Welcome dear students! Today we are going to learn about Rational Numbers from Class 7 Maths.
8.1 INTRODUCTION You began your study of numbers by counting objects around you. The numbers used for this purpose were called counting numbers or natural numbers. They are 1, 2, 3, 4, and so on. By including 0 to natural numbers, we got the whole numbers, i.e., 0, 1, 2, 3, and so on. The negatives of natural numbers were then put together with whole numbers to make up integers. Integers are negative 3, negative 2, negative 1, 0, 1, 2, 3, and so on. We, thus, extended the number system, from natural numbers to whole numbers and from whole numbers to integers. You were also introduced to fractions. These are numbers of the form numerator over denominator, where the numerator is either 0 or a positive integer and the denominator, a positive integer. You compared two fractions, found their equivalent forms and studied all the four basic operations of addition, subtraction, multiplication and division on them. In this Chapter, we shall extend the number system further. We shall introduce the concept of rational numbers alongwith their addition, subtraction, multiplication and division operations.
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8.2 NEED FOR RATIONAL NUMBERS Earlier, we have seen how integers could be used to denote opposite situations involving numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then the distance of 5 km to the left of the same place could be denoted by negative 5. If a profit of rupees 150 was represented by 150 then a loss of rupees 100 could be written as negative 100. There are many situations similar to the above situations that involve fractional numbers. You can represent a distance of 750 meters above sea level as 3/4 km. Can we represent 750 meters below sea level in km? Can we denote the distance of 3/4 km below sea level by negative 3/4? We can see negative 3/4 is neither an integer, nor a fractional number. We need to extend our number system to include such numbers.
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8.3 WHAT ARE RATIONAL NUMBERS? The word rational arises from the term ratio. You know that a ratio like 3:2 can also be written as 3/2. Here, 3 and 2 are natural numbers. Similarly, the ratio of two integers p and q where q is not equal to 0, i.e., p:q can be written in the form p/q. This is the form in which rational numbers are expressed. A rational number is defined as a number that can be expressed in the form p/q, where p and q are integers and q is not equal to 0. Thus, 4/5 is a rational number. Here, p = 4 and q = 5. Is negative 3/4 also a rational number? Yes, because p = negative 3 and q = 4 are integers. You have seen many fractions like 3/8, 4/8, 12/3. All fractions are rational numbers. Can you say why? How about the decimal numbers like 0.5, 2.3? Each of such numbers can be written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 5/10, 0.333 = 333/1000.
Is the number negative 2/3 rational? Yes, it is rational. List ten rational numbers. 1/2, 3/4, negative 5/6, 7/8, negative 2/9, 0/1, 10/11, negative 13/14, 15/16, negative 17/18.
In p/q, the integer p is the numerator, and the integer q which is not equal to 0 is the denominator. Thus, in negative 3/7, the numerator is negative 3 and the denominator is 7.
Mention five rational numbers each of whose (a) Numerator is a negative integer and denominator is a positive integer. negative 1/2, negative 3/4, negative 5/6, negative 7/8, negative 9/10. (b) Numerator is a positive integer and denominator is a negative integer. 1/negative 2, 3/negative 4, 5/negative 6, 7/negative 8, 9/negative 10. (c) Numerator and denominator both are negative integers. negative 1/negative 2, negative 3/negative 4, negative 5/negative 6, negative 7/negative 8, negative 9/negative 10. (d) Numerator and denominator both are positive integers. 1/2, 3/4, 5/6, 7/8, 9/10.
Are integers also rational numbers? Any integer can be thought of as a rational number. For example, the integer negative 5 is a rational number, because you can write it as negative 5/1. The integer 0 can also be written as 0/1 or 0/2 or 0/7. Hence, it is also a rational number. Thus, rational numbers include integers and fractions.
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Equivalent rational numbers A rational number can be written with different numerators and denominators. For example, consider the rational number 2/3. 2/3 = (2 times 2)/(3 times 2) = 4/6. We see that 2/3 is the same as 4/6. Also, 2/3 = (2 times negative 5)/(3 times negative 5) = negative 10/negative 15. So, 2/3 is also the same as negative 10/negative 15. Thus, 2/3 = 4/6 = negative 10/negative 15. Such rational numbers that are equal to each other are said to be equivalent to each other. Again, negative 10/negative 15 = 10/15. By multiplying the numerator and denominator of a rational number by the same non zero integer, we obtain another rational number equivalent to the given rational number. This is exactly like obtaining equivalent fractions. Just as multiplication, the division of the numerator and denominator by the same non zero integer, also gives equivalent rational numbers. For example, 10/15 = (10 divided by 5)/(15 divided by 5) = 2/3, negative 12/24 = (negative 12 divided by 12)/(24 divided by 12) = negative 1/2. We write 2/3 as negative 2/negative 3, 10/15 as negative 10/negative 15.
8.4 POSITIVE AND NEGATIVE RATIONAL NUMBERS Consider the rational number 2/3. Both the numerator and denominator of this number are positive integers. Such a rational number is called a positive rational number. So, 3/8, 5/7, 2/9 are positive rational numbers. The numerator of negative 3/5 is a negative integer, whereas the denominator is a positive integer. Such a rational number is called a negative rational number. So, negative 5/7, negative 3/8, negative 9/5 are negative rational numbers.
Fill in the boxes: (i) negative 5/4 = negative 25/20 = negative 15/12 (ii) negative 3/7 = negative 9/21 = negative 6/14.
Is 5 a positive rational number? Yes, because 5 can be written as 5/1, where both numerator and denominator are positive. List five more positive rational numbers: 1/2, 3/4, 7/8, 9/10, 11/12.
Is negative 8/3 a negative rational number? We know that negative 8/3 = (8 times negative 1)/(3 times negative 1) = negative 8/3, and negative 8/3 is a negative rational number. So, negative 8/3 is a negative rational number. Similarly, 5/negative 7, 6/negative 5, 2/negative 9 are all negative rational numbers. Note that their numerators are positive and their denominators negative. The number 0 is neither a positive nor a negative rational number. What about negative 3/negative 5? You will see that negative 3/negative 5 = (negative 3 times negative 1)/(negative 5 times negative 1) = 3/5. So, negative 3/negative 5 is a positive rational number. Thus, negative 2/negative 5, negative 5/negative 3 are positive rational numbers.
Which of these are negative rational numbers? (i) negative 2/3 is negative. (ii) 5/7 is positive. (iii) 3/negative 5 is negative. (iv) 0 is neither. (v) 6/11 is positive. (vi) negative 2/negative 9 is positive.
Is negative 8 a negative rational number? Yes, negative 8 can be written as negative 8/1, so it is negative. List five more negative rational numbers: negative 1/2, negative 3/4, negative 5/6, negative 7/8, negative 9/10.
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8.5 RATIONAL NUMBERS ON A NUMBER LINE You know how to represent integers on a number line. Let us draw one such number line. The points to the right of 0 are denoted by + sign and are positive integers. The points to the left of 0 are denoted by negative sign and are negative integers. Representation of fractions on a number line is also known to you. Let us see how the rational numbers can be represented on a number line. Let us represent the number negative 1/2 on the number line. As done in the case of positive integers, the positive rational numbers would be marked on the right of 0 and the negative rational numbers would be marked to the left of 0. To which side of 0 will you mark negative 1/2? Being a negative rational number, it would be marked to the left of 0. You know that while marking integers on the number line, successive integers are marked at equal intervals. Also, from 0, the pair 1 and negative 1 is equidistant. So are the pairs 2 and negative 2, 3 and negative 3. In the same way, the rational numbers 1/2 and negative 1/2 would be at equal distance from 0. We know how to mark the rational number 1/2. It is marked at a point which is half the distance between 0 and 1. So, negative 1/2 would be marked at a point half the distance between 0 and negative 1. We know how to mark 3/2 on the number line. It is marked on the right of 0 and lies halfway between 1 and 2. Let us now mark negative 3/2 on the number line. It lies on the left of 0 and is at the same distance as 3/2 from 0. In decreasing order, we have, negative 1/2, negative 2/2 which equals negative 1, negative 3/2, negative 4/2 which equals negative 2. This shows that negative 3/2 lies between negative 1 and negative 2. Thus, negative 3/2 lies halfway between negative 1 and negative 2. Mark negative 5/2 and negative 7/2 in a similar way. Similarly, negative 1/3 is to the left of zero and at the same distance from zero as 1/3 is to the right. So as done above, negative 1/3 can be represented on the number line. Once we know how to represent negative 1/3 on the number line, we can go on representing negative 2/3, negative 4/3, negative 5/3. All other rational numbers with different denominators can be represented in a similar way.
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8.6 RATIONAL NUMBERS IN STANDARD FORM Observe the rational numbers 3/5, 5/8, 2/7, negative 7/11. The denominators of these rational numbers are positive integers and 1 is the only common factor between the numerators and denominators. Further, the negative sign occurs only in the numerator. Such rational numbers are said to be in standard form. A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1. If a rational number is not in the standard form, then it can be reduced to the standard form. Recall that for reducing fractions to their lowest forms, we divided the numerator and the denominator of the fraction by the same non zero positive integer. We shall use the same method for reducing rational numbers to their standard form.
EXAMPLE 1: Reduce negative 45/30 to the standard form. SOLUTION: We have, negative 45/30 = (negative 45 divided by 3)/(30 divided by 3) = negative 15/10 = (negative 15 divided by 5)/(10 divided by 5) = negative 3/2. We had to divide twice. First time by 3 and then by 5. This could also be done as negative 45/30 = (negative 45 divided by 15)/(30 divided by 15) = negative 3/2. In this example, note that 15 is the HCF of 45 and 30. Thus, to reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign, if any. (The reason for ignoring the negative sign will be studied in Higher Classes) If there is negative sign in the denominator, divide by negative HCF.
EXAMPLE 2: Reduce to standard form: (i) 36/negative 24 (ii) negative 3/negative 15 SOLUTION: (i) The HCF of 36 and 24 is 12. Thus, its standard form would be obtained by dividing by negative 12. 36/negative 24 = (36 divided by negative 12)/(negative 24 divided by negative 12) = negative 3/2. (ii) The HCF of 3 and 15 is 3. Thus, negative 3/negative 15 = (negative 3 divided by negative 3)/(negative 15 divided by negative 3) = 1/5.
Find the standard form of (i) negative 18/45 (ii) negative 12/18 SOLUTION: (i) HCF of 18 and 45 is 9. negative 18/45 = (negative 18 divided by 9)/(45 divided by 9) = negative 2/5. (ii) HCF of 12 and 18 is 6. negative 12/18 = (negative 12 divided by 6)/(18 divided by 6) = negative 2/3.
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8.7 COMPARISON OF RATIONAL NUMBERS We know how to compare two integers or two fractions and tell which is smaller or which is greater among them. Let us now see how we can compare two rational numbers. Two positive rational numbers, like 2/3 and 5/7 can be compared as studied earlier in the case of fractions. Mary compared two negative rational numbers negative 1/2 and negative 1/5 using number line. She knew that the integer which was on the right side of the other integer, was the greater integer. For example, 5 is to the right of 2 on the number line and 5 is greater than 2. The integer negative 2 is on the right of negative 5 on the number line and negative 2 is greater than negative 5. She used this method for rational numbers also. She knew how to mark rational numbers on the number line. She marked negative 1/2 and negative 1/5 as follows: She converted negative 1/2 to negative 5/10 and negative 1/5 to negative 2/10 to get a common denominator. She found that negative 1/5 is to the right of negative 1/2. Thus, negative 1/5 is greater than negative 1/2 or negative 1/2 is less than negative 1/5. Can you compare negative 3/4 and negative 2/3? negative 1/3 and negative 1/5? We know from our study of fractions that 1/5 is less than 1/2. And what did Mary get for negative 1/2 and negative 1/5? Was it not exactly the opposite? You will find that, 1/2 is greater than 1/5 but negative 1/2 is less than negative 1/5. Do you observe the same for negative 3/4, negative 2/3 and negative 1/3, negative 1/5? Mary remembered that in integers she had studied 4 is greater than 3 but negative 4 is less than negative 3, 5 is greater than 2 but negative 5 is less than negative 2. The case of pairs of negative rational numbers is similar. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order. For example, to compare negative 7/5 and negative 5/3, we first compare 7/5 and 5/3. We get 7/5 is less than 5/3 and conclude that negative 7/5 is greater than negative 5/3. Take five more such pairs and compare them. Which is greater negative 3/8 or negative 2/7? negative 3/8 is greater. Which is greater negative 4/3 or negative 3/2? negative 4/3 is greater.
Comparison of a negative and a positive rational number is obvious. A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a negative rational number will always be less than a positive rational number. Thus, negative 2/7 is less than 1/2. To compare rational numbers negative 3/negative 5 and negative 2/negative 7 reduce them to their standard forms and then compare them. negative 3/negative 5 = 3/5 and negative 2/negative 7 = 2/7. Since 3/5 is greater than 2/7, negative 3/negative 5 is greater than negative 2/negative 7.
EXAMPLE 3: Do negative 4/9 and negative 16/36 represent the same rational number? SOLUTION: Yes, because negative 4/9 = (negative 4 times negative 4)/(9 times negative 4) = 16/negative 36 = negative 16/36 or negative 16/36 = (negative 16 divided by 4)/(36 divided by 4) = negative 4/9.
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8.8 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS Reshma wanted to count the whole numbers between 3 and 10. From her earlier classes, she knew there would be exactly 6 whole numbers between 3 and 10. Similarly, she wanted to know the total number of integers between negative 3 and 3. The integers between negative 3 and 3 are negative 2, negative 1, 0, 1, 2. Thus, there are exactly 5 integers between negative 3 and 3. Are there any integers between negative 3 and negative 2? No, there is no integer between negative 3 and negative 2. Between two successive integers the number of integers is 0. Thus, we find that number of integers between two integers are limited (finite). Will the same happen in the case of rational numbers also? Reshma took two rational numbers negative 3/5 and negative 1/3. She converted them to rational numbers with same denominators. So negative 3/5 = negative 9/15 and negative 1/3 = negative 5/15. We have negative 9/15 is less than negative 8/15 is less than negative 7/15 is less than negative 6/15 is less than negative 5/15 or negative 3/5 is less than negative 8/15 is less than negative 7/15 is less than negative 6/15 is less than negative 1/3. She could find rational numbers negative 8/15, negative 7/15, negative 6/15 between negative 3/5 and negative 1/3. Are the numbers negative 8/15, negative 7/15, negative 6/15 the only rational numbers between negative 3/5 and negative 1/3? We have negative 3/5 = negative 18/30 and negative 8/15 = negative 16/30. And negative 18/30 is less than negative 17/30 is less than negative 16/30. i.e., negative 3/5 is less than negative 17/30 is less than negative 8/15. Hence negative 3/5 is less than negative 17/30 is less than negative 8/15 is less than negative 7/15 is less than negative 6/15 is less than negative 1/3. So, we could find one more rational number between negative 3/5 and negative 1/3. By using this method, you can insert as many rational numbers as you want between two different rational numbers. For example, negative 3/5 = (negative 3 times 30)/(5 times 30) = negative 90/150 and negative 1/3 = (negative 1 times 50)/(3 times 50) = negative 50/150. We get 39 rational numbers negative 89/150 to negative 51/150 between negative 90/150 and negative 50/150 i.e., between negative 3/5 and negative 1/3. You will find that the list is unending. Can you list five rational numbers between negative 5/3 and negative 8/7? We can find unlimited number of rational numbers between any two rational numbers.
Find five rational numbers between negative 5/7 and negative 3/8. Convert to common denominator 56. negative 5/7 = negative 40/56, negative 3/8 = negative 21/56. Five rational numbers between them are negative 39/56, negative 38/56, negative 37/56, negative 36/56, negative 35/56.
EXAMPLE 4: List three rational numbers between negative 2 and negative 1. SOLUTION: Let us write negative 1 and negative 2 as rational numbers with denominator 5. We have, negative 1 = negative 5/5 and negative 2 = negative 10/5. So, negative 10/5 is less than negative 9/5 is less than negative 8/5 is less than negative 7/5 is less than negative 6/5 is less than negative 5/5 or negative 2 is less than negative 9/5 is less than negative 8/5 is less than negative 7/5 is less than negative 6/5 is less than negative 1. The three rational numbers between negative 2 and negative 1 would be, negative 9/5, negative 8/5, negative 7/5.
EXAMPLE 5: Write four more numbers in the following pattern: negative 1/3, negative 2/6, negative 3/9, negative 4/12, ... SOLUTION: We have, negative 2/6 = (negative 1 times 2)/(3 times 2), negative 3/9 = (negative 1 times 3)/(3 times 3), negative 4/12 = (negative 1 times 4)/(3 times 4). Thus, we observe a pattern in these numbers. The other numbers would be (negative 1 times 5)/(3 times 5) = negative 5/15, (negative 1 times 6)/(3 times 6) = negative 6/18, (negative 1 times 7)/(3 times 7) = negative 7/21, (negative 1 times 8)/(3 times 8) = negative 8/24.
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EXERCISE 8.1 1. List five rational numbers between: (i) negative 1 and 0: negative 1/6, negative 2/6, negative 3/6, negative 4/6, negative 5/6. (ii) negative 2 and negative 1: negative 11/6, negative 12/6, negative 13/6, negative 14/6, negative 15/6. (iii) negative 4/5 and negative 2/3: Common denominator 60. negative 48/60 and negative 40/60. Five: negative 47/60, negative 46/60, negative 45/60, negative 44/60, negative 43/60. (iv) negative 1/2 and 2/3: Common denominator 6. negative 3/6 and 4/6. Five: negative 2/6, negative 1/6, 0/6, 1/6, 2/6.
2. Write four more rational numbers in each pattern: (i) negative 3/5, negative 6/10, negative 9/15, negative 12/20: Next are negative 15/25, negative 18/30, negative 21/35, negative 24/40. (ii) negative 1/4, negative 2/8, negative 3/12: Next are negative 4/16, negative 5/20, negative 6/24, negative 7/28. (iii) negative 1/6, negative 2/12, negative 3/18, negative 4/24: Next are negative 5/30, negative 6/36, negative 7/42, negative 8/48. (iv) negative 2/3, 2/negative 3, 4/negative 6, 6/negative 9: Next are 8/negative 12, 10/negative 15, 12/negative 18, 14/negative 21.
3. Give four rational numbers equivalent to: (i) negative 2/7: negative 4/14, negative 6/21, negative 8/28, negative 10/35. (ii) 5/negative 3: 10/negative 6, 15/negative 9, 20/negative 12, 25/negative 15. (iii) 4/9: 8/18, 12/27, 16/36, 20/45.
4. Draw the number line and represent the following rational numbers on it: (i) 3/4: Divide the segment between 0 and 1 into 4 equal parts. Mark the 3rd point from 0. (ii) negative 5/8: Divide the segment between 0 and negative 1 into 8 equal parts. Mark the 5th point to the left of 0. (iii) negative 7/4: Divide the segment between negative 1 and negative 2 into 4 equal parts. Mark the 3rd point to the left of negative 1. (iv) 7/8: Divide the segment between 0 and 1 into 8 equal parts. Mark the 7th point from 0.
5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S. From the textbook diagram, point T is at negative 2 and U is at negative 1. The segment TU is divided into 3 equal parts, so each part is 1/3 unit. Moving right from T, R is at negative 5/3 and S is at negative 4/3. Similarly, point A is at 2 and B is at 3. The segment AB is divided into 3 equal parts of 1/3 unit each. Moving right from A, P is at 7/3 and Q is at 8/3. Thus, P = 7/3, Q = 8/3, R = negative 5/3, S = negative 4/3.
6. Which of the following pairs represent the same rational number? (i) negative 7/21 and 3/9: negative 1/3 and 1/3. Not same. (ii) negative 16/20 and 20/25: negative 4/5 and 4/5. Not same. (iii) negative 2/negative 3 and 2/3: 2/3 and 2/3. Same. (iv) negative 3/5 and negative 12/20: negative 3/5 and negative 3/5. Same. (v) 8/negative 5 and negative 24/15: negative 8/5 and negative 8/5. Same. (vi) 1/negative 3 and negative 1/9: negative 1/3 and negative 1/9. Not same. (vii) negative 5/negative 9 and 5/negative 9: 5/9 and negative 5/9. Not same.
7. Rewrite the following rational numbers in the simplest form: (i) negative 8/6 = negative 4/3. (ii) 25/45 = 5/9. (iii) negative 44/72 = negative 11/18. (iv) negative 8/10 = negative 4/5.
8. Fill in the boxes with the correct symbol out of greater than, less than, and equal to. (i) negative 5/7 is less than 2/3 (ii) negative 4/5 is less than negative 5/7 (iii) negative 7/8 is equal to negative 14/16 (iv) negative 8/5 is greater than negative 7/4 (v) 1/negative 3 is less than negative 1/4 (vi) negative 5/11 is equal to negative 5/11 (vii) 0 is greater than negative 7/6
9. Which is greater in each of the following: (i) 2/3 or 5/2: 5/2 is greater. (ii) negative 5/6 or negative 4/3: negative 5/6 is greater. (iii) negative 3/4 or negative 2/3: negative 2/3 is greater. (iv) negative 1/4 or 1/4: 1/4 is greater. (v) negative 3/2 and 7/3: 7/3 is positive and negative 3/2 is negative. A positive rational number is always greater than a negative rational number. Therefore, 7/3 is greater.
10. Write the following rational numbers in ascending order: (i) negative 3/5, negative 2/5, negative 1/5. Already ascending. (ii) negative 1/3, negative 2/9, negative 4/3: Convert to denominator 9. negative 3/9, negative 2/9, negative 12/9. Order: negative 4/3, negative 1/3, negative 2/9. (iii) negative 3/7, negative 3/2, negative 3/4: Convert to denominator 28. negative 12/28, negative 42/28, negative 21/28. Order: negative 3/2, negative 3/4, negative 3/7.
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8.9 OPERATIONS ON RATIONAL NUMBERS 8.9.1 Addition Let us add two rational numbers with same denominators, say 7/3 and negative 5/3. We find 7/3 + (negative 5/3). On the number line, the distance between two consecutive points is 1/3. So adding negative 5/3 to 7/3 will mean, moving to the left of 7/3, making 5 jumps. We reach at 2/3. So, 7/3 + (negative 5/3) = 2/3. Let us now try this way: 7/3 + (negative 5/3) = (7+(negative 5))/3 = 2/3. We get the same answer. Find 6/5 + (negative 2/5) and 3/7 + (negative 5/7), in both ways and check if you get the same answers. 6/5 + (negative 2/5) = 4/5. 3/7 + (negative 5/7) = negative 2/7. Similarly, negative 7/8 + 5/8 would be (negative 7+5)/8 = negative 2/8 = negative 1/4. Also, negative 7/8 + 5/8 = (negative 7+5)/8 = negative 2/8. Are the two values same? Yes. So, we find that while adding rational numbers with same denominators, we add the numerators keeping the denominators same. Thus, negative 11/5 + 7/5 = (negative 11+7)/5 = negative 4/5. How do we add rational numbers with different denominators? As in the case of fractions, we first find the LCM of the two denominators. Then, we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator. Then, add the two rational numbers. For example, let us add negative 7/5 and negative 2/3. LCM of 5 and 3 is 15. So, negative 7/5 = negative 21/15 and negative 2/3 = negative 10/15. Thus, negative 7/5 + (negative 2/3) = negative 21/15 + (negative 10/15) = negative 31/15.
Additive Inverse What will be negative 4/7 + 4/7? negative 4/7 + 4/7 = (negative 4+4)/7 = 0. Also, 4/7 + (negative 4/7) = 0. Similarly, negative 2/3 + 2/3 = 0 = 2/3 + (negative 2/3). In the case of integers, we call negative 2 as the additive inverse of 2 and 2 as the additive inverse of negative 2. For rational numbers also, we call negative 4/7 as the additive inverse of 4/7 and 4/7 as the additive inverse of negative 4/7. Similarly, negative 2/3 is the additive inverse of 2/3 and 2/3 is the additive inverse of negative 2/3. What will be the additive inverse of negative 3/9, 9/11, negative 5/7? They are 3/9, negative 9/11, 5/7 respectively.
EXAMPLE 6: Satpal walks 2/3 km from a place P, towards east and then from there 15/7 km towards west. Where will he be now from P? SOLUTION: Let us denote the distance travelled towards east by positive sign. So, the distances towards west would be denoted by negative sign. Thus, distance of Satpal from the point P would be 2/3 + (negative 15/7) = (2 times 7)/(3 times 7) + (negative 15 times 3)/(7 times 3) = 14/21 + (negative 45/21) = negative 31/21 = negative 1 10/21. Since it is negative, it means Satpal is at a distance 1 10/21 km towards west of P.
Find: negative 13/7 + 6/7 = negative 7/7 = negative 1. 19/5 + (negative 7/5) = 12/5. Find: (i) negative 3/7 + 2/3 = negative 9/21 + 14/21 = 5/21. (ii) negative 5/6 + (negative 3/11) = negative 55/66 + (negative 18/66) = negative 73/66.
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8.9.2 Subtraction Savita found the difference of two rational numbers 5/7 and 3/8 in this way: 5/7 minus 3/8 = 40/56 minus 21/56 = 19/56. Farida knew that for two integers a and b she could write a minus b = a + (negative b). She tried this for rational numbers also and found, 5/7 minus 3/8 = 5/7 + (negative 3/8) = 19/56. Both obtained the same difference. Try to find 7/8 minus 5/9 and 3/11 minus 8/7 in both ways. Did you get the same answer? Yes. So, we say while subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number. Thus, 5/3 minus 14/5 = 5/3 + additive inverse of 14/5 = 5/3 + (negative 14/5) = 25/15 + (negative 42/15) = negative 17/15 = negative 1 2/15. What will be 2/7 minus (negative 5/6)? 2/7 minus (negative 5/6) = 2/7 + additive inverse of negative 5/6 = 2/7 + 5/6 = 12/42 + 35/42 = 47/42 = 1 5/42.
Find: (i) 7/9 minus 2/5 = 35/45 minus 18/45 = 17/45. (ii) 2/1 minus (negative 5/13) = 2 + 5/13 = 31/13.
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8.9.3 Multiplication Let us multiply the rational number negative 3/5 by 2, i.e., we find negative 3/5 times 2. On the number line, it will mean two jumps of 3/5 to the left. We reach at negative 6/5. Let us find it as we did in fractions. negative 3/5 times 2 = (negative 3 times 2)/5 = negative 6/5. We arrive at the same rational number. Find negative 4/7 times 3 and negative 6/5 times 4, using both ways. negative 4/7 times 3 = negative 12/7. negative 6/5 times 4 = negative 24/5. So, we find that while multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged. Let us now multiply a rational number by a negative integer, negative 2/9 times (negative 5) = (negative 2 times (negative 5))/9 = 10/9. Remember, negative 5 can be written as negative 5/1. So, negative 2/9 times negative 5/1 = 10/9. Similarly, 3/11 times (negative 2) = 3/11 times negative 2/1 = negative 6/11. Based on these observations, we find that, negative 3/8 times 5/7 = (negative 3 times 5)/(8 times 7) = negative 15/56. So, as we did in the case of fractions, we multiply two rational numbers in the following way: Step 1 Multiply the numerators of the two rational numbers. Step 2 Multiply the denominators of the two rational numbers. Step 3 Write the product as Result of Step 1 over Result of Step 2. Thus, negative 3/5 times 2/7 = (negative 3 times 2)/(5 times 7) = negative 6/35. Also, negative 5/8 times negative 9/7 = ((negative 5) times (negative 9))/(8 times 7) = 45/56.
What will be (i) 3/5 times (negative 7/6) = negative 21/30 = negative 7/10. (ii) 5/2 times (negative 2/5) = negative 10/10 = negative 1. Find: (i) negative 3/4 times 1/7 = negative 3/28. (ii) 2/3 times negative 5/9 = negative 10/27.
TRY THESE: Find: (i) 2/3 times negative 7/8 = negative 14/24 = negative 7/12. (ii) negative 6/7 times 5/7 = negative 30/49.
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8.9.4 Division We have studied reciprocals of a fraction earlier. What is the reciprocal of 2/7? It will be 7/2. We extend this idea of reciprocals to non-zero rational numbers also. The reciprocal of negative 2/7 will be 7/negative 2 i.e., negative 7/2; that of negative 3/5 would be negative 5/3. What will be the reciprocal of negative 6/11 and 8/5? They are negative 11/6 and 5/8.
Product of reciprocals The product of a rational number with its reciprocal is always 1. For example, negative 4/9 times reciprocal of negative 4/9 = negative 4/9 times negative 9/4 = 1. Similarly, negative 6/13 times negative 13/6 = 1. Try some more examples and confirm this observation.
Savita divided a rational number 4/9 by another rational number negative 5/7 as, 4/9 divided by negative 5/7 = 4/9 times 7/negative 5 = 28/negative 45 = negative 28/45. She used the idea of reciprocal as done in fractions. Arpit first divided 4/9 by 5/7 and got 28/45. He finally said 4/9 divided by negative 5/7 = negative 28/45. How did he get that? He divided them as fractions, ignoring the negative sign and then put the negative sign in the value so obtained. Both of them got the same value negative 28/45. Try dividing 2/3 by negative 5/7 both ways and see if you get the same answer. This shows, to divide one rational number by the other non-zero rational number we multiply the rational number by the reciprocal of the other. Thus, 7/2 divided by 4/3 = 7/2 times reciprocal of 4/3 = 7/2 times 3/4 = 21/8.
[CHECKPOINT]
EXERCISE 8.2 1. Find the sum: (i) 5/4 + (negative 11/4) = (5 minus 11)/4 = negative 6/4 = negative 3/2. (ii) 5/3 + 3/5 = 25/15 + 9/15 = 34/15. (iii) negative 9/10 + 22/15 = negative 27/30 + 44/30 = 17/30. (iv) negative 3/11 + (negative 5/9) = negative 27/99 + (negative 55/99) = negative 82/99. (v) negative 8/19 + (negative 2/57) = negative 24/57 + (negative 2/57) = negative 26/57. (vi) negative 2/3 + 0 = negative 2/3.
2. Find: (i) 7/24 minus 17/36 = 21/72 minus 34/72 = negative 13/72. (ii) 5/63 minus (negative 6/21) = 5/63 + 18/63 = 23/63. (iii) negative 6/13 minus (negative 7/15) = negative 6/13 + 7/15 = negative 90/195 + 91/195 = 1/195. (iv) negative 3/8 minus 7/11 = negative 33/88 minus 56/88 = negative 89/88 = negative 1 1/88.
3. Find the product: (i) 9/2 times (negative 7/4) = negative 63/8. (ii) 3/10 times (negative 9) = negative 27/10. (iii) negative 6/5 times 9/11 = negative 54/55. (iv) 3/7 times (negative 2/5) = negative 6/35. (v) 3/11 times 2/5 = 6/55. (vi) negative 3/5 times negative 5/3 = 15/15 = 1.
4. Find the value of: (i) (negative 4) divided by 2/3 = negative 4 times 3/2 = negative 12/2 = negative 6. (ii) negative 3/5 divided by 2 = negative 3/5 times 1/2 = negative 3/10. (iii) negative 4/5 divided by (negative 3) = negative 4/5 times negative 1/3 = 4/15. (iv) negative 1/8 divided by 3/4 = negative 1/8 times 4/3 = negative 4/24 = negative 1/6. (v) negative 2/13 divided by 1/7 = negative 2/13 times 7/1 = negative 14/13. (vi) negative 7/12 divided by (negative 2/13) = negative 7/12 times negative 13/2 = 91/24.
[CHECKPOINT]
WHAT HAVE WE DISCUSSED? 1. A number that can be expressed in the form p/q, where p and q are integers and q is not equal to 0, is called a rational number. The numbers negative 2/7, 3/8, 3 are rational numbers. 2. All integers and fractions are rational numbers. 3. If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer, we get a rational number which is said to be equivalent to the given rational number. For example negative 3/7 = (negative 3 times 2)/(7 times 2) = negative 6/14. So, we say negative 6/14 is the equivalent form of negative 3/7. Also note that negative 6/14 = (negative 6 divided by 2)/(14 divided by 2) = negative 3/7. 4. Rational numbers are classified as Positive and Negative rational numbers. When the numerator and denominator, both, are positive integers, it is a positive rational number. When either the numerator or the denominator is a negative integer, it is a negative rational number. For example, 3/8 is a positive rational number whereas negative 8/9 is a negative rational number. 5. The number 0 is neither a positive nor a negative rational number. 6. A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1. The numbers negative 1/3, 2/7 are in standard form. 7. There are unlimited number of rational numbers between two rational numbers. 8. Two rational numbers with the same denominator can be added by adding their numerators, keeping the denominator same. Two rational numbers with different denominators are added by first taking the LCM of the two denominators and then converting both the rational numbers to their equivalent forms having the LCM as the denominator. For example, negative 2/3 + 3/8 = negative 16/24 + 9/24 = negative 7/24. Here, LCM of 3 and 8 is 24. 9. While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number. Thus, 7/8 minus 2/3 = 7/8 + additive inverse of 2/3 = 7/8 + (negative 2/3) = 21/24 + (negative 16/24) = 5/24. 10. To multiply two rational numbers, we multiply their numerators and denominators separately, and write the product as product of numerators over product of denominators. 11. To divide one rational number by the other non-zero rational number, we multiply the rational number by the reciprocal of the other. Thus, negative 7/2 divided by 4/3 = negative 7/2 times (reciprocal of 4/3) = negative 7/2 times 3/4 = negative 21/8.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]