8 4. Find the value of 7 6 3 5 - ´ and 6 7 5 3 - ´ 7 6 3 5 - ´ = 7 6 3 5 - ´ ´ = 42 15 - 6 7 6 ( 7) 5 3 53 - ´- ´ = ´ = 42 15 - 7 66 7 3 553 - - \ ´= ´ You will find that multiplication is comulative for rational numbers in general, a ´ b = b ´ a for any two rational. 5. Because Divison is not a binary operation, no need to check, wether it is commulative for rational numbers are not we can directly say it is not commutative. III. Worksheet 1. Find the values of 2 5 7 8 + and 5 2 8 7 + and compare the values. What do you observe. 2. Find the values of 5 3 2 4 æ ö - + ç ÷ è ø and 3 5 4 2 æ ö - ç ÷ + è ø compare them what do you observe. 3. Find the values of 5 3 2 4 æ ö - - ç ÷ è ø and 3 5 4 2 æ ö - ç ÷ - è ø compare them what do you observe. 4. Find the values of 1 3 2 4 - -æ ö ´ ç ÷ è ø and 3 1 4 2 æ öæ ö - - ç ÷ç ÷ ´ è øè ø compare them what do you observe. IV. What I have learnt Partially Perfectly Can't do 1. Rational number are commutative under the operations of addition, substraction and multiplication.
9 I. Learning Outcomes : 1. After completing this worksheet students understood. 2. Rational numbers are associative under the operations of addition and multiplication. II. Conceptual understanding /Model Problem /Example / Activities : l We have seen that integers satisfy associative property under addition and multiplication. Now we shall check it for Rational Numbers. 1. Find the value of 23 5 356 - - æ ö æ ö + + ç ÷ ç ÷ è ø è ø and 23 5 35 6 æ öæ ö - - ç ÷ç ÷ + + è øè ø and compare them. What do you observe. Sol. 23 5 356 - - æ ö æ ö + + ç ÷ ç ÷ è ø è ø 2 18 25 3 30 - - é ù = + ê ú ë û 2 7 3 30 - -æ ö = + ç ÷ è ø 60 21 81 3 30 90 -- - = = ´ 9 10 - = 23 5 35 6 é ùæ ö - - ê ú + + ç ÷ è ø ë û 10 9 5 15 6 é ùéù -+ - = + ê úêú ë ûëû 1 5 15 6 æöæö - - = + ç÷ç÷ èøèø 6 75 15 6 - - = ´ STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 4 Topic / Concept: Rational Numbers - Associative property ACADEMIC YEAR 2020-21 (LEVEL 2)
10 81 90 - = 9 . 10 - = So 2 3 5 23 5 3 5 6 35 6 - -- - æ ö æö æ öæö ++ = + + ç ÷ ç÷ ç ÷ç÷ è ø èø è øèø we find that addition is associative. For rational numbers that is for any three rational numbers a, b and c, a + (b + c) = (a + b) + c 2. Find the value of 7 52 3 49 - æ ö ´ ´ ç ÷ è ø and 75 2 , 34 9 æ ö - ç ÷ ´ ´ è ø compare then what to do inger. Sol: 7 52 3 49 - æ ö ´ ´ ç ÷ è ø 7 (5 2) 3 49 - ´ = ´ ´ 7 10 3 36 - = ´ 7 10 3 36 - ´ = ´ 70 108 - = 35 54 - = 75 2 34 9 æ ö - ç ÷ ´ ´ è ø 75 2 34 9 æ ö - ´ = ´ ç ÷ è ø ´ 35 2 12 9 - = ´ 70 108 - =
11 35 54 - = We observe that multiplication associative for rational numbers, that is for any three rational numbers. a, b and c, a ´ (b ´ c) = (a ´ b) ´ c. III. Worksheet 1. Find the value of 2 41 3 52 - - é ù æ ö + + ê ú ç ÷ è ø ë û and 2 41 3 52 é ù æ öæ ö - - ç ÷ç ÷ + + ê ú è øè ø ë û compare them what do you observe. 2. Find the value of 53 6 22 7 11 æ ö æ ö - ´ ´ ç ÷ ç ÷ è ø è ø and 53 6 22 7 11 æ öæ ö - ç ÷ç ÷ ´ ´ è øè ø compare them. What do you observe. IV. What I have learnt Partially Perfectly Can't do 1. Rational numbers are associative under the operations of addition, substraction and multiplication. Instructions 1. Take some more rational numbers and check. for yourself. 2. Because substraction is not commutative no need to check. Wheather it is associative or not.
12 I. Learning Outcomes : After completing this worksheet students understood 1. Know the identity element of addition and multiplication. II. Conceptual understanding /Model Problem / Example / Activities : l We have seen that zero is called the identify for the addition of integers and when you multiply an integer with 1, you will get back same integer. So zero is called additive identify and "1" is called multiplicative identity l Now we can check for Rational numbers. 1. Find 2 0 7 - + , 2 0 7 æ ö - + ç ÷ è ø and compare them. What do you inger. 2 2 0 . 7 7 - - + = 2 2 0 . 7 7 æ ö - - + = ç ÷ è ø 2 22 00 . 7 77 - -- æ ö \ += + = ç ÷ è ø Remember that '0' is also a rational number. \ Zero is called is an Identity of addition in rational numbers a + 0 = 0 + a = a " a ÎQ. STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 5 Topic / Concept: Rational Numbers - Identity property ACADEMIC YEAR 2020-21 (LEVEL 2)
13 2. Find the values of 2 1 7 - ´ and 2 1 7 æ ö - ´ ç ÷ è ø and compare them. What do you observe. Sol. 2 1 7 - ´ = 2 7 - 2 2 1 7 7 - - ´ = 2 22 1 1 7 77 æ ö æö - -- \ ´=´ = ç ÷ ç÷ è ø èø You will find that when you multiply any rational number with 1 you get back the same rational number as the product. \ We can say that a ´ 1 = 1 ´ a = a. For any rational number a. We say that 1 is the mulitplicative identity for rational numbers. III. Worksheet 1. Find the values of 7 0 2 - + and 7 0 2 æ ö - + ç ÷ è ø compare then. What do you observe. Fill in the blanks. 2. 4 4 .................. 5 5 - - ´ = 3. 3 3 .................. 2 2 - - ´ = 4. Find the value of 7 1 8 ´ and 7 1 8 ´ compare them. What do you observe. IV. What I have learnt Partially Perfectly Can't do 1. Know the identity element of addition and multiplication. Instructions 1. Do a +ev more such additions and multiplcations.
14 I. Learning Outcomes : After completing this worksheet students are able to Know the additive inverse of a Rational numbers. II. Conceptual understanding /Model Problem / Example/ Activities : l Negative of a number (or) additive inverse of a rational number. 1. Find the value of 1 + (-1) and (-1) + 1 compare them what do you observe. Sol : 1 + (-1) = 0 and (-1) + 1 = 0. \ 1 + (-1) = (-1) + 1 = 0. What is the negative of (-1)? It will be 1 also 2 + (-2) = (-2) + 2 = 0. So we say 2 is additive inverse of (-2) and vice versa. In general for any integer a, we have a + (-a) = (-a) + a = 0. So a is the additive inverse of (-a) and (-a) is the additive inverse of a. For the rational number 2 3 , we have 2 2 2 ( 2) 0. 33 3 æ ö - +- += = ç ÷ è ø Also 2 2 0. 3 3 æ ö æö - ç ÷ ç÷ + = è ø èø STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 6 Topic / Concept: Rational Numbers - Inverse property (in addition) ACADEMIC YEAR 2020-21 (LEVEL 2)
15 In general for a rational number a b , we have a a aa 0 bb b b æ ö æ ö æö - - + = += ç ÷ ç ÷ ç÷ è ø è ø èø we say that a b - is the additive inverse of a b and a b is the additive inverse of a b æ ö - ç ÷ è ø . III. Worksheet 1. Fiill in the blank (i) 8 8 ... ..... 0 9 9 - -æ ö += + = ç ÷ è ø (ii) 11 11 ...................... .......... 0. 7 7 æ öæ ö - - + =+= ç ÷ç ÷ è øè ø 2. What is the additive Inverse of b a ? 3. What is the additive inverse of p q - . IV. What I have learnt Partially Perfectly Can't do 1. I understood the concept of additive inverse of a given Rational Number. Instructions 1. Go through the example and exercise sums similar to the learned concept and practice regularly. 2. Collect similar problems. Share and discuss with your friends
16 I. Learning Outcomes : After completing this worksheet you are able to 1. Know the Reciprocal of Rational number other than 0. II. Conceptual understanding /Model Problem /Example / Activities : l By which Rational number would you muliply 8 21 to get product 1. Obviousully by 21 8 . We say that 21 8 is the reciprocal of 8 21 . Since 8 21 1 21 8 ´ = . Q. Can you find what is the reciprocal of '0' ? Is there a rational number which when multiple by 0 gives 1 ? Obviously No ? Thus, zero has No reciprocal. We say that a rational number c d is called the reciprocal or multiplicative inverse of another rational number a b is a c 1. b d ´ = III. Worksheet Fill in the blanks 1. Reciprocal of 1 , x where (x ¹ 0) is ........... STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 7 Topic / Concept: Finding Reciprocal of Rational Numbers ACADEMIC YEAR 2020-21 (LEVEL 2)
17 2. The Reciprocal of a positive rational number is ......... 3. Is 8 9 the multiplicative inverse of 1 1 8 - ? Why or why not ? 4. What must be multiplied by 7 -5 so as to get the product 1 ? 5. Write a rational number that does not have a reciprocal ? 6. Write the rational number that are equal to their reciprocals. IV. What I have learnt Partially Perfectly Can't do 1. Know the Reciprocal of Rational number other than 0. Instructions 1. For more practive of concept collect more example problems, share among your friends and discuss.
18 I. Learning Outcomes : After completing this worksheet students are able to known the representation of Rational numbers on the number line. II. Conceptual understanding /Model Problem / Example / Activities : (i) Natural numbers (ii) Whole numbers (iii) Integers (iv) Rational numbers (v) 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 -4 -3 -2 -1 0 1 2 3 4 -1 1 / 2 0 1 / 2 1 0 1 / 3 2 / 3 1 STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 8 Topic / Concept: Representation of Rational Numbers on Number Line ACADEMIC YEAR 2020-21 (LEVEL 2)
19 Rational numbers can be represented by some points on the number line. Draw a line mark a points on it which represents 0 (zero) set equal distances on both sides of 0 each point on the division represents an integer as shown in above (iii) the length between two successive integers is called as unit length. Let us concider a rational number 1/2 divide unit length between 0 and 1 into 2 equal parts call each them. Sub-divisions. The point at the line indicating 1st sub-division represents 1/2. Similarly in figure (v) we made three. subdivisions. Any rational number can be represented on the number line in this way. In a rational number the numerical below the bar, i.e., the denominator, the number of equal parts into which the first unit has been divided. The numberal above the bar i.e.l the numerator tells 'how many of these parts are considered. III. Worksheet 1. Write the rational number for each point lebelled with a letter. (i) (ii) 2. Represent these numbers on different number lines (i) 7/4 (ii) -5/6 3. Represent 259 , , 11 11 11 --- on the number line. IV. What I have learnt Partially Perfectly Can't do representation of Rational numbers on the number line. Instructions For more practice of concept correct some more. examples problems, share among your friends. 0 / 5 A 2 / 5 3 / 5 ? ? 6 / 5 7 / 5 10/ 5 12/ 5 11/ 5 D E ? ? -12/ 6 J -10/ 5 -9/ 5 ? ? -6/ 5 ? ? 0 / 6 -1/ 6 f -4/ 3 -3/ 6 ? ? I H G
20 I. Learning Outcomes : After completing this worksheet students understood How to obtain rational numbers between two rational numbers by using like fractions method. II. Conceptual understanding /Model Problem / Example / Activities : 1Q. Can you tell the natural numbers between 1 and 5 ? Ans: They are 2, 3 and 4. 2Q. How many natural number are there between 7 and 9 ? Ans. There is only one and it is 8. 3Q. How many natural numbers are there between 10 and 11 ? Ans. No natural number exist between 10 and 11. 4Q. List the integers that are between -5 and 4. Ans. They are -4, -3, -2, -1, 0, 1, 2, 3 5Q. How many integers are three between -1 and 1. Ans. One and it us 0. 6Q. How many integers are there between -9 and -10 ? Ans. None From the above we will find a definite number of natural numbers (Integers) between two natural numbers (Integers). STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 9 Topic / Concept: Rational number between two rational numbers using like fraction method ACADEMIC YEAR 2020-21 (LEVEL 2)
21 7Q. How many rational numbers are three between 3 10 and 7 10 ? Ans. 4 5 , 10 10 and 6 10 . But in the above we can also write 3 10 as 30 100 and 7 10 as 70 100 Now the numbers 31 32 33 , , 100 100 100 ...... 68 69 , 100 100 are all. between 3 10 and 7 10 . The number of these rational numbers is 39. Also 3 10 can be expressed as 3000 10,000 and 7 10 as 7000 10,000 Now, We see that the rational numbers 3001 3002 6999 , ,..... 10,000 10,000 10,000 are between 3 10 and 7 10 . These are 3999 number in all. In this way, we can go on inserting more and more rational numbers between 3 10 and 7 10 . So unlike natural numbers and integers the number of rational numbers between two rational numbers are infinite. Here is one more example. 8Q. Find any ten rational numbers between 5 6 - and 5 8 Ans. We first convert 5 6 - and 5 8 to rational numbers with the same denominators. 5 4 20 6 4 24 -´ - = ´ and 5 3 15 8 3 24 ´ = ´ (24 is the L.C.M of 6 and 8)
22 Thus we have 19 18 17 16 14 , , , .... 24 24 24 24 24 ---- as the rational numbers between 20 24 - and 15 24 . You can take any ten of these. III. Worksheet 1. Write any three rational numbers between -2 and 0. 2. Find ten rational numbers between 2 5 - and 1 2 3. Find the five rational numbers between 3 2 - and 5 3 . 4. Find five rational numbers between 2 3 and 4 5 . 5. Write five rational numbers greater than -2. IV. What I have learnt Partially Perfectly Can't do 1. How to obtain rational numbers between two rational number using like fractions method. Instructions For more practive of concept cllect some more example problems share among you friends and discuss.
23 I. Learning Outcomes : After completing this worksheet Students understood 1. How to obtain rational numbers between two rational numbers by using Average method. II. Conceptual understanding /Model Problem / Example / Activities : l Let us find rational numbers between 1 and 2 one of them is 1.5 or 1 1 2 or 3 2 . This is the mean of 1 and 2 we have studied in earlier classes. l We can use the idea of mean also to find rational numbers between any two given ratonal numbers. Q1. Find a rational number between 1 4 and 1 2 Sol. We find the mean of the given rational numbers. 1 1 2 4 2 æ ö ç ÷ + ¸ è ø 1 2 2 4 æ ö + = ¸ ç ÷ è ø 3 13 4 28 =´= 3 8 lie between 1 4 and 1 2 . This can be seen on the number line also STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 10 Topic / Concept: Rational number between two rational numbers using average method ACADEMIC YEAR 2020-21 (LEVEL 2)
24 1 1 2 4 2 æ ö ç ÷ + ¸ è ø = 3 8 We find the midpoint of AB which is C represented by 1 1 2 4 2 æ ö ç ÷ + ¸ è ø = 3 8 we find that 1 31 4 82 < < If a and b are two rational numbers, then a b 2 + is a rational number Lies between a and b. Such that a b a b 2 + < < This again shows that there are Infinite number of rational numbers between any two given rational numbers. Q2. Find three rational numbers between 1 4 and 3 8 Sol : We find the mean of the given rational numbers. As given in the above example, the mean is 3 8 and 131 482 < < we now find another rational number between 1 4 and 3 8 . For this, we again find the mean of 1 4 and 3 8 that is 1 3 2 4 8 æ ö ç ÷ + ¸ è ø 2 3 2 8 æ ö + = ¸ ç ÷ è ø 5 1 8 2 = ´ 5 16 = Now find the mean of 3 8 and 1 2 1 / 4 1 / 2 3 / 4 1 0 1 / 4 1 / 2 3 / 8 1 / 4 1 / 2 5 / 16 3 / 8 A C B
25 we have 3 1 2 8 2 æ ö ç ÷ + ¸ è ø 3 4 2 8 æ ö + = ¸ ç ÷ è ø 7 1 8 2 = ´ 7 16 = Thus 537 , , 16 8 16 are the three rational numbers between 1 4 and 1 2 . Thus can clearly be shown on the number line as follows 13 5 2 4 8 16 æ ö ç ÷ + ¸= è ø 7 31 2 16 8 2 æ ö =+ ¸ ç ÷ è ø By this average method we can find infinite number of Rational Numbers between given two rational numbers. III. Worksheet 1. Find a rational number between 2 3 and 3 4 . 2. Find three rational numbers between 2 3 and 3 4 . 3. Find three rational numbers between 1 8 and 3 4 . IV. What I have learnt Partially Perfectly Can't do 1. How to obtain rational numbers between two rational numbers by using Average method. Instructions 1. For more practice of concept collect some more example problems, share among your friends and discuss. 1 / 4 7 / 16 5 / 16 3 / 8 1 / 2 0 1 / 4 3 / 8 1 / 2 3 / 4 1
26 I. Learning Outcomes : After completing this worksheet Students are able know the decimal representation of rational numbers. II. Conceptual understanding /Model Problem / Example/ Activities : A rational number can be expressed as terminating or a non-terminating recurring decimal. In order to express a fraction p q in decimal form we divide p by q. Take a rational number 25 16 Division Method : Step 1. Divide Numerator by Denominator. Step 2. Continue division untill we get a remainder less than division. Step 3. Put a decimal Right side of the Quotient and devidend. Step 4. Put a zero right side of the devided after decimal and right side of the remainder and continue the divison process. Step 5. Continue step 4 untill you get the remainder 0 (zero) or up to a required number of places. 16)25(1 16 9 16)25.0(1 16 90 STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 11 Topic / Concept: Decimal representation of rational numbers (terminating) ACADEMIC YEAR 2020-21 (LEVEL 2)
27 Q.2. Write 17 5 in decimal representation. Sol. 5 ) 17.0 (3.4 15 20 20 0 III. Worksheet 1. Write the decimal representation of (i) 1 2 (ii) 13 25 (iii) 8 125 (iv) 1974 10 IV. What I have learnt Partially Perfectly Can't do 1. I have learnt about decimal representation of rational numbers. Instructions For more practice of concept collect, some example problems, share among your friends and discuss. 16) 25.0000 (1.5625 16 90 80 100 96 40 32 80 80 0
28 I. Learning Outcomes : After completing this worksheet Students are able to know the decimal representation of rational numbers. (non terminating - recurring) II. Conceptual understanding /Model Problem / Example/ Activities : A rational number can be expressed as terminating or a non-terminating recurring decimal in order to express a fraction p (q 0) q ¹ in decimal form we divide p by q. Q.1. Convert 5 3 in decimal form \ 5 1.6 3 = It is non - terminating, recurring decimal. Bar on the top of the 6 represents that 6 is repeating. Since one digit is repeating its period 6 and periodicity is 1. By observing above division always we are getting remainder as 2 and Quotient as 6. 3) 5.0 (1.66.... 3 20 18 20 18 2 STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 12 Topic / Concept: Decimal representation of rational numbers (non terminating) ACADEMIC YEAR 2020-21 (LEVEL 2)
29 Q.2. Convent 1 7 in to the decimal form division method. 1 0.142857142857... 7 \ = 1 0.142857 7 Q = The numbers bellow the bar 142857 are repeating It is non terminating and recurring decimal. It's period is 142857 and periodicity is 6 III. Worksheet 1. Communicate the following in decimal form and write their period and periodicity. (i) 1 3 (ii) 17 6 (iii) 11 19 (iv) 20 19 IV. What I have learnt Partially Perfectly Can't do 1. Representing rational Numbers as non-terminating and recurring decimals. Instructions 1. For more practice of concept collect, some more example problems, share among your friends and discuss. 16)1.0 0 0 0 0 0 0 0 0 0(0.142857 01 10 7 30 28 20 14 60 56 40 35 50 49 10 7 30 28 20 14 60 56 40 1428...
30 I. Learning Outcomes : After completing this worksheet Students are able to know the conversion of a decimal number into a fraction. II. Conceptual understanding /Model Problem /Example / Activities : Every terminating decimal or every non-terminating repeating decimal can be expressed in the form of p q . A. Conversion of terminating decimal into the Form of p q . Example 1. Convert 15.75 in the form of p q . Solution: Step 1. Count the number of places after decimal. Step 2. take 1 and right side of it write the number of zeros equal to the number of places ofter decimal, divide and Step 3. Multiply the decimal number by the number obtained in step (2) 15.75 = 63 4 315 1575 100 20 15.75 = 63 4 STATE COUNCIL OF EDUCATION RESEARCH AND TRAINING TELANGANA, HYDERABAD Class: 8 Medium: English Subject: Mathematics Name of the chapter: Rational numbers (Q) Worksheet No.: 13 Topic / Concept: Conversion of decimal number to fraction ACADEMIC YEAR 2020-21 (LEVEL 2)
31 B. How to convert the non-terminating and recurring decimals into p q form. Example 2. Convert 15.075 in p q form. Solution. Step 1. remove the bar and decimal from the given number = 15075. Step 2. Write the number that is not having bar = 150 Step 3. Subtract the number obtained in step (2) From the number obtained in step (1) this is the numerator of out p q ie, p = 15075 - 150 p = 14925. Step 4.Count the digits after decimal = 3 Step 5. Count the periodicity = 2. Step 6. Write number of "9" s equal to periodicity and write zeros (0) equal to digits which are not having bar in the denominator. Step 7. Now, 15.075 = 15075 150 14925 990 990 - = III. Worksheet 1. Convert 0.54 in the form of p q . 2. Convert 15.732 in the form of p q 3. Convert 1.24 in the form of p q 4. Convert 14.5 in the form of p q 5. Convert 0.9 in the form of p q 6. Convert 10.363636 ... in the form of p q
32 7. Convert 12.12343434 ... in the form of p q IV. What I have learnt Partially Perfectly Can't do 1. I can convert of a decimal number . into a fraction Instructions For more practice of concept collect, some more example problems, share among your friends and discuss.
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