., , * Two quantities can be compared only if they are in, the same unit. We cannot find the ratio of the weight, of orange to the length of banana., , + We can compare two ratios by expressing them as, fractions. To compare 1:2 to 2:3, we will compare, , Biot 152, 7” 3 and check. Here, 3 <7 1:2 < 2;3., , * Ifwe multiply or divide the numerator and the, denominator of a fraction by the same number, we, get equivalent fractions, and thus, equivalent ratios., , * When two ratios are equal in value, then they are, said to be in proportion., , Consider, 10 km : 30 km = 1: 3 and, , 15 minutes : 45 minutes = 1: 3., , Since, both the ratios are equal; we say that the two, ratios are in proportion., , Here, 10:30 = 15:45 => 10:30 :: 15:45,, , If two ratios are in proportion, then,, , Product of extremes = Product of means., , That is, 10 x 45 = 30 x 15 = 450., , * Three quantities a, b, and care said to be in, continued proportion, if a: b=b:¢ :, , * In unitary method, we first find the value of one unit, and then the value of the required number of units., Sajid bought 3 apples at 733. How much should he, Pay to buy 9 apples?, , The cost of “parity = %33 s, Therefore, the cost of | apple at =U, , Now, the cost of 9 apples = 9 x 11 = 299, , SAW, , , , ROS ETH aaa, , Prag Vitel pyre, , , , Example 1; From the following, identify the ratios, , that are in proportion., a. 2:7 and 6:21 —_b. 3:6 and 5:8, Solution:, , a. Product of extremes = 2 x 21 = 42 and, Product of means = 7 x 6 = 42., Since, Product of extremes = Product of means., The ratios 2:7 and 6:21 are in proportion., , b. Product of extremes = 3 x 8 = 24 and Product of, means = 6 x 5 = 30., Since, Product of extremes + Product of means., The ratios 3:6 and 5:8 are not in proportion., , Example 2: Check whether the following are in, , continued proportion or not., a. 5km, 10km,15km_ — b. %8, 224, 272, Solution:, , a. Ratio of the 1* and the 2 quantity = 5:10 = 1:2, Ratio of the 2°and the 3™ quantity = 10:15 = 2:3, 5km, 10 km, and 15 km are not in continued, proportion as the ratio between the first and the, second quantities is not equal to the ratio between, the second and the third quantities, , b. Ratio of the 1* and the 2” quantity = 8:24 = 1:3, Ratio of the 2“and the 3“ quantity = 24:72 = 1:3, Thus, %8, 224, and 272 are in continued proportion,, , Example 3: If a car travels 100 km with 5 litres of, , diesel, then how much distance will it cover with 2.5, litres of diesel?, , Solution: Distance travelled by the car with 5 litres of, diesel = 100 km, , Therefore, the distance travelled by the car with t litre, of diesel is = 1 = 20 km, , , , Hence, the distance travelled by the car with 2.5 litres, , of diesel = 20 x 2.5 = 50 km,

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PERL., , , , , , Exercise, , 1. Write true or false against each of the following:, & 14:16 2 24:26 & 13:26 =: 51:102, & 3.9 26:2 d. 8:40 =: 10:50, 2. Fill up the empty boxes given that the following are, , , , , , 3. Complete the following table., See See, , 3 Pens a SRS ae, Ry sense sane, viens, , , , DIRECT PROPORTION, , Observe how the change, , in one quantity leads to a, , corresponding change in the, , other quantity., , * If we blow more air into a, balloon, it expands more., , * More the number of books in, the bag more will be the weight, of the bag., , * Less questions attempted, correctly in an exam, lesser the, marks obtained., , If the weight of a notebook is 2.5 kg then, what, , will be the weight of three or five or more such, , notebooks?, , Let us represent this information in the form of a, , table:, , pes 4a, , ‘, |, , ea!, A, , , , aaa 1322-2, 1 25 kg 25-745, a G 3 2, 3 3x 2Skg=75 kg 77% ;, S . 5 : = Ss %, ; 5 5x25 kg = 125 kg ms 3, Begg ero eae Seen et ¢ HAPS tale aa, Caeg 1425 ges ek Ss rem, , , , 4. If the cost of 5 kg of ric, , 5. Check whether the, , , , , , , , , , ¢ is 7180, then what wj, , of rice? . :, the cost of 7 BE following are in proportion g, , 3 : 2kg:36kg, & SEO cada 10cm: 40cm, c. 3.5 litres : 21 litres and 6 cm 736 cm, 6. The prize money of ©2000 is divided in the ra, for the winner and the runner up. How much does, runner et?, 7. th foiling oat in continued proportion, Find, missing numbers., , a. 5,20, b. 3, 12, __.,, , 1. When the number of notebooks increase, the, combined weight also increases. Hence the two», , , , quantities exhibit a direct variation with respect, , to each other. ae, 2. The ratio of the number of notebooks to the , total weight of the notebooks remains constant,, , that is, 2 ., Thus, when two variables x and y are in direct, , Proportion or vary directly, then,, , * the ratio « remains constant, say k (+ 0), where _, , kis the constant of Proportionality for x and y., x, , *: 5 Pe, y — k, for different values of x and the, corresponding values of y., , * The relation between the, be expressed as x oc y, saying that, x = ky., , two variables can, which is equivalent to, , Two quantities are said to follow direct, Sisesheed or direct variation ifan increase/, , fe in one causes an ji :, fatale $ an increase/decrease, , Tin @ Way that their rati i, Constante Bose their ratio remains ©, , et SI ees

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xample 4: Check whether x and y are in direct, Orion,, , , , Here, 20 a? 82 35 5 183 (constant), , 0 12°°16>- a, Therefore, x and y are in direct proportion., , Sha WO 3S, see, , Therefore, x and y do not vary directly., 3: The following quantities are in direct, , , , . x) wy, By using the relation, —* = =,, H *, , , , , , , , , , ad, , ok an het ALib hsb, Example 6: A hostel cant, , a week, How much quantity of rice is needed for 20, , days?, , Solution: Let the quantity of rice needed for 20 days, , bey kg. s, , , Look at the following table:, , Be st, egies ati ALES, , , , , , teen requires 42 kg of rice for, , If the mimber of days increase, the amaunt of ree, , required also increases,, , Hence, they are in direet proportion, Sa, we can ise, , the relation, * = 2, NH, , Here x, = 7 days yy 42 Kus ty = 20 days yy = he, , Therefore, *h = “+ peeattays 42M20 . 129, Wo pans Werk 7, Thus, the quantity of rice needed for 20 days Is, 120 kg., Example 7: Ona certain map of Indl. the distanee, between Uttar Pradesty and Gujarat is stiown as 5 ent, whereas the actual distance between the (we staten Is, 1.400 kin. What scale is used to dria the map? Using, the scale, find the actual distance between two pices, ata distance of LS em on the map,, Solution: To find the seate we need to find the rato, of the distance between the 2 cities an the map to the, actual distance between the giles,, Let the ratio be given by wsy,, Hence, we can write way © $; 1400 10, , {Since L100 kins 1400 LO" om), , ¥ 4‘, , y 1800x10" 2KO% 10” 28000000, Thus, the scale used to draw the map is 1: 28000000,, We can see that | em on the map represents the netual, distance of 280 kin (2.80,00,000 em),, Therefore, actual distance between two cities shown at, a distance of 15 em on the map = 280 x 1S = 4200 km, (42,00,00,000 em)., Example 8: 9 identical bottle making machines can, produce a total of 270 bottles in a day, How many, bottles will 15 such machines produce in one day?, Solution: Let the number of bottles produced by 15, such machines be y., Let us tabulate the given information,, , RECS 9 27, SSI «ss, , If more machines are there, more number of, , ; bot, can be produced in a day, sia, Hence, it is a case of direct Proportion., , 3 Isx2, 8 0 y. = y= S22 2 480, , , , Therefore, |, si fore, 15 machines can Produce 450 bottles in a

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ETERS,, , , , CPR, , , , , , , , , , , , , , , , , { 1. Check whether the numbers in the following, i cases vary directly or not., \ a. ‘ b. a ?, | 04° 8 12 3 34 6 15, | 2 40 60 15 6 7 12 2, 2. Find the missing values given that the following, numbers are in direct Proportion,, S84 a 16, 3: ley 5, , , , , , , , , , , , , , * ‘Two quantities are said to be directly, , Proportional, if with the increase (or, decrease) in one (quantity, the other quantity, increaves (ur decreases} to the same extent, When two quantities « and y vary directly,, they can be written as ua y, which is same ay, x= ky (where & ¢ Ois a constant, , 1, Check whether the following quantities are in direct, Proportion or not, , , , , , , , 2. Find the missing values,, in direct proportion., . Be ieee, , , , , , aah length, , 7 ‘di is 20 mand the er i, , 3. a height fe ted the height of the tower 4, of its shadow, , casts a shadow of 12 m under the same, that casts, ea :, 4 ee cage T books is 35 kg how many, * pooks would weigh 5008? ie, 5. If Sth of a vessel is filled with water in I minute,, oe, , fis filled in, then the rest of the vesse! sot, minutes under the same oo, , 5 = a, al bg 23, , ie, , * cand y are in direct proportion, if they have a, constant ratio, such that,” © 4 (#0). Here, bis, ’, the proportionality conctant., * Whea cand » are chrectly proportional to cach ;, other, we have, Dw 12 a, , rhooOD, , , , *, \ Wpandy vary deteetly with cach other, such that, when pb 20, qui,, , areigss, , DpeWqatt, © pe l§igato ,, , dpelsquis, , Application tased Questions, , 4 Qeatiey of sop for 6 peopte requires 3, tablespoon of satr How, many tablespoons of, : Aco at make the soup foe 24 people?, . MENS Grey 140 L of mi

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yma cc, , |The dimension of a photo is32in x4qin. The, sin., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , photo is enlarged proportionatel, , : gcd p jy such that the, longer dimension is now 10 in. Fi i i, es hatoradat in. Find the dimension, ig. The cost of p m of a wire is %t, At the, , 8%, same rate,, , what will be the cost of f ii :, aa qm of the wire of the same, , ig, The cost of 4 dozen pens is 7960. What i, Bbppertaitte . What is the cost, 10, Ramesh completes : of his job in 20 days. At the, same rate, how many more days will i i, ek da Wott ys will it take him to, , INVERSE PROPORTION, , When an increase in one, quantity results in a decrease, in other quantity i.¢., if one, <j quantity increases: the other, quantity decreases and vice, versa, the two quantities are said to vary inversely., Consider the examples:, 1. The time taken to finish a regular size pizza, decreases if the number of people sharing the, pizza increases., 2, More the number of people working, lesser will be the time taken to complete it., regularly supplied 10 kg of, rice for an office canteen with a strength of 200, people. If on a Saturday the office invited 50 more, a seminar, what will be the quantity of, head on that day?, , Let us tabulate this information:, , SoRae 2, 410000, _ 200, , ter una’ rier pet, pT TM Bla,, , Observe that,, 1. As the number of pec!, head reduces. The two, , , , ona task,, , A food contractor, , , , quantities show, 2. 200 x 50 = 104, , \Two quantities are ‘said to follow inverse proportion - or inverse variation. if an increase in one causes a, decreases in the other and vice versa ina way that their, , | product remains constant. aera, , , , (TS re, , LL. On a map, the unit of 0.5 cm represents 5.5 km., , The distance between the points A and B on the, map is 6.5 cm. What is the actual distance between, , A and B?, , 12. There are two wheels P and Q with 6 cogs and 14, , cogs respectively. P is meshed with Q. What is the, , number of revolutions made by the wheel Q. when, , the wheel P makes 30 revolutions?, 13. In 2 years a sum of £20,000 earns a simple interest, , of 26000. At the same rate of interest, what will be, , the interest earned for 235,000 in 2 years? What, can you infer about the relation between the simple, interest earned and the amount invested for the, same period at the same rate?, , ables x and y are in inverse, , Thus, when two vari, her or vary inversely, then,, , proportion to each ot, + xx y= constant,, or 2b = 22, where x)..%, , an) xy, =a eM Th mh, represent the values of x: ¥y.¥2 Fepresent the, corresponding values of yand kis the constant, of proportionality., , + the relation between « and y, asx = which is equivalent to saying that,, , can be represented, , re=s, , Example % Check whether the following quantities, , , , , , , are in inverse proportion of nol., a., 6, b, 10°60 «200, eis, , , , Solution:, a. If two quantities p and q are in inverse proportion,, , then p, X 9, =P: * 4x, Here, 24x 1 =8x3=3x8=12x2=4x6, = 24 (constant), , Therefore, the quantities are in inverse proportion., b. Two quantities a and b will be in inverse proportion,, , if a, x by = a x by., Here, 20x 6 #40 x 2# 10x 3+ 60x 18 = 200 x 60., , Therefore, the quantities are not in inverse, proportion.