Math5

In front of every fifth house there is a lamp post, every third house there is a neem tree and every second house there is a pet animal. Draw pictures and complete the given grid. 1. Locate houses having both lamp post and pet animal and hence, find the common multioles of 2 and 5. 2. Locate houses having both lamp post and neem tree and hence, find the common multiples of 3 and 5. 3. Locate houses having lamp post, neem tree and pet animal and hence, find the common multiples of 2. 3 and 5. 4. Write two more common multiples of 3 and 5 not found in the grid. 5. Write the lowest of all common multiples of 2, 3 and 5. Prime Factorisation Every composite number can be expressed as the product of its factors. When all the lactors are prime, it is called prime factorisation. lx,f mple 1: 36=2xL8=3xL2=4x9=6x6 In each of the above factorisations, all the factors are not prime. Thus, we decompose the given number into factors till all the factors are prime n u m Ders. So, we write 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3. l,rke a look at the following factorisations. l.(.16 ) ?I \"(1t?\\ 3x3x2x2 GT])\\'(-O\\-.. al6\\),\\.G-),-. 1trt\\\"1'K 2x2x3x3 2 x 3 x 2x 3 2x2x3x3 Tho above arrangements are also known as factor trees. Hance, prime factorisation of a composite number is unique. ltarrrplc 2: Find the prime factorisation of 120. Step 1: Start with the smallest prlme factor of 120. t2O=2xGQ Step 2: Again start with the smallest prime factor of 60. LzO=2x2x30

Step 3: Continue the process till all the factors are prime. t2O=2x2x2x15 12O=2x2t2t3x5 Thus,the primefactorisation of 120is2x 2 x 2x3x 5. 2x36 3:Example Find the prime factorisation of 72 with the help of a factor tree. 72 is factorised into 2 and 36. 2x2x 36 is factorised into 2 and 18. 'ir) 18 is factorised into 2 and 9. 9 is factorised into 3 and 3. , i'2i x2x t2'^\\_1.,..?, Thus, the prime factorisation of 72 is 2x2x 2x3x 2x2x2x3x3. Coprime Numbers Coprime Consider the numbers 18 and 20. ore not necessorily The factors of 18 a rc 1,2, 3,6, 9 and 18. prime numbers. For The factors of 20 a re 1-,2, 4,5, 10 and 20. example, tB ond 25, The common factors of 18 and 20 are L and 2. Now, consider the numbers 1.8 and 19. one coprime. The factors of L8 are 'J., 2, 3,6, 9 and 18. The factors of 19 are l and 19. The only common factor of 18 and L9 is 1. Two numbers which have no common factor other than l are called coprime numbers. 18 and 1\"9 are coprime numbers. Whereas 18 and 20 are not coprime because they have common factors other than 1. Other examples of coprime numbers are 25 and 34,7 and 45, and so on. 't List coorime numbers between 11 and 25.

1. Find the prime factorisations ofthe followins numbers. a. 112 b. 64 c. 88 d. 125 e. 96 f. 1r.0 2. Draw the factor tree of the followins numbers. a. 56 b. 98 c. 250 d. L00 e. lJ5 f. 242 3. Check if the following numbers are coprime or not. a. 42 and 39 b. 19 and 23 c. 1.8 and 33 4. Find the prime numbers between 150 and 2OO. Highest Common Factor The highest common factor of two or more numbers is the rargest number which divides the given numbers without leaving a remainder. Highest common factor is abbreviated as HCF. lt is also known as the greatest common divisor, denoted by GCD. Let us learn to find the HCF of two or more numbers. To Find HCF by Prime Factorisation Method 1:Example Find the HCF of 12 , 24 and 36. We know that, 2 36= The circled prime factors are the common factors of 72,24 and 36. But the highest common factor is obtained by multiplying all the common prime factors. Therefore, the HCF of 12,24 and 36 is 2 x 2 x 3 = 12 This method of finding HCF is called prime factorisation method. To Find HCF by Short Division Method HCF of 72,24 and 36 can also be found using division by prime factors. stcp 1: Divide the numbers by the smallest common prime factor and write the quotients below the respective numbers. \\lep 2: Repeat Step l with the quotients obtained. stcp 3: Continue the process, till there are no common prime factors.

t2, 24, 36 6, L2, 18 Step 4r HcF of the given numbers is the product of all the common prime factors. lnerelore, HLI- = z x z x 5 = Lz This method of finding HCF is called short division method. Circle Example 2: Find the HCF of 36, 64 and 96 using prime only the commbn focfors of Ihe given factorisation and short division method. numbers. 3 is o Using prime factorisation method, foctor of 36 ond 96 36=3 x3 '.(rAJ.,,{Az) x2x2 buf not of 64. 5o 64=2 x, do not circle 3. 96=3 x 2 xVx\\4 x2x2 Therefore. the HCF of 36, 64 and 96 is 2 x 2 = 4- Using short division method, 18, 32. 48 9, 16 and 24 have no other common prime factors. Therefore. the HCF of 36, 64 and 96 is 2 x 2 = 4. Example3: FindtheHCFof42, 105and 63 using prime factorisation and short division method. Using prime factorisation method, of two or more numbers connot be lorger thon ony of the given . Therefore, the HCF of 42, L05 and 63 is 3 x 7 = 2t. Using short division method, 42 1.05, 63 L4, 35, 27 Therefore. the HCF of 42, 105 and 53 is 3 \\ 7 = 2L.

Word Problems Example 1: Find the greatest possible length of a wire which can be used to measure exactly two wires of length 18 m and 24 m, respectivelv. Required length = HCF of 18 and 24 One method to find the required length of wire is by using prime factorisation method. 2\"4== VA Thus, the HCF of L8 and 24 is 2 x 3 = 6. Another method to find the required length of wire is by using short division method. 2178, 24 Step 1: Divide by the smallest common prime factor 2. Step 2: Divide by the smallest common prime factor 3. 3,4 Thus, the HCF of 1.8 and 24 is 2 x 3 = 6. Therefore, the greatest possible length of a wire which can be used to measure exactlv two wires of length 18 m and 24 m is 6 m. Example2: Find the greatest number that divides 149and 101, leaving 5 as remainder. The number that divides 149 leaving 5 as a remainder must divide 149-5 = L44 without leavinga remainder. Similarly, it must divide 101- 5 = 96 without leaving a remainder. Therefore, we have to find the HCF of 144 and 96. 1AA q6 72, 48 18, t2 Thus, the HCF of 144 and 96 is 2 x 2 x 2 x 2 x 3 = 48 Hence, the greatest number that divides 149 and 101 leaving 5 as remainder is 48.

1.. Find the HCF of the following numbers by prime factorisation and short division method. 28,42 h ?6 q6 c. 24,48,72 d. 16,36,56 t. 70, 105, 35 g. 16,48,80 f. 2L,49,63 k. 18,27,54 h. 32, L20,64 j. 34,26, s8 l. 125, 100 2. Find the greatest number that willdivide 108, 144 and 216 without leaving any remainder. 3. Find the greatest length of a scale that can be used to measure exactly 35 m and 63 m of cloth. 4. There are three drums containing 42 litres,70 litres and 98 litres of water. Find the capacity of the largest r[tcontainer that can be used to measure water in the three drums by the exact number of times. Find the biggest number which leaves a remainder l when it divides 151, leaves a remainder 2 when it divides 127 and leaves a remainder 3 when it otvloes r /6. Lowest Common Multiple In a school, both Nursery and KG classes start at 8:40 a.m. KG classes have each period of 30 minutes and Nurserv classes have each period of 40 minutes. The principal of the school wants to give both Nursery and KG a juice break at the same time. But, she does not want them to take a break in the middle of anv class. At what time can she give them the break? Do you think it is possible for her to give a break to both the classes at the same time and that does not fall in the middle of anv class? Before finding an answer, let us have a look at the multiples of some numbers. Multiples of 4: 4, I, L2, 16, 20,24, 28,32,36, 40, 44, 48, 52, 56, 60 ... Multiples of 5: 5, 10, 15, 20,25,30,35,40, 45, 50,55, 60... . -baFl Multiples of 1\"0: 70, 20,30,40, 50, 60, 70, 80 ... l46 lE

Let us now write down the common multiples of 4. 5 and 10. They are 20, 40, 60... The smallest of all these multiples is 20. Therefore, 20 is called the lowest common multiple of 4, 5 and LO. Lowest common multiple is denoted by LCM. Likewise, if we can find the lowest common multiple of 30 (duration of KG period) and 40 (duration of Nursery period), we will be able to find a common time for both the classes. Multiples of 30: 30, 60, 90, 120, 150, 780,21O,240 ... Multiples of 40: 40, 80, L20, L6O,2OO,240 ... So, if they take a break after 120 minutes from the starting time, they wilj have a common break. After 120 minutes, KG would trave finished = 4 periods and Nursery #'l'r.l ffwould have finished = 3 perioOs. So, they need not take a break in the middle of anv class. It is not possible to list multiples of big numbers Iike 254, 328, etc., and then find the LCM. So, we will now learn to find the LCM of numbers bV prime factorisation method. To Find LCM by Prime Factorisation Method Example 1; Find the LCM of 12,36 and 60. Step 1: Find the prime factors of all the given numbers. Step 2: Circle the factors common to at least two of the given numbers and ta ke onlv one. 'A1'\\A.xlz\\2//x'\\l3a/xj,. s 5 36 = 2Take only one Take onlV one 3 60= Take only one2 Step 3: Multiply the one common factor from each group with the product of the remaining factors. LCM = 2 x 2 x 3 x productof remaining prime factors =2x2x3x3x5=180 Hence, the LCM of 12,36 and 60 is 180.

or=A's'AExample 2: Find the LCM of 45, 75, and 125. zs=\\y'\"A,.lr] 125= 5 x \\7 , \\7 LCM = 5 x 5 x 3 x product of remaining prime factors =75x3x5=11.25 Hence, the LCM of 45, 75 and 1.25 is 1125. Example 3: Findthe LCM ol1,!2,256 and32O. 7 2x2 256 = 5 A'A'W'V/'32O = LCM = 2 x 2 x 2 x 2 x 2 x 2 x product of remaining prime factors =o+x zx 2x5x / = 8960 Hence, the LCM of 112,256 and 320 is 8960. Remember LCM is the product of common and non-common prime factors. Only one factor should be taken in the place of factors common to at least two of the etven numbers. Word Problem Example: Chris, Sunil and Jose were practising for a hurdle race. Chris crosses a hurdle after every 20 metres, Sunil after every 25 metres and Jose after every 30 metres. What is the least distance after which all of them will face a common hurdle? Till that distance, how many hurdles will be faced by Chris, Sunil and .lose? The least distance after which all of them will face a common hurdle is the LCM of 20, 25 and 30. xA\\/A1) x 2 \\,,4.\\) x 3t '\\,Ylt/ =2x5 x2x3x5=300 Thus, the least distance is 3OO m.

Let us see how many hurdles will be faced bv them. Number of hurdles for Chris = 300 Number of hurdles for Sunil 300 Number of hurdles for Jose 300 30 =10 Therefore, the least distance for them to face a common hurdle is 300 m. Number of hurdles faced by Chris, Sunil and Jose are 15, 12 and 10, respectively. 1. Find the LCM ofthe following numbers. a. 72,96 b. 49,147 c.35, 75 o. LO8,I44 g. 64, 120, r28 e. 10,25,30 f. 48,64,96 k. 96, t44, L26 h. 65, L30, 260 i. 72, r08,120 j. 20,35, 10s 55, 72t, 165 _, _2. Three common multiples of 15 and 25 are ano 3. Traffic lights at three different crossings on a road change after every 10 minutes, 15 minutes and 20 minutes, respectively. lf they change together at 9 a.m. first, when will they change together again? 4. Swati wants to distribute sweets to some of her friends. She wants to give either 14, 15 or 10 sweets to her friends and keep 8 for herself. What is the least number of sweets she must have in order to distribute? 1.. What number should be subtracted from l\"5O to make it exactly divisible by 16, 18 and 12? 2. I am the smallest five-digit number which when divided bv 11 leaves 5 as remainder. Who am l? What number should be added to me to make me exactly divisible by 13?

Target OlymPiod Find the values ofA and B if the given number 74579888 is divisible by 33' John has some marbles. lf he arranges them in groups of five, 4 are left' lf he arranges in groups of four, 3 are left lf he arranges them in groups of three, 2 are left and if he arranges in groups of two, 1 is left. lf the number of marbles with John is less than 60, then how many marbles does he have? L. Write allthe prime numbers between 10 and 40' 2. Write allthe composite numbers between L5 and 45' 3. Which of the following are coprimes? c. 3,25 d' 11,48 b. 16, 18 g' 4,36 h' 33'45 a. 5,!2 e. 8,27 f' L5,25 4. Write the prime factorisation of 63, 81 and 125' 5. Check if 25788 is divisible by 9 or not. iw @ Crossword Puzzle Top to bottom: 1. LcM of 25 and 30. 2. HcF of L00 and 175. Left to Right: 3. LCM of 64 and 256 4. HcF of 66 and 110. 5. HcF of two numbers is 3 and their LCM is 36. What is the product of the two numbers?

Write all the numbers from 2 to LOO on cards and put in a box. Call two students. One student picks up a card. That student will get as many points as the number. His/her partner writes all the factors of the number on the board and he/she will get as many points as the sum of all the factors. Note 7: Do not include L ond the number itself os foctors. Example: lf a student picks up 10, he/she gets 10 points. His/her partner gets 7 points. (Factors of L0 are 2 and 5. Thus, 2 + S = 7\\ Note 2: lf a number is picked, thot card is to be kept dside. Sdme number cannot be used dgatn, Note 3: lf a child forgets d foctor, 2 points will be deducted from the total score as penalty. For example, if the number is 54 dnd the chitd forgets to write g os its factoti 2 points will be deducted from the total score. flMlei\"rrug Las Obiective: To find the LCM of two or three numbers, say 11, 5 and 7 Materials required: Grid sheet and colour pencils. Method: On a grid sheet or a number grid, circle the multiples of 11 in red, the multiples of 5 in blue and the multiples of 7 in green colour. The first number which is circled in all the three colours is the LCM. MY Proiect a oranges and 60 of equal l;ffi ,;;;;\"-i',u:::'::,:\"nilT:.::J,'ff\"**m\"annaq^oe,\"s\".\"H, etowantsfrtuoitasrrHanegheastn3e0maipnphleesa'p4s5 s't::\"^::\"\"t tha\"theaps but lessnlill;.1\",'irr.rl\" \"*o'\",\" ff :::X'Ji;' Drawreouirementsa colourtul ol *\" arransement' wli\" [.n\"titt ot \"ating fruits everv dav'

T!4 1. You know that the t_cM of 25 and 45 is 225 and their HcF is 5. Now what is 25 x 45? 25 i< qS = 11ZS What is 225 x 5? 225 x 5 = 1125 What do you observe? Product of two numbers = product of their HCF and ICM Check this for 36 and 64. Let us use this result to find one of the two numbers, if one of the numbers and the HCF and LCM of the two numbers are gtven. Example: The HCF and LCM of two numbers are 9 and 504, respectivety. lf one of the numbers is 63, find the other number. Let the other number be A. Product of the numbers = HCF x LCM + Wehave63xA=9x504 + A= q-'Y;;(-n/t =72 Thus, the other number is 72. 2. To find the HCF of small numbers, we use prime factorisation, How can we find the HCF of 3-digit or 4_digit numbers? A method known as Euclid,s algorithm is very useful in such cases. Example: Find the HcF of 210, 525 and 3L5. :li : i:r :: Let us first find the HCF 21oIs 25 1osl3 1 s of 21.0 and 525. HCF of 210 and 525 is 105. : ::p 2: Find the HCF of 105 and the third number. The remainder is O and the divisor is 105. Therefore, the HCF of 210, 525 and 315 is 105. !q'-----

i :'1 Write all the prime numbers between 10 and 40. Which of the following are coprime? a. 5, L2 b. 16, 18 c. 3,25 d. 11,48 e.8,2L e.4,36 h. 33,45 t, !), zJ 3. Write 16, 36,42 and 2g as the sum of two prime numbers. 4. Find the prime factors of45, !27 and72. 5. Check whether 25782 is divisible bV 3 or not. 6 Find the HcF of the folrowing using prime factorisation and short division method. a. 42,182 b. 3s, 17s c.70,720,1.80 d. 65, t6g, L43 7. Draw the factor tree of 42, 70, 2g and 1j.0. 8. Check if 35724 is divisible by 3 and 4. tf yes, can we say it is also divisible by 6? 9. Check if 572972 is divisible by 18 or not. 10. Find the LCM ofthe following: b. 42,77 c. 84, 1L2, 49 d. 100,525, 650 1,J,. Find the maximum size of each of the two bottles, if oil contained In two cans of sizes 84 litres and 144 litres have to be filled in I them exactly without leaving any oil behind. i 1,2 Find the LcM of 225, 324 and 450. i!L\"^J Find if the given numbers are divisiblebV2,3,4, 5, 6, 9 and 10 2 3 4 5 6 9 10 2342 4020 83250 9932 oo oo

o woRKsHEErd A farmer wants to grow sunflower plants in his farm. He wants to plant them in horizontal and vertical rows such that the number of plants in the horizontal row is a multiple of 25 and the number of plants in the vertical row is a multiple of 15' He also wants the number of plants in both vertical and horizontal rows to be equal' a. What is the least number of plants he must plant in each row to satisfy the above given condition? Ttshhaeele.pfalHarmentresermhpaolvavenetsgsrttohhweenfl,ierstahtsetthnfrauermembreeorrwdsoefcopifdlaepnslatsntotwsrhheiocmhroizvsoeantittshafylelytphaleanndgtsitvhewenhficicroshtnadtwrietoiornecoaWdluymhfenonsr of plants verticallY. b. How many plants does he have for sale? He decides to tie the removed plants into small bundles of 9 each to take them to the marKet. c. How many bundles of plants will he tle? d. Suppose he wants to plant sunflower plants horizontally in the multiple of 10 in least number of plants and vertically in the multiple of 16 What should be the each row if the number ot plants in both vertical and horizontal rows are equal? i. 40 ii 80 iii. 160 iv. 320 F*-E II-TT_ F_E LE E E F f,rl tl i F a E ! !r E fl-'Lt iirHHHHHrt ilii ilirrHHHrl!lJiirretBgss llii'.iiirretLrHeBr --l1f;Tfi

4 l--) 1j Wnen a whole is divided into equal parts, each equal part is called a fraction. t1i '3 j-The figure represents ii Here, 1 is the numerator and 3 is the denominator. I Let us revise the following fractions: i 248t6i;-,.Equlvalent lractions have the same value: I11=Aa = I a _!_ = ,i LtKe rractrons have the same denominator: iE1,?iEC, i, 8, \"t.. i *..l, ],,;i iii -,i,' unlike fractions have different denominators: 5619, Proper lracttons are those fractions whose numerator ?q is less than tls 7 denominator: :6, 9 etc. II ll lmproper fractions are those fractions whose numerator isgreaterthan orequal to ;i. rls li. 138 oenomtnator: -- -,J -- etc, i)v6 li ij Mixed fractions are a combination of a whole number and a fraction: f]3, Z], etc. 1. Express the shaded regions as fractions. 4 i\"r,|i|||,,f\\a. T- T--f-Tr-l lL_lJI D.|l.__+,r+r+r_r.1lH_J| c. lY-^sr4l\\yAlLl|_lI ' 2. Ti.k the like fractions. 1 Io. 1s. 15 - 1s2or --r Lii a.Aiz''-f -1 rv r5. 22 22L- J | \"\" I .d-8t .

3. Convert the following: o' Mixed into imProPer fraction a. lmproper into mixed fraction i 1: ll. f- )? .. 19 2o 4) Evaluate. v. !+L+! !7 12*8 32 32 32 c. ^. 18 18 2' Zt i s. ;i+ rino ot tr'e following: i a. 280 b d.!. I24 c. 48 192 ii o. on , mango tree, there were 12 mangoes 5 mangoes fell on the ground' What fraction Moths Around Us Anurima decided to make fruit salad for dessert Shecutsanapple into 1,6 equal pieces and one banana into 8 equal pieces She also ta\"aonkdeosfm\"a,aink\"ero\"sraaongb,*eo\"wwtlh\"oifcfhbsaahnlabaasadnn1aTa2hnaeeanqndau,noasdnhl eepoorptaarieknticeosesensoodfuostoehrsteahnsrme8heeeixefhposaiervecehveeersorrsoyretfhhl[einr-sg..elf? ff;t;;;;;^:;ple, 1 15hs hs5116p6ortion of aPPle, 3 Potiot n of banana and 12 portion of orange' Finding Equivalent Fractions Two fractions are said to be equivalent if they have the same value either by multiplying both the Equivalent fractions for a given fraction can be found by dividing both the numerator ano denominaior by the same number or \"dlern\"orm\"ain\"art.o\"rabY the same number' i'Let us see the equivalent fractions of 1x 5 5 lMultiplying numerator 'J, lx 2 ] a=3\"2=l ilMndultdiPelnYoinmginnautomr ebrY\"tozrl 3x5 15 ind denominator bY 5l - t2-Theretore, -, and j are equivalent tracflons' IJ 48+ 4 - !l [Dividing numerator - 3AQ 48+ 3 IO IDividing numerator 5-,l 48 20 and denominator DY 60 60+ 4 15 and denominator Similarly, = 60+ bv 4l *,1* 4.\"0 are equivalent fractions' .-6ThJust,.:.. -

:Example: Vikram watched TV for of an hour and 5 ::Riya watched TV t.oIJr. ot an hour. Who 25 watched TV for a longei span of time? Method 1: t2 x gO'minutes = 36 minutes i5 ho\"r=4Vikram watched TV for of an 3 .- . - t115 n ::)<hrya warcned I V tor ot an hour = 6d minutes = 3G minutes r; -- D1 Thus, both watched TV for the same span of time. Method 2: 5 _15 are 1q 12--q5.:-q+5=15? - and - equivalent fractionsa2sl5= : -Thus,q 15 01 an hour. of an hour is the same as Therefore, both watched TV for the same span of time. Checking for Equivalent Fractions To check whether two fractions are equivalent or not, we can use cross multiplication, that is, multiply the numerator of the first fraction by the denominator of the second and the denominator of the first fraction by the numerator of the second. rf both the oroducts are equal, then the given fractions are equivalent, otherwise they are not equivalent. ifl IExample 1: To check and are equivalent fractions. v.L6 4 t \",9x8=72 9.--\\a188 * q,.re= zz The products are equal. Therefore, the fractions are equivalent. Lowest Form of a Fraction A fraction is said to be in its lowest form if the only common factor of the numerator and the denominator is 1. 1?CA ;' i, g, i are In tnerr lowest forms. :r4zq, (=2t ,: are not in their lowest forms. 25

Reducing a Fraction to its Lowest Form Method 1: be reduced to its lowest form bv dividing both the numerator and the A fraction can denominator bv a common factor' A\\ Example: Reduce] to its lowest form' bU 45 45+3 15 60 60+3 20 !eut is not in its lowest form, so repeat the process till the numerator and 2U the denominator have no common factor except 1' 'c\"^' L5 _ 1250++55- I 20 4 Therefore, the lowest form of ;f, is 1 4 Method 2: Find the HcF of the numerator and the denominator and divide both of them by their HCF' The fraction will get reduced to the lowest form in one step' Examole l.: Reduce 49 to its lowest form' b4 The HCF of 48 and 64 is 16. H=ffi=irhererore, -. -I nus, 48 reduced to its lowest form is 1. *Examp' le 2: Reduce l2u to its lowest form' Method 1: T2 72+2 6 ts;6 6+2 3 3 2:? 1 24 24+2 t2 =#=at720 120+2 60' n= 3o-l-= G= 15+ 3 5 Thus.'7-:2l-0reduced to its lowest torm .1\" ' ls -r Method 2: 24 24+ 24 L HCF of 24 and !2o is 24. Therefore, r20 720+24 5 I11'u5, ?n reduced to its lowest form is

fraction.Multiply the numeratorandthedenominator by 3 in each of the followrng an equiva lent fractions to get ) a\"i.4 oh3f g-1.1!! a. 19 \".+ r.4t n4 L3 .*Match each part in tt riot U 3. i 3.1 *J* .*a. r \" *,lt'n tn\",, .\"rrt\"., _\" \"^' \"\"i7- iv. 3- l!I r_..' 1L c. :s)of \"ooo\"t\" thour i. 40 minutes ,.,.. €75 iii. mixed fraction Fill in the boxes. ia=l=---l=I+'ll=gI=qun Reduce the following fractions to their lowest form. 'j: b.+ 13 a.? \" t4 .+5 30 .-. 60 -' 42 ' 10 169 50 tr5. Reduce the foflowing fractions to # ;# \" :;4''. ;.*z\"a\"'6o'Jjl''n1o5'*0.thse'7i7rHcF 9s@Ecr na,o;;;L+;;;;Divide the crass into three groupsofstudents.Theteacherpreparescards with fractions written on them. The teacher can € of 1 w,h ;ilffTilTl;:::il::, jff:nas,he denom o board.To play the game, onechildfrom1lhefirst group should come and the fraction on the picK a card to represent r-r|_fl o,oooooo j=+,.;=;jExample: Equivalent fraction of with 6 as the denominator can be represented as: I ne group with maximum points is the winner.

I Comparing Like Fractions Fractions with the same denominator are called like fractions' Like fractions are easY to compare since they have the same denominator' So' we can compare them by comparing the numerators' Example: Shyam, Ananya and Vikky are reading books with 100 pages each Shyam has ,eao ? of the book, Ananya has reaO fittr of the book and Vikky has read frtr, ath of th\" book in a day. Who has read the maximum number of pages? 10 Method 1: -fIn one dav, Shvam has read x 100 = 30 pages LU '10 LAnanva has read r loo = 70 Pages '10 IVikkv has re\"d t LOO = 90 pages Thus, Vikky has read the maximum number of pages' Method 2: Comparison can also be done without finding the actual number of pages read' To compare like fractions with the same denominator, compare the numerators' Greater the numerato[ greater is the fraction. Here,9 > 7 > 3' q7? T'\"h'A--r-A'f1^r0AL>-110>=10 So, Vikky has read the maximum number of pages' Comparing Unlike Fractions Fractions with different denominators are called unlike fractions' Two cases arise while comparing unlike fractions 5 5 55Case l.: Fractions with the same numerator. Example; Arrange i'i'O,Ain descending order' case, the fraction with the larger denominator is smaller' Hence' in the is the largest' fractions, isthesmallest Next, -:-,then s .nd In this 1I t5 c = Iabove T\"h''a-rof-^r-A' !1;L1 12s-11 72=2.3

Case 2: Fractions with different numerators. convert these type of unrike fractions into rike fractions by making the denominator same. ;,; iExample: Arrange J) and b in ascending order. The easiest way to do this is to find the LCM of all the denominators. LCM of 3, 5 and 6 is 30. NoW multiply both the numerator and the denominator of all the fractions bv a suitable number to get the denominator as 30. 4ra s sx5 2s ?3_z3\"xto1o=-3200.' a!=- 5\"6 =_?30q' e= 6l.5=30 #'ftus' ?9 < #[?r#\"nd 4 are like rractions] Therefore, 245 5)b 1. Insert the appropriate sign (r, . o|. =1 in the boxes. c, -2t 0l--J1t-0 6 3;-1 21 q-.-? 8 !. -7t \"7 t- ' 2 -55 .'. 5r-r3 9*18l- 11u11 -3 r\"..E3Ur-1-r31 o 1^ R- . -1t 3L-rt-11 ,;, 11o3p1-=1It5 '. 1115r----7---t-3 r,. 6 r--r 6 11u 13 1.L* 15 L3*L3 ' 6 1--5 17 13 2. Arrange the fractions in ascending order. 827. 't'h 1s 3 11' 13 L7 Lt 1.1' 7L 6^' + 6lo E?7.1 1?q? t 2 L3 LL, 5'11' 8 2t68A 5 310 s

?) 4 7L I 3. Arrange the fractions in descending order' hi-:-7-=3 t a. -3s.5-1,1-,01-143 c, 5' r.0 15 5 8 816 10 5 13 7 \",7A'5s2'aL', 55r 't t 12' 12' 4' 12 e, 8' 4'3'2 Addition of Fractions Addition of Like Fractions Toaddtwoormore|ikefractions,addthenumeratorsandputtheanswerovertne same denominator. b. 72 3LExamples: Add. a. 5 + 5 --; -35+5-5=15-=3-+1 4 72 7+2 9 Thus?1,:4+:=- -1+? -1? 13 13 5)) _.7 29 13lhUS, .r- 13 -13 Addition of Unlike Fractions To add two or more unlike fractions, first change them into like fractions and then add. 1)q Example 1: Add ;5, to- ano -. LCM of 5, 7 and 5 is 210. 5x35 $x35 I !x42 42 2- 2x3O 60 5 t75 5 5x 42 27O' 7 7x30 ZIO' 5 Z1O 1)E 42 60,t75- 42 + 60 + 775 277 Therefore, 5to 270 2L0 2to 2to 2lo Example 2: Seema ate 1 of a pizza while her brother ate :1th of the oizza. lrd 4 How much did theY eat altogether? lWe need to add and 1' Let us convert them into like fractions and then add' :. +=Altosether 1* 1 Z of the pizza' thev = \"te 5+ 7 Ft Thus, they ate 12th ofthe pizza altogether' tl6f2l lf -'rgr.ptr

Addition of Mixed Fractions Example: ndd 51 and 21. There are two ways of adding mixed fractions. Method 1: j wtethod 2: First add the whole numbers I Convert the mixed fractions into improper I fractions and then add. i.e.,5+2=7 lNoW add the fractions, and '|18)-8-L=-a4nLoz.--=34-4 11 f.LcMof8and4is8. 8 I Now, LCM of 8 and 4 is 8. 4 3x2 9 j r. -46 |a.1= Lr L'J.x 2 _ 22 [Converting into 4xz 1 7x'J- 1 .3 8 8 \"n6 4 4x2 8 like fractionsl 8 8x1= -8and-= so, 1* 3 = 1* 67 rII8T4h8us8. 51+ 23 - 4L + 22 - 63 848 88 8 77 _l 8 8r thus, s8l 4+ zl = 7+7- 8 Example: MrRaobought 21 kgof sugarand 31 kgof rice. Findthetotal weightof sugar and rice bought by him. Weight of sugar = 211g Weisht of rice = 3i ke :l)rotar weisht = (zf - rs 1,1,. 7 1-1- t4 _tq= b_41 -Kp lAs7-=-'lt4 4244 Thus, Mr Rao bought o.L* kg of sugar and rice. 4 4 1. Add the following: t 3.2 q11 rl z^-1+ ^3L- 10-10 - 11 3 2 , .^15.3-1 s. r1a0*251 I. -vf 50-f - E. J^-1t'/_- L 5+

2. Fill in the circles so that both arms have the same sum One has been done for you' \" ,+{ ,, 3?,3!, a1 o. t#, ,*, +}, zi' s} , z}, t!, *' +1' 5: 8 3. itrrJessica, Mehreen and piyali decided to fill a bucket with milk. Jessica filled 5 ot ttre ir .. 1bucket, Mehreen filled rd ofthe bucket and Piyalifilled; th ofthe bucket Howmuch of the total bucket is fill;d with milk? L5 High Valley school organised a trip to Darjeeling in the summer vacation for class V students. The group of students along with their teachers decided to go for trekking from Ghoom to sandakphu and back The total distance to be covered is 104 km. They cou\"red ]th of the distance on the first day 8 and camped at Sukhia Pokhari at night. They covered lt1h of the distance on the second I5 day. What fraction of the total distance did the group cover in the first two days? Do you think such trips and group activities help you bond with others better? support with examples. Manvi, Preeti and Prabhu decided to paint all the 40 pots in their garden' Manvi painted L7 pots, Preeti painted L1 pots and Prabhu painted 7 pots' What fraction of the total pots is painted? Subtraction of Fractions Subtraction of Like Fractions To subtract two or more like fractions, subtract the numerators in order and put the answer over the same denominator. _l?bToa

1 1.Example: Subtract from Remember ib 10 10 convert ihe onswer _ 3-7 _ 2 _ L to its lowest or 10 L0 10 105 simplesf forh Subtraction of Unlike Fractions To subtract two or more unlike fractions, first convert the fractions inro like fractions by finding the LCM of their denominators and then subtract. j58Example: Subtract )1 1. from LCM of 8 and 5 is 40. 7 7x5 35 3 3x8 8 8x5 40' 5 5x8 ;40 lLonverungo; 5and into like fractionsl -7 J) 24 35- _ 1,1, Inus,-- 40 40 40 40 8 Subtraction of Mixed Fractions First convert the mixed fractions into improper fractions with the same denominator and then subtract. Example L: Subtract L2j 4from2!. Wehave,2414= 29 2and 11 =9= 5X Z g_q= g zx 2 44 rhus'4.21_112= 4 Example 2: Renuka had LO metres of red ribbon. She used 49 metres W of the ribbon. How much of the ribbon ts left witf; her? Length of the ribbon = 10 m 4lLength of the ribbon used = m 8 Thus, length of the ribbon left with Renuka =1\\/ 81/0-4?:l\\ _ 10_35_ 80_35 18 88 [Changing into like fractions] 80-35 8 88 Hence, J_ m of the ribbon is left with Renuka. 8

1. Subtract the following: ,a c, 7 _r d.:-8-:7- e. L _I 11 8 a. \\L LI i. _t->--2 50 5 i -3t-4- 8Z- .2 -L L b-2--+ 10-51: 5 32 3 AAA2. IFill each with the correct mixed number' s1l,6s2l' ^1 ,'61' .'65' 1-21' -*31 ^'i1 \"a' 3. lrdAaria's mother baked a cake for her birthday' Aaria ate of the cake lthwhile her sister ate of the cake' How much of the cake is left? 4. From a tin containing L0 titres of paint, 71 litres was used for painting. tin?How much Paint is left in the 6 5. Revathy travels 21 km by walking and 31 km by car' How much more distance does she cover bv car than bY walking? lEypl oettri vretro is a comfortable and safe mode of transportation' In each Delhi Metro coach, the authorities have reserved 1,h of,h\" seats for physically challenged fand women, and th of the seats ior elderly How many more seats are reserved for the former group than the latter? What value does this act of Delhi Metro Rail authorities dePict? 1t tMr Patil spends jrd of his salary on his children's education and of his remaining salary on household expenses' What fraction of his salary is left with him?

Mental Maths Long time ago, an old man decided to divide his property among his three sons. He left one house for each of his three sons and decided to divide his fifteen cars in the following manner: f th of his total cars to his first jth,d son, of the remaining cars to his second son and whatever is left should go to his third son. After the old man's death, his sons did not know how to divide the cars as per their fathert wish. They decided to seek the help ofa wise man in the village. How do you think the wise man would have found the solution? Multiplication of Fractions Multiplying a Fraction by a Whole Number Example: There were twelve monkeys on a tree. Each monkey was given a banana. But each monkey ate only f,th of the banana given to it. How many bananas were eaten rn a ll? Total number of bananas eaten by the 12 monkeys AA -+ ... lz Umes Of',4Lz^- (since multiplication is repeated addition) t4 Step 1:Write the whole number as a fraction with 1as the denominator. t7- z Step 2; Cancel out the common factors in the numerator and the L At denominator. =9 Step 3: Multiply the numerators and the denominators. Therefore,9 bananas were eaten in all.

1\". AnY fraction x Remember ]*o=o; I'o=o 2. Any fraction x 1 = the fraction itself ]\"t= j; i\"=i Multiplying a Fraction by Another Fraction A whole number is To multiply a fraction by another fraction' follow the steps ony counting number given below. of zefo. Step 1: Multiply the numerators and the denominators respectivelY. Step 2: Simplify to reduce it to the lowest form' Example 1: Multiply: ; x? 9 3 2 3x 2 4 9 4x9 36 rh\"',?\"3=+ MExua|mtipp|leica2t:ioMnoufltfirnalVct:iofznsxcfa),na|sobedonebyEcxaanmcpel|e|in3g:thM1e1uclotimplmy:o49n-fxa'3c1t6ors. 11 A1 1x 1 1 t2 1x 1 1 ia 2t3 6 916 3x 4 72 34 Multiplying a Fraction by a Mixed Fraction and cancel out the common Step 1: Change the mixed fraction into an improper fraction factors in the numerator and the denominator' Step 2: Multiply the numerators and the denominators respectively Step 3: Simplify to get a mixed fraction' ,IL Example 1: MuhiPlY: 3: x - ?-3:1x4-L5O= -3 x 4 2x 4 Y93'=)3 2= d 3x1 L

73Example 2: Multiply: g1x 1 971\" 1= 64x 1_ 64 _ e- ':L 3 7 3 2L 21- Example 3: Multiply: !12*3?, z! 11 Ir s2!\"34\"2LL\"=LZl^L,( '23,2113 =2333='t? e5n 1. Simplify the followi ng: a.iq'o b. 9X- c. t:X1 83 -L52-x35^-L l1 d. 94 o 5 i 1-) -2x 2^^u f- :x 0 1 IIJX. 2 T3 1,1, 25 5 h. t. 3 Q 1t n. 4.-14,1X 7-3 20 | k AA m. '1 z-1 o. ^281-X53-^1 o '11 t-1xa -6xr-v 2. Solve the following: \". zl ot +-1 o. !* a! .-.8-1OIJ. -^L5 ')t o ^5-1OT.- L A 3- Of Z- iz5 t5 |th3. The distance between Alka's house and her schoor is 24 km. she warks of the distance and then goes by car with her friend. How much distance does she waik every day? ') Saeed ate :rd of a cake weighing 330 g. How many grams of cake did Saeed eat? j i1't 5. kg of flour and kg of sugar are needed to bake a cake. How much flour and sugar are needed to make 48 such cakes? 6. Sehar goes for jogging every day. lf she can jog a stretch of 1 i10 m in 1 minute, then how much distance will she cover in 25 minutes?

! fA book contains 312 Pages. o\"c\"t of the book have pictures. How many pages do not 7. have pictures in the book? 8. 3iA car travels at a speed of 80 km per hour. lf it takes hours to reach Delhi from Mathura, then what is the distance between Delhi and Mathura? (Hint: dis13n6s = spsgd x time) The cost of 1 metre of lace is { L1ib. What is the cost of 30 metres of lace? 10. 50 students went to a circus. lt the cost of 1 ticket is ( 15 +, then find the total money needed to buy tickets for all ofthem' Fill in each A with one of the given fractions. A 27 g' 16' LOO' 45' 205 Multiplicative Inverse Two numbers are said to be multiplicative inverse of each other if their product is 1' consider the following Products. 34t2 43 t2 250 't'.31371j1x7?=?=1:=717.. =x-=- 520 In each case, the product is 1. I IThus, we can say tt'.t .nO .r\" multiplicative inverse of each other' Also' 1 and 3 are multiplicative inverse of each other, and so on Let us find the multiplicative inverse of a mixed fraction' -tConsider the mlxed lractlon J-. ^1convert 31 into an improper fraction, i.e., 7 222 ^5t.n7c2e2-,-7x-= r Therefore, -7 rs tne multiplicative inve ,r. otlor zL

,1,I,z|,siFind the multiplicative inverse of 41, ] ^a Division of Fractions Division of a Whole Number by a Fraction :Example 1: lf 1 litre of oil is filled in one bottle, then how many bottles will be needed to fill 2 litres of oil? Let us represent this pictorially. trtffiFggg gggg,,@a,tar7,a,g,8a.f3fqiffiffirfr \\ ,-/ \\ ,-/ \\--l \\_,-/ '--) ,,) 1 litre l litre tWe find that 8 bottles are needed to fill 2 litres. In short, we are trying to find ;that how many litres are there in 2 litres, which is given by 2; 1. 2+-i isnothingbut-i x muttipticative inverse of 1,r\".,1\",t= ? =, Thus,8 bottles will be needed. Steps of dividing a whole number by a fraction Step 1: Find the multiplicative inverse ofthe divisor. step 2: Multiply the dividend with the multiplicative inverse of the divisor and write the answer in the lowest form Example 2: Divide 9 bv'49. -;3.4;. t -Step 1:The multiplicative inverse ot ls 9 lHere, = divlsor and = dividendl 3 step 2: Muttipty !,t ,4 4 1,2 10, \" \" 3t g 1 Thus, 9 + is equal to 12.

) Division of a Fraction by a Whole Number The steps of dividing a fraction by a whole number are the same as dividing a whole number by a fraction. The only difference is that you need to write the whole number as a fraction with denominator 1. ]fxamnle, litre of milk is to be divided equally among 5 children' How much milk will each child get? jEach child will get 1 + 5 litres of milk' i1* s i' , ITherefore, l)1= 1 * 1 lsince, ) is the multiplicatlve inverse of 1 -10 Thus, each child will get -1i- litres of milk' Division of a Fraction by a Fraction Example 1: Divide jq89bv a1q. !5 f) pJ,l.. P is the multiplicative inverse of f; ) 89816 (since, p1 =1 8 c. 1q ? i i i.a9aThus, + is equal to Example 2: Divide It4b2Y -:-' 3 8 3 M6 L'qs 9 7 42714 8 I4 Division of a fraction bY a fraction is nothing but multiplying the dividend fraction with the multiPlicative inverse of the divisor fraction. _*

1. Divide the following: a. 7+.5aq;75 b. !+'- L,? d. 4+: q-1 q1 4 f. 1.021:+12-=52 e. =+2= 11 3 e. 84 i. 14+=) ,K. 5.2 l. J..-5--1 ? 7 1*1 -97-33 h. =+2L n- 2.-^I 55 o'5,531I:6+42j1 o. :lq::: ? 3 7 ^418. \"1 ,, -L=!24L2, 2 !: !- s. )_-31i-9. 8 t. \"- . \" 1 1? x. J-AO-IA)- L1. ^2 . -'J. L1 37II_ - .LJ_ 3i2. The cost of 1 kg of rice is t 140. Find the cost of L kg of rice. 3. The cost of 51; litres of petrol is { 315. Find the cost of 1 litre of petrol. 4. A man walks 8zj 6km in 2* hours. How much distance will he cover in j_ hour? 5. 12 kg of sweets are packed lnto small packets each containing; kg of sweets. How many such small packets can be made? 6. somu needs 2; m of cloth to stitch one shirt for himsell How many shirts can he stitch out of 131 metres of cloth? 7. lf jrd can of paint is used to paint a window then how many windows can be painted with 6 cans of paint? 1. The product of two numbers is 12j. lfoneofthe numbers is af , tno tt e other number. 2. 3 There are 30 girls in a class. lf ;th of the class is girls, then what is the total strength of the class? 3. 1 A man spends aq th of his salary on house rent and half of the remaining amount on tran:portation. lf he spends {5OOO more on transportation, then what is his salarv?

Torget OlYmPiod 1-'. A box contains 72 apples' Eight apples in each dozer are good' and the rest are rottln. What fraction ofthe total apples are rotten? 2. A merchant pays duty on certain goods at three different places' At the first 'l of the worth of goods' at the second elace' he clv:s-i:|:f^:h:. place, he gives ]rd and at the third place' he gives 1th of th\" remaining amount' lf remaining amotint the total duty paid by him is ( 1020, what is the total worth of the goods? Solve the following: 97 L1 L1 88 q7 11 = x -:- = ajlt4u!/lt/lastrsl!-eg 1') i ;Objective: Multiply: x Materials required: colour pencils, beads' strips Method 1: Using striPs The denominators of the two fractions are 3 and 5' Hence' draw a rectangle measuring 3 x 5 and divide it into unit squares' colour 1 out of 3 columns, red and 2 out of 5 rows' yellow' Count the number of squares which have both the colours Colour them orange. We find that 2 out of 15 squares are coloured orange' rL22 Thus, the answer is fr, that is,3 x a = G

Method Using beads Take 15 beads. Since, 3 x 5 = 15, 61u;6\" them into 3 groups of 5 beads each. Take 1 group j.out of 3 groups, since the first fraction is You have 5 beads with you. Take 2 out of 5 beads, since the second fraction is 35.' NoW you have 2 beads out of 15 beaos. rneretor5e,*\"53=t35 My Proiect L. Find out from your social studies teacher the area of different states in tndia and the total area of India. Find the fraction of the area of each state to the whole area of India. 2. Find the fraction of area covered bv water to the total area of land in India. Submit a project report to your maths teacher. Did you Know? I fthnbout of the human body is made up of water and other fluids About f;th of our body is filled with bones. The number of bones in a new born baby is 270. While growing, arouno r of $the bones fuse togetheq and so the number of bones in an adult is 206. I ne otameter ot the moon is _! th of that of the earth. I feround th ot the world,s population is in India. I . Around lOth ofthe earth,s surface is covered with water. )

77L27. Find the multiplicative inverse of 4, ] , 1. boxes.2. Fill in the \"na 9 3 f_lTl 2r +lzroLl -725 -Reduce 2J.5a5n-d198 into their lowest forms. 4. Answer the following: a. Arrange in ascending order. b. Add the following. \",3131'37_' 3:.33' ,3,5?' 723!5\" g' 1,1,' 3' 22 t.- 4.L-35+ 2-t-L+ 2^1-, .-8+-9+6- c. Subtract the following. .l.7--L- 1, 3 .I..5-^-t21- ^1 ..t..-5-- 1 7 I 7 d. Solve the following. i. -32o3r.-41 ,ii',2Lo.l 5_1-, ...7.49 '.u''3(.-1t8 2 't' i- tA 5. lf the cost of one glass of juice is { 12; what will be the cost of 20 glasses ofjuice? 6. lf a car runs 7j km with 1 litre of petrol, then how much distance can it cover with rtf, litres of petrol? 7. lf a dog-6eats ;th of a packet of dog food, then how many dogs can be fed with 15 packets of dog food? 8. Aisha needs 21j bundles of wool to knit a sweater, how many sweaters can she knit with 25 bundles of wool? 9. Pranav made L2 bowls ofsoup. lf hewantstoserve L8 people,then how much soup will each person get? 10. How much offruit salad is to be made to serve 22 people, if each person has to be ;given kg of fruit salad? oo

Long long ago, there lived a king who was very impressed with the sense of humour of a poet, who could make anybody forget his worries. The king offered to give the poet whatever he asked for. Instead of asking for a bag of gold or silvet the poet asked for 100 whiplashes. cuess why? Yes, the gatekeepers of the palace had let him in only on the condition that he should share with them whatever he would get from the king. The first gatekeeper asked for )\" ith of what he would get. The second one asked for fth of what was remaining after paying the first gatekeeper. The third gatekeeper said that whatever the poet was left with, must be given equally to him and his wife. The poet decided to teach the greedy gatekeepers a lesson and hence asked for whiplashes. NoW answer the following questions. a. How many whiplashes did each gatekeeper get? b. lf the poet decided to give the whiplashes equally to everyone, what fractions would each one get? What would be the total share ofthe third gatekeeper and his wife then? c. lf the first gatekeeper got 60 whiplashes, how many had the poet asked for? How many did the second o^rdVda^ar oat? d. lf the share of the first gatekeeper was 8 more than the share of the second galeKeeper, how many whiplashes had the poet asked for? What was the share of the third gatekeeper and his wife? e. What is the moral of the storv? I - - .t- -rt---- ..--- I | llr r r r I t fr r r Lf,-r r r r_-l_.-r_ rLr.ra_FE .t.r-t.. E E r r__n.. I Ifl J 9 _Lt'.,:L |st. E l| E A H.

Moths Around Us Rahul was down with fever. His father took him to the doctor. The doctor measured his body temperature and gave him some medicine. Rahul Papa ! The doctor said that my temperature is 102.4'F. What did he mean by point 4? Father 102.4 is a number which is bigger than 102 but less than 103. The point is known Rahul as decimal point. How do we measure point 4? Father Rahul! Did you notice the small markings between two big markings on the thermometer? These small markings help us to measure point 4. Rahul, where else have you seen these small markings? Rahul I have seen it on my ruler and the weighing machine in our house. Father Very good ! Rahul But Papa! Can we have 102.45 also? Or only 1.02.4? Father Yes, we can have 102.45 also. Ask your teacher to explain the details to you. What is a Decimal? Recall the place value concept that you learnt in your previous classes. Hundreds Tens Ones Place Value [Place value of 5 is 500] 5 lPlace value of 6 is 60] 6 7 5x100 lPlace value of 7 is 7] 6x10 @ 7 7x! 6 @

The value of the place of a digit decreases to its one_tenth as you move from left to right on the table. 10= -l1of100;1=: of10 What happens if we move furttier right from the ones place? a a.The value tenths.called ofthe place further decreases to its one-tenth. i.e. of t = this place is L0 Look at the place value chart. Hundrcds Tens Ones zF Tenths Hundredths Thousandths o fL oflooo otroo :1 a10.-r1- 1^, 1 r \"^',1o!o o. 10 of 10 J j.o \"' 10 10 10 Lo=|ll' =100 =10 111 ru Luu 1000 The digits to the right of the ones place are separated using a dot called the decimal point. Consider the numbet 345.2I i10The Place value of 2 is2 x 1) -l- = 10 The Place value of 1 k 1r -1:1010- --l- 100 Such numbers are called decimal numbers. The given number is read as three hundred forty-five point two one- 345 is called the whole part of the decimal number and 2i. is caled the fractionar or decimal part of the decimal number. Tenths When 1is divided into j.O equal parts, then each part is called one-tenth. Fractionalform Decimal form L 10 0.1 Hundredths When one-tenth is further divided into 10 equal parts, each part is called one-hundredth. 4Fractional form of one-hundredth = 100 Decimal form of one-hundredth = O.O1

Thousandths Alwoys reod the digits ofter decimol When one-hundredth is divided into 10 equal parts, then each part is called one-thousandth. lohFractional form of one-thousandth = Decimal form of one-thousandth = 0.001 Expanded Form of Decimals Consider the decimal number 5179.85. The place value of 8 is :i8x 8 = - = 0.8 ]100 :The placevalue of 5 is x s = 100 = o os Therefore, the expanded form of 5L79'85 is - ; i;5OOO + 1OO +70 + 9. = 5OOO+ 100 + 70+9 +0 8 + 0 05 Number Name: lt is read as five thousand one hundred seventy-nine point eight five' Example 1: Expand 78.785. 7a.785=70+7t*8r5*r**rO* Number Name: lt is read as seventy-eight point seven eight five Example 2: Write 25.705 in the expanded form. f7t*51o*25.705 = 20 + 5 + [obtetu\" that there is zero in the l00 th place] Number Name: lt is read as twenty-five point seven zero five' Note that the digits ofter the decimal point sre reod os \"seven zero Iive\" dnd not os seven hundred five. Short Form of Decimals ,L* *l.:ExamDle Consider 300 + 80 + 3 57 10 100 1000 Write its short form.

Here, 383 is the whole part. Now, if we add the decimal part with the whole part, we get, 457 5drt + id + io * 1n- = t383.4571 L, -.J wnole part Decimol port Short form of Hence, the short form is 383.457. decimal number * #Example 2: Write the short form of 2OOO + 300 + 3 - - -# ++#-+Here, 2ooo+ 3oo +, = 2303 + #th= 2303.103 [here is zero in the place here.] Hence, the short form is 2303.103 Pictorial Representation of Decimals Example 1: How will you represent 2.78 pictoriallv? -ss;.\".2.78 has two whole parts, seven tenths and eight hundredths. Example 2: Represent 1.52 pictorially. 10 8 100 L.52 has one whole part, five tenths and two hundredths. t: 2 100 10 Writing Decimal Numerals Example: Write the decimal numeral of -'qha. 20+3+0.3+0.005 itu? 5 -We know that, 0.3 = and O.OoS = 1000

Therefore, 20 + 3+ 0.3+ 0.005= zz+1+ 5 = 25.5U> 10 2-r+.00_170+09 1000 b.700+50+2+0.7+ 0O09 = 700 + 7s2.709 50+ _3c. -= 80+ 5+ 0.03 = 80+ 5+-= 55.U5 100 1. Read and write the number names of the following: d. 272.5L c. 781.001 h. 287 .222 a. 25.12 b. 138.123 e. 345.105 f. 292.292 g. 78.078 2. Write the expanded form of the following decimal numbers. d. 509.09 c. 428.85 h. 1s1.105 a. 23.7 b. 528.13 e. 420.03 f. 108.28 g. 325.91 3. Write the short form of the following and write the number names. a. 3oo+40+!.#.# b. 80+2+- c. tooo+g+a+-a d. 200+50+6+-=- IUUU e. 900+80+L+a f. 700+30+5+ j1^0+]1:+00 1.-0J00 4. A number is broken into thousands, hundreds, tens, ones, one-tenths, one-hundredths and one-thousandths, which are jumbled in the square box. sort them and write the corresponding decimal numeral and its number name. one has been done for you' 100 -1: 100 L0 L L 10 10 L Number Number Name 10 1 100 10 L00 - 1, 22 1, L 223.71 Two hundred 100 twenty-three poant one one.

ID. 10 c. 1, ^d. 1000 1 1000 L000 1000 10 .tUU 1000 1OO 1000 1 10 100 1 1'l 1oo 100 1oo 1OO 100 10 1 1, 1, 1 10 10 10 100 10 10 1oo 10 t L r L0 100 10 10 100 10 Represent pictorially the decimal numbers 3.08, 5.19 and 1.11. Converting Fractions into Decimals Case 1: When the denominator is a multiple of 10, write the numerator and place the decimal point after as many praces starting from the right of the number as the number of zeros in the denominator. 7 o'l; 782 328 nExa mples: =oszs = too =7'ez; rooo case 2: when the denominator is not a murtiple of 10, convert the given fraction into its equivalent fraction with denominator i.o or multiple of 10. Then convert it into decimal numDer. Convert the following into decimar numerats. Exa mple 1: a. 3 3x2 there is no whole number-- 5 wrile ze?o in the ploce of 5x2 10 h 13 I3xQ 52 the whole number, = 0.52 25 25x zf 199 i.e. 3 is writfen ,/ 1o os 0.3. Example 2: convert 411 into decrmat numeral. 20 Consider the fractional part and convert it into its decimal form. 11 l1x 5 55 20 20x5 199 0.55

Combine the whole part and the decimal part to form the decimal number. a2ll=0a5150=0 +.ss Converting Decimals into Fractions Example 1: Write 4.3 as a fraction. 4.3= 4+ 0.3 = {1-11?=0,LN-1I?10T?11?=0 -1::0= 4L 10 Example 2: Write 2.08 as a fraction. 2.08 = 2+ 008 ^ 8 200 8 208 ^8 ^2 100 100 100 L00 L00 25 Remember The number of digits after the decimal point will tell you the number of zeros in the denominator, when we want to convert a decimal number into fraction. You can use a J?R? short cut, 2.83 = 100'(since there are two digits after the decimal point, so we will have two zeros in the denominator.) q17q #Similarly', 5.175 = IUUU (three digits after the decimal point means three zeros in the denominator.) Like and Unlike Decimals Decimals with equal number of digits after the decimal point are called like decimals and the decimals with unequal numbers of digits after the decimal point are called unlike decimals. 1.32,2.1,5,7.89, etc., are like decimals whereas,2,32,7.1,50.038, etc., are unlike decimals. But unlike decimals can be converted into like decimals by adding zeros to the extreme right of the decimal numbers. Adding zeros to the extreme right of the decimal numbers does not change the value of the decimal. Example: L.7, 2.31 and 7.325 can be written as like decimals by adding zeros to the right of the decimal numbers as 1.700, 2.310 and 7.325. NoW these numbers have equal number of digits after the decimal point.

L. Convert the following fractions into decimals. a. z- D, . -t-Lt0 ,? d. 1 e. L7- ^1 10 25 100 25 20 K. - Z^-t- 1 1 h.35_7 1? 100 1000 \"^2 100 1000 100 -10 2. Convert the following decimals into fractions. 7.O3 b. 8.1 c. 100.01 d. 32.L7 L8.007 t. 65.65 c. 3L.013 h. 81.08 i. 59.2 j. 93.09 k. 75.075 55.52 3. Convert the following into like decimals. a. 0.8,0.81,0.719 b. 2.3,2.03,2.OO3 c. 71.23, 68.73,5.73I Comparing and Ordering of Decimals 48.326 and 48.Iis eqr\"r*J I Decimals can also be compared like whole numbers. Let us compare the decimals 49.321 and 32.775. S , r.-l First compare the whole parts, that is,4g and 32. Here,48 > 32. Therefore, 48.321 > 32.175. NoW consider the numbers 48.326 an d 48-L75. The whole part is same for the given numbers. So, consider the first digit to the right of the decimal point. We have, 3 > 1. Therefore, 48.326 > 48.L75. Now compare, 48.32 and 48.325. and 48.325 Equal First convert the unlike decimals into like decimals bv writing 48.32 as 48.320. :--rlllEqual We find,48.32 < 48.325. [See the adjacent representation] Equal We can also adopt a short cut method for comparison. 0<5 Write 48.32 as 48.320 and remove the decimal point from both the numbers. Compare the numbers 48320 and 49325. Since,48320 < 48325, Therefore, 48.32 < 48.325.

Example: Arrange the following decimal numbers in ascending order using the short cut method. 4.82, 4.932,3.28,2.97 Write the numbers as like decimals and remove the decimal points. The numbers will be 4820, 4992,32aO, 2970. The numbers in ascending order are 2970 < 3280 < 4820 < 4932. Therefore, 2.97 < 3.28 < 4.82 < 4.932. Do not compare the decimal parts of the numbers, like the way you compare the whole parts, That is/ 4.37 is not greater than 4.5. Compare them only after writing the given decimal numbers as like decimals. So, after converting them into like decimals, we get 4.37 < 4.50, since 437 < 450. 1. Fill in the boxes with >, < or = sign. 70.5 6.829 a. 3.03 3.33 b. 9.099 9.99 8 d. 4.32 4.g2 e. 0.3 0.13 g. 0.3 0.6 h. 0.333 0.33 2. Arrange the following in ascending order. a.7.29,7.2,7.21.,7.3 b. 72t.3, 120.8, 120.7 9, 727.29 d. 7.O4, 1..I4, 0.O4, L.004 c. 37.6L,37 .5, 42.9, 4S.o 3. Arrange the following in descending order. b. 81..32, 81\".23, 81.03, 8L.3 a. 13.45, t2.77, 17.27, 74.35 A 11 111 tz,L, La,zL, Lz.zz c. 48.5, 38.5, 17.5,27.5 Ajay, Amit and Arjun took part in a longjump event on the sports day. Ajay covered 3.8 m, Amit covered 3.81 m and Arjun covered 3.18 m. Arrange them for the first, second and third prizes based on the distances covered by them. 5. The height of Sonal, Kunal, Usha and Kavya are 90.29 cm, 91.82 cm, 90.92 cm and 92.18 cm, respectively. Arrange them in increasing order of their heights.

Colour the squares and compare the decimals. 0.8L 0.08 Addition of Decimals Convert all the decimals into like decimals. Write all the decimal numbers to be added such that the digits with the same place are written exactly one below the other and the decirial points should also lie one below the other, and then add. Step 3: ln the total, put the decimal point directly below the decimal point of the given nu mbers. Exa mple 1; Add 341.08,871.121 and 55.1 @o Step 1: Write 341.08 as 341.080 and 55.1. as 55.100. 34L.080 Step 2: Write the decimal numbers one below the other and add. 87 t.1,2L step 3: Put the decimal point in the same place as for the eiven + 55.100 numbers. 1,267.301 Thus, 341.08 + 871 .!2! + SS.t = L267.3OL o Example 2: Mansi walked 3.14 km on Monday,2.g2 km on 3. L4 Tuesday and 3.2 km on Wednesday. Find the total 2.82 distance she covered in all the three days. + 3.20 Total distance covered in all the three days v.Ib = 3.t4 km + 2.82 km + 3.2 km Step 1: Write 3.2 as 3.20 (converting given decimal numbers into like decimals). Step 2: Add the decimal numbers. Step 3: put the decimal point directly below the decimal point of the gtven numbers. Thus, Mansi covered 9.16 km in the three days.

Subtraction of Decimals \" ' Step 1: Convert the given decimal numbers into like decimals. Step 2: Write the smaller decimal number below the larger decimal number such that the digits with the same place are exactly one below the other and the decimal points also lie one below the other' Step 3: Subtract the smaller number from the bigger number. Step 4: In the answer, put the decimal point in the same place as for the given n u m bers. Example 1: Subtract 187.29 from 350.7. oz6BDg@.@7 @g Step 1: Write 350.7 as 350.70. -787 .29 Step 2: Write the decimal numbers one below the other and r63.4r subtract the smaller number from the bigger number. Step3: Putthedecimal point in the same place as for the given numbers. Thus, 350.7 - 187.29 = 1-63.41 Example 2: ln a Loo-metre race, Neha ran the distance in 38.2 seconds and Simmi ran the distance in 42.1.2 seconds. Who ran faster and by how much? Time taken by Simmi 42.12 seconds Time taken by Neha - Ja.zu seconos Difference in time 3.92 seconds Thus, Neha ran faster by 3.92 seconds. 7. Add the following: b. 72.19 + 45.54 + 54 d. 39.L6 + 42.47 + 8I.I2 a. 48.7t2 + 32.415 + 12.213 f. L81-.L + 132.22 + 17.19 c. L2t.84 + 72.1+ 55.13 h. 4.32L + 7.923 + 18.t25 + 723.98L e. 77.77 + 99.99 + 55.5 + 22.2 g. 48.712 + 44.12 + 49.423 2. subtract the following: 253.55 - t78.792 a.7a5}r-42.721 b. 183.45 - 81.2

99 .99 - 77 .777 -e. 927 432.8L f. 529.L - 428.53 74.$ - 23 t. 285.72 - 35.293 n. 72.1,3 - 49 3. From the sum of L27.78 and38.57, subtract 57.932. 4. Frcm78.12, subtract the sum of 18.2 and 30.17. 5. Subtract the sum of 92.37 and 121.92 from the sum of 235.2 and 87.19. 6. Add the difference of65.23 and 38.97 to 32t.49. 7. The distances between three villages A, B and C are given below. From A to 8-13.78 km From B to C-28.31 km From A to C-35.91 km a. A person goes from village A to B and then from B to C and then from C to A. What is the total distance travelled bv him? b. lf he goes from village A to C without going to B, will he travel less or more than what he would travel if he goes from A to C through B? 8. A fruit vendor bought 30 kg of apples and then added 12.8 kg of apples to it. lf he sold 40.75 kg of apples, how much is left with him? 9. In a supermarket, 78.25 kg of rice and 92.7 kg of wheat is sold on Monday. L25.5 kg of rice and 128.5 kg of wheat is sold on Tuesday. Which item is sold more in the two days-rice or wheat, and bv how much? l0.2.35kgofwheal,L3kgofjoworand3.45kgofriceismixed,andgrindedformaking porridge. What is the total weight of the mixture? lf it is desired to have 9.5 ks of the mixture, then how much more lowar is to be added? lygl n water tanker carries 1O,OOO titres of water. tt gives 4325.75 litres of water to building A and - 3275.35 litres of water to building B. lf 325.8 litres of water leaks out on the road, how much water would be left in the tanker? Discuss the importance of conservation of water. Also list two ways by which you can conserve water at home.

Multiplication of Decimals Multiplication bY a Whole Number Step 1: lgnore the decimal point and multiply the digits' Step 2: Place the decimal point in the answer so that it has the same number of digits after the decimal point as given in the decimal number' Example 1: MultiplY:45.27 x 6 digits after the decimal point 45 27 x6 277.6-22<-2 digits after the decimal point in the answer Thus,45.27 x 6=27!.62 Example 2: Multiply:0 0005 x 8 5x8=40 decimal number' from the right end There are four digits after the decimal point in the given So, we place the decimal point after four places starting of the answer' Thus, 0.0005 x 8 = 0.0040 or 0 004 MultiPlication bY 10, 100 or 1000 right by multiple of 10' shift the decimal point to the To multiply a decimal number by a number of zeros in the multiplier' the number of places equal to the ExamPle 1: MultiPlY: 6'84 x 10 6.84 x L0 (The multiplier 10 has I' zero) = 68.4 (Move the decimal point by 1 place to the right) Example 2: Multiply:13 468 x 100 L0013.468 x (The multiplier 100 has 2 zeros) 1346.s= ivou\" the decimal point by 2 places to the right) Example 3: MultiplY:93 28 x 1000 100093.28 x (The multiplier L000 has 3 zeros) 93280= (Move the decimal point by 3 places to the right)