14 Perfect Mathematics Class 8 Teaching Hours : 45 2 ARITHMETIC 2.1 Whole Numbers 2.1.1 Introduction of Whole Numbers In our daily life, we use decimal number system. We feel it easy to use this number system. The decimal number system has base 10 as it has ten different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. There are other number systems like Binary number system, (base 2), Quinary number system (base 5), etc. Now, see the basic characteristics of each of these number systems. Binary number system It has a base 2. It uses two symbols 0 and 1 as its digits. Quinary number system It has a base 5. It uses five symbols 0, 1, 2, 3 and 4 as its digits. Decimal number system It has a base 10. It uses ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 as its digits. Decimal (Base Ten) Numeration System As the decimal number system has ten different symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, it is also called base 10 numeration system. The value of a digit in a numeral depends upon its place. For example, let's take 13 pencils and regroup them into the group of 10 pencils. 13 = 10 + 3 = 1 × 10 + 3 × 1 = 1 × 101 + 3 × 100 Again, take 34 pencils and regroup them into the group of 10 pencils each. 34 = 30 + 4 = 3 × 10 + 4 × 1 = 3 × 101 + 4 × 100
Whole Numbers 15 Example 1: Express 324810 in the expanded form. Solution: Here, 3248= 3 (thousands) + 2 (hundreds) + 4 (tens) + 8 (ones) = 3 × 1000 + 2 × 100 + 4 × 10 + 8 × 1 = 3×103 + 2×102 + 4×101 + 8×100 The numeral represents three thousands two hundreds and forty eight. Binary (Base Two) Numeration System The binary number system has two symbols, 0 and 1 as its digits. It is also called base 2 numeration system. For example, let's take 15 pencils and regroup them into the group of 2 pencils each. Again, arrange the groups of 2 pencils into the base of 2 with maximum possible powers. So, 15 = 8 + 4 + 2 + 1 = 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 We have the following place value notation in base 2 numeration system. Number Sixteen Eight Four Two One Notation in base 2 100002 10002 1002 102 12 Complete the following table of binary number conversion. Number Names Decimal Number Groups of 2 into the base 2 Binary Number One 1 1 12 Two 2 2 + 0 102 Three 3 2 + 1 112 Four 4 4 + 0 + 0 1002 Five 5 4 + 0 + 1 Six 6 4 + 2 + 0 Seven 7 4 + 2 + 1 Eight 8 8 + 0 + 0 + 0 Nine 9 8 + 0 + 0 + 1 Ten 10 Eleven 11 Activity - 1
16 Perfect Mathematics Class 8 Properties of Binary Numeration System The binary numeration system is a base-2 number system, meaning that it only uses two digits : 0 and 1. In this system, each digit represents a power of 2, starting from 20 , which represents 1, and increasing by a factor of 2 for each successive digit. The properties of the binary numeration system are : (i) The binary numeration system uses only two digits, 0 and 1. (ii) The value of each digit in a binary number depends on its position within the number. (iii) Each digit in a binary number represents a power of 2. Example 2: Convert 101112 into decimal number. Solution: We have, 101112 = 1 (sixteen) + 0 (eight) + 1 (four) + 1 (two) + 1 (one) = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20 = 16 + 0 + 4 + 2 + 1 = 23 Therefore, 101112 = 2310. The following table gives an outlook to the numerals used in decimal and binary numeration systems. Number Names Decimal Number Base 2 grouping Binary 24 23 22 21 20 Number Zero 0 0 0 One 1 1 1 Three 3 1 1 11 Four 4 1 0 0 100 Five 5 1 0 1 101 Seven 7 1 1 1 111 Twelve 12 1 1 0 0 1100 Fifteen 15 1 1 1 1 1111
Whole Numbers 17 Number Names Decimal Number Base 2 grouping Binary 24 23 22 21 20 Number Sixteen 16 1 0 0 0 0 10000 Nineteen 19 1 0 0 1 1 10011 Twenty two 22 1 0 1 1 0 10110 Twenty three 23 1 0 1 1 1 10111 Twenty eight 28 1 1 1 0 0 11100 Twenty nine 29 1 1 1 0 1 11101 Thirty one 31 1 1 1 1 1 11111 Example 3: Convert 24310 into binary number. Solution: We can convert 24310 into binary number by dividing it by 2 as follows : 2 243 Remainder 2 121 1 2 60 1 2 30 0 2 15 0 2 7 1 2 3 1 2 1 1 0 1 The required number is 111100112. The required number is 111100112. Alternately, We can convert decimal number 243 into binary number just making the groups of power of 2 as follows : 243 = 128 + 64 + 32 + 16 + 0 + 0 + 2 + 1 = 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 = 111100112
18 Perfect Mathematics Class 8 Quinary (Base Five) Numeration System In the Quinary Numeration System (QNS), we use five digits 0, 1, 2, 3 and 4. As the system uses 5 digits, it is also called base 5 numeration system. For example, let's take 22 pencils and regroup them into the group of 5 pencils each. So, 22 = 20 + 2 = 4 × 5 + 2 × 1 = 4 × 51 + 2 × 50 Complete the following table of quinary number conversion. Quinary Number Expanded form Decimal Number Number Name 1 1 1 One 10 1 × 51 + 0 × 50 5 Five 11 1 × 51 + 1 × 50 6 Six 23 2 × 51 + 3 × 50 13 Thirteen 123 1 × 52 + 2 × 51 + 3 × 50 38 Thirty eight 204 2 × 52 + 0 × 51 + 4 × 50 342 3 × 52 + 4 × 51 + 2 × 50 1234 1 × 54 + 2 × 52 + 3 × 51 + 4 × 50 2340 2311 23014 Activity - 2 Properties of Quinary Numeration System The quinary numeration system is a base-5 number system, meaning that it only uses five digits : 0, 1, 2, 3, and 4. In this system, each digit represents a power of 5, starting from 50 , which represents 1, and increasing by a factor of 5 for each successive digit. The properties of the quinary numeration system are : (i) The quinary numeration system uses five digits : 0, 1, 2, 3, and 4. (ii) The value of each digit in a quinary number depends on its position within the number. (iii) Each digit in a quinary number represents a power of 5.
Whole Numbers 19 We have the following place value notation in base 5 numeration system. Number Six Hundred Twenty Fives One Hundred Twenty Fives Twenty Fives Fives Ones Notation in base 5 100005 10005 1005 105 15 In the Quinary Number System (QNS), the numerals 40235 represent 4 hundred twenty fives, 0 twenty fives, 2 fives and 3 ones. The following table gives an outlook to the numerals used in decimal and quinary numeration systems. Number Names Decimal Number Base 5 grouping Quinary 5 Number 3 52 51 50 One 1 1 1 Two 2 2 2 Three 3 3 3 Four 4 4 4 Five 5 1 0 10 Six 6 1 1 11 Seven 7 1 2 12 Eight 8 1 3 13 Nine 9 1 4 14 Ten 10 2 0 20 Eleven 11 2 1 21 Sixteen 16 3 1 31 Nineteen 19 3 4 34 Twenty 20 4 0 40 Seventy 70 2 4 0 240 Hundred Sixty 160 1 1 2 0 1120 Example 4: Convert 24315 into decimal number. Solution: Here, 24315 is converted into decimal number by taking the place values of the digits as follows : 24315 = 2 × 53 + 4 × 52 + 3 × 51 + 1 × 50 = 2 × 125 + 4 × 25 + 3 × 5 + 1
20 Perfect Mathematics Class 8 = 250 + 100 + 15 + 1 = 366 (decimal number) \ 24315 = 36610. Example 5: Convert 48910 into quinary number. Solution: We can convert 48910 into quinary number dividing it successively by 5 as follows: 5 489 Remainder 5 97 4 5 19 2 5 3 4 0 3 The required number is 34245. Hence, 48910 = 34245. Alternately, We can change decimal number 489 into quinary number by making the groups of power of 5 as follows : 489 = 375 + 100 + 10 + 4 = 3 × 125 + 4 × 25 + 2 × 5 + 4 × 1 = 3 × 53 + 4 × 52 + 2 × 51 + 4 × 50 = 34245 \ 48910 = 34245. Example 6: Convert 101102 into quinary number. Solution: We can convert 101102 into quinary number by converting it into decimal number first and then into quinary number as follows : 101102 = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 = 16 + 0 + 4 + 2 + 0 = 22 Hence, 101102 = 2210. Now, we convert 2210 into quinary number as follows. 2210 = 20 + 2 = 4 × 51 + 2 × 50 = 425 Therefore, 101102 = 425.
Whole Numbers 21 Example 7: If and denote 1 and 0 respectively, which of the following circle should be shaded to denote 45 in the binary number system? Show by shading. a b c d e f Solution: We can convert 4510 into the binary number dividing it successively by 2 as follows : 2 45 Remainder 2 22 1 2 11 0 2 5 1 2 2 1 2 1 0 0 1 a, c, d, f should be shaded. ⸫ The required number is 1011012. Shading are on a b c d e f Exercise 2.1.1 1. Express the following numbers in the expanded form. a) 29410 b) 214510 c) 210410 d) 1120110 2. Express the following binary numbers in the expanded form. a) 112 b) 1012 c) 11112 d) 111012 3. Convert into decimal number. a) 1012 b) 11102 c) 10010012 d) 10011112 4. Convert into binary number. a) 1010 b) 5610 c) 10610 d) 21410 5. Express the following quinary numbers in the expanded form. a) 245 b) 2135 c) 10245 d) 32135 6. Convert the following quinary numbers into decimal numbers. a) 12045 b) 22005 c) 10345 d) 42425 7. Convert into quinary number. a) 24010 b) 10010 c) 67810 d) 1003210 8. Convert the following binary numbers into quinary numbers. a) 1012 b) 10112 c) 100112 d) 1111012 9. Convert the following quinary numbers into binary numbers. a) 125 b) 1035 c) 2215 d) 10215 10. If and denote 1 and 0 respectively, which of the following circle should be shaded to denote 194 in the binary number system? a b c d e f g h
22 Perfect Mathematics Class 8 11. If and denote 1 and 0 respectively, which of the following blocks should be shaded to denote 25 in the binary number system? a b c d e 12. Represent 153 in terms of a and b, if a denotes 0 and b denotes 1. 13. In an exam Sharmila obtained 1101112 marks and Pramila obtained 2115, marks. a) Write the marks obtained by Sharmila in decimal number system. b) How much more or less marks did Pramila obtain than Sharmila in decimal number system? 14. In an exam Ramdev secured 1111102, marks and Bamdev secured 2345, marks. a) Write the marks obtained by Ramdev in decimal number system. b) How much more or less marks did Bamdev obtain than Ramdev in decimal number system? 15. a) Which of the numbers 11012 and 245 is larger? Also, find their difference. b) Which of the numbers 1100112 and 1445 is larger? Also, find their difference. a) Let's collect some peas and prepare a table for the decimal, quinary and binary numerals for the numbers five, eight, eleven and fifteen. Peas Decimal numerals Quinary numerals Binary numerals b) Write the total number of students in your school and the students in class 8. Convert these numbers of students into binary and quinary number system and present it in your class. c) Ask the price of any ten goods at your nearby shop. Convert all the prices into binary and quinary number systems. Project Work Exercise 2.1.1 Introduction of Whole Numbers 1. Show it to your teacher. 2. Show it to your teacher. 3. a) 510 3. b) 1410 3. c) 7310 3. d) 7910 4. a) 10102 4. b) 1110002 4. c) 11010102 4. d) 110101102 5. Show it to your teacher. 6. a) 17910 6. b) 30010 6. c) 14410 6. d) 57210 7. a) 14305 7. b) 4005 7. c) 102035 7. d) 3101125 8. a) 105 8. b) 215 8. c) 345 8. d) 2215 9. a) 1112 9. b) 111002 9. c) 1111012 9. d) 100010002 10. a, b, g 11. a, b, e 12. baabbaab 13. a) 55 13. b) 1 more 14. a) 62 14. b) 7 more 15. a) 11012 < 245; 1 15. b) 1100112 > 1445; 2
Rational and Irrational Numbers 23 2.2 Rational and Irrational Numbers Introduction A number is a mathematical value used for counting or measuring objects. Numbers are used to performing arithmetic calculations. Examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value. We define the integers clearly through the following set of numbers. Natural Numbers The set of counting numbers {1, 2, 3, 4, ... } is called the set of Natural numbers.We denote the set of natural numbers by N. Thus, N = {1, 2, 3, 4, ...}. Whole Numbers Counting numbers including zero is the set of Whole Numbers. The set of whole numbers is generally denoted by W. i.e. W = {0, 1, 2, 3, .... }. Therefore, N∈W. Integers The set of the whole numbers together with negative of all natural numbers is called the set of Integers. The set of integers is denoted by I. Thus, I = { ..., – 3, – 2, – 1, 0, 1, 2, 3, ... } Natural Numbers Whole Numbers –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Integers (i) Positive numbers are placed at the right side of zero and they are greater than 0. (ii) Negative numbers are placed at the left side of zero and they are smaller than 0. (iii) Zero is kept at the centre which is neither positive nor negative. QUICK TIPS
24 Perfect Mathematics Class 8 2.2.1 Rational Numbers The set of numbers which can be expressed in the form of p q , where p and q are integers and q ≠ 0, is called the set of Rational Numbers. This set of rational numbers is denoted by Q. Examples of rational numbers include : 3 4 , 2, –5 6 , 0.25 Rational numbers can be positive, negative, or zero. They can also be expressed in different forms, such as fractions, mixed numbers, or decimals. Rational numbers can be added, subtracted, multiplied, and divided just like any other type of number. When performing arithmetic operations on rational numbers, it is important to ensure that the denominators are the same, so that the fractions can be combined. Consider the following fractions, root of a number and their decimal numbers : Terminating Non-terminating Recurring Non-terminating and Non-recurring 6 3 = 2 1 3 = 0.3333 ... ... 2 = 1.4142135623... ... 7 2 = 3.5 1 6 = 0.1666 ... ... 3 = 1.7320508075... ... 13 5 = 2.6 13 11 = 1.181818 ... ... 5 = 2.2360679774... ... 7 4 = 8.25 3 11 = 0.2727 ... ... ... 8 = 2.8284271247... ... 3 8 = 0.375 22 7 = 3.142857142857 ... ... 12 = 3.4641016151... ... In the table, we see that 6 3 , 7 2 , 13 5 , ... ... have fixed number of digits in corresponding decimal fraction. These fractions are called terminating decimals. The decimal fractions of 1 3 , 1 6 , 13 11, ... ... have no fixed number of digits. So, these fractions are expressible into non-terminating decimals. In these decimals, a digit or a block of digits are repeated after decimal places, so these are the fractions of recurring decimals. The decimal values of 2, 3, 5, 8, .... ..., etc. are non-terminating. The decimal numbers are non-terminating decimals in which the digits are not repeated in the same sequence after a fixed digit. Such decimals are called non-terminating and non-recurring decimals.
Rational Numbers 25 Flowchart of Decimal Numbers Decimal Numbers Rational Numbers Irrational Numbers Terminating Decimal Numbers Non-terminating Decimal Numbers Non-terminating Recurring Decimals Non-terminating Non-recurring Decimals Finding Fractions for Recurring Decimals Consider a fraction 1 6 . By division, we get 1 6 = 0.166 ... ... . Here, 6 is repeated indefinitely. So, 0.1666.... is written as 0.16 is a recurring decimal. In the same way, the number 0.2 is also a recurring decimal because 0.2 = 0.20 in which 0 is repeated indefinitely. A recurring decimal may be terminating or non-terminating. A recurring decimal is terminating if '0' repeats indefinitely, otherwise, it is non-terminating. We can find a fraction for every non-terminating recurring decimal number. Study the following examples. Example 1: Find a fraction that gives 0.6. Solution: Let x = 0.6666 .... ... ... (i) Multiply it both sides by 10, 10x = 6.6666 .... ... ... (ii) Subtracting (i) from (ii), we get 10x – x = (6.6666....) – (0.6666....) or, 9x = 6.0 or, x = 6 9 or, x = 2 3 Hence, 2 3 is the required fraction.
26 Perfect Mathematics Class 8 Example 2: Find a fraction corresponding to the decimal 3.42. Solution: Let x = 3.424242 .... ... (i) Multiplying equation (i) both sides by 100, we get 100x = 100 × 3.424242.... or, 100x = 342.4242 ... ... (ii) Subtracting equation (i) from (ii), we get 100x – x = 342.4242... – 3.4242... or, 99x = 339 or, x = 339 99 or, x = 113 33 = 314 33 \ The required fraction is 3 14 33. Exercise 2.2.1 1. Express the following fractions into decimals and decide which are terminating, non-terminating and recurring. a) 7 8 b) 4 9 c) 5 1 7 d) 17 11 e) 56 13 f) 103 16 2. Find the fraction corresponding to the following recurring decimals. a) 0.3 b) 0.2 c) 0.7 d) 0.25 e) 0.27 f) 4.38 g) 1.108 h) 0.654 Exercise 2.2.1 Rational Numbers 1. a) 0.875 → Terminating 1. b) 0.4444... → Non-terminating recurring 1. c) 5.1428571428... → Non-terminating recurring 1. d) 1.5454... → Non-terminating recurring 1. e) 4.3076923076... Non-terminating recurring 1. f) 6.4375 → Terminating 2. a) 1 3 2. b) 2 9 2. c) 7 9 2. d) 25 99 2. 3) 3 11 2. f) 434 99 2. g) 41 37 2. h) 218 333
Irrational Numbers 27 2.2.2 Irrational Numbers The numbers that cannot be expressed as a ratio of two integers and cannot be expressed as a terminating or repeating decimal are irrational numbers. Examples of irrational numbers include π (pi) and 2 (the square root of 2). These numbers cannot be expressed in the form p q , where p and q are integers. The denominator q is not equal to zero (q ≠ 0). The set of all such numbers is called the set of Irrational Numbers. Hence, the set of rational numbers and the set of Irrational numbers are disjoint sets. Diagrammatic Presentation of Set of Numbers Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Terminating decimals and non-terminating recurring decimals are rational numbers. The non-terminating and non-recurring decimal numbers are irrational numbers. Consider the numbers 2, 3, 5. Now, 2 = 1.414213 ... ... is a non-terminating and non recurring decimal number. 3 = 1.7320508 ... ... is a non-terminating and non recurring decimal number. 5 = 2.236068 ... ... is a non-terminating and non recurring decimal number. As in the above examples, 2, 3, 5, 8 etc. are the non-terminating and non-recurring decimal numbers. Properties of Irrational Numbers Properties of irrational numbers help us to pick up irrational numbers out of a set of real numbers. Given below are some of the properties of irrational numbers : Irrational numbers consist of non-terminating and non-recurring decimals. These are real numbers only. When an irrational number and a rational number are added, the result or their sum is an irrational number only. For an irrational number x, and a rational number y, their result, x + y = an irrational number.
28 Perfect Mathematics Class 8 When any irrational number is multiplied by any non-zero rational number, their product is an irrational number. For an irrational number x and a rational number y, their product xy is an irrational number. For any two irrational numbers, their least common multiple (LCM) may or may not exist. Addition, subtraction, multiplication, and division of two irrational numbers may or may not be a rational numbers. The table illustrates the list of some of the irrational numbers. Irrational number Value π 3.14159265.... 2 1.414213562... 3 1.73205080... 5 2.23606797... 7 2.64575131.... 11 3.31662479... 13 3.605551275... – 3 2 –0.866025.... 33 2 0.721124.... 47 3 3.60882608… 2 3 1.25992104... Rational vs Irrational Numbers The table illustrates the difference between rational and irrational numbers. Rational numbers Irrational numbers It can be expressed in the form of a fraction or ratio i.e., p q , where q ≠ 0. It cannot be expressed in the form of a fraction or ratio. The decimal expansion is terminating or non-terminating recurring (repeating). The decimal expansion is non-terminating and non-recurring at any point. Example : 2, 0.333.., 0.6565.., 1.75 Example : π, 3, 11, etc.
Irrational Numbers 29 Flowchart of Real Number System Real Numbers Rational Numbers (Q) Integers (Z) Whole Numbers (W) Natural Numbers (N) Irrational Numbers (Ir) Fraction (F) Negative Numbers (Z) Zero (0) Representation of irrational number on a number line Irrational number 2 can be represented on a number line as shown below. On a number line XOX', take OA = 1 unit. Draw a right angle OAB and mark AB = 1 unit. Join OB. From Pythagoras theorem, OB2 = OA2 + AB2 or, OB2 = 12 + 12 = 1 + 1 = 2 \ OB = 2 X' O A C B X –1 0 1 2 2 Again, draw a square OABM on above figure and produce MB upto N. Taking O as the centre and OB as the radius, draw an arc which cuts the number line XX' at point C. Hence, OC = OB = 2. X' O A C E B D F X –1 0 1 3 2 3 2 5 6 M N Draw a right angled DOCD in between MN and X'X at C, then CD = 1 unit. In right angled DOCD, OD2 = OC2 + CD2 [⸪ By Pythagoras theorem] or, OD2 = 2 2 + 12 = 2 + 1 = 3 \ OD = 3
30 Perfect Mathematics Class 8 Taking O as the centre and OD as the radius, draw an arc which cuts the number line XX' at point E. Hence, OE = OD = 3. Draw a right angled DOEF at E, then EF = 1 unit. In right angled DOEF, OF2 = OE2 + EF2 [ By Pythagoras theorem] or, OF2 = 3 2 + 12 = 3 + 1 = 4 \ OF = 2 Similarly, we can represent 5, 6, etc. on the number line. Introduction to Surds A surd is a number written in the root form to express its exact value. An irrational number has an infinite number of non-terminating and non-recurring decimals. Therefore, surds are irrational numbers with root signs if we cannot remove the root sign to get a rational number. For example, 2 (square toot of 2) cannot be simplified further; so, it is a surd. 9 (square root of 9) can be simplified (to 3); so, it is not a surd. 43 (cube root of 4) cannot be simplified further; so, it is a surd. In a surd an , the sign ' ' is called radical sign, a is called radicand and n is called the order (or degree) of the surd. The roots of a rational number which cannot be expressed as a rational number is called a surd. 2, 3, 43 , 5 etc. are the examples of surds. Definition Note : Every surd is an irrational number but every irrational number is not a surd. Let's take some more examples. Number Simplified As a decimal Remarks 9 3 3 Not a surd 2 2 1.41421.... Surd 3 3 1.73205.... surd
Irrational Numbers 31 Number Simplified As a decimal Remarks 1 4 1 2 0.5 not a surd 4 9 2 3 0.66666.... not a surd 53 53 1.70997.... surd 11 3 11 3 2.22398.... surd 125 3 5 5 not a surd From the above table, a surd is a number with a root sign which, when changed into decimal number goes on without repeating and it is an irrational number. Different Types of Surds Like and Unlike surds The surds with the same order and same radicand are called like surds. 3, 2 3, – 4 3 etc. are like surds. 4 53 , 53 , –3 53 etc. are like surds. 2 and 8 are like surds. The surds which have a different radicand or a different order are unlike surds. For example, 2 and 3 are unlike surds. 53 and 5 are unlike surds. Pure and Mixed Surds A surd having only one irrational factor is known as a pure surd. For example, 2, 53 , 11 4 etc. are pure surds. A surd having a rational and an irrational factor is known as a mixed surd. For example, 3 2, 2 53 , 10 7 etc. are mixed surds. Example 1: Convert 125 into a mixed surd and 3 23 into a pure surd. Solution: Here, 125 = 5 × 5 × 5 So, 125 = 5 × 5 × 5 = 5 5 Hence, the mixed surd of 125 is 5 5.
32 Perfect Mathematics Class 8 Again, 2 33 = 3×(2×2×2) 3 = 24 3 Hence, the pure surd of 23 is 24 3 . Simple and Compound Surds A surd having only term is called a simple surd. For example, 2, 10, 3 53 , etc. are simple surds. A surd containing two or more terms is called a compound surd. For example, 2 + 5, 7 – 4 3, 5 + 6 – 3 53 , etc. etc are compound surds. Laws of Surds Let a and b be two positive real numbers and m and n be integers. Then, (i) a = a 1 2 e.g. 36 = 36 1 2 = (62 ) 1 2 = 62 × 1 2 = 6 (ii) an = a 1 n e.g. 125 3 =125 1 3 = (53 ) 1 3 = 53 × 1 3 = 5 (iii) an . bn = ab n e.g. 3 × 5 = 3 × 5 = 15 (iv) an n b = a b n e.g. 3 12 33 = 12 3 3 = 43 (v) an m = a mn e.g. 53 4 = 5 4×3 = 5 12 = 54 3 QUICK TIPS Operation of Surds Addition and Subtraction of Surds Two or more like surds can be added or subtracted. To add or subtract surds, follow the given steps : a) Express all the given like surd into simplest form. b) Add or subtract the coefficients of like terms keeping the irrational factor same. Examples, (i) 5 + 3 5 = (1 + 3) 5 = 4 5 (ii) 5 3 – 2 3 = (5 – 2) 3 = 3 5 (iii) 27 + 2 12 = 3×3×3 + 2 2×2×3 = 3 3 + 2 × 2 3 = (3 + 4) 3 = 7 3
Irrational Numbers 33 Example 2: Add or subtract the following. a) 4 5 + 7 5 b) 8 2 – 3 2 Solution: a) Here, 4 5 + 7 5 = (4 + 7) 5 = 11 5 b) Here, 8 2 – 3 2 = (8 – 3) 2 = 5 2 Example 3: Simplify the following : a) 10 5 + 7 2 – 6 5 – 3 2 b) 20 + 5 32 – 3 18 – 6 45 Solution: a) Here, 10 5 + 7 2 – 6 5 – 3 2 = 10 5 – 6 5 + 7 2 – 3 2 = (10 – 6) 5 + (7 – 3) 2 = 4 5 + 4 2 b) Here, 20 + 5 32 – 3 18 – 6 45 = 2 × 2 × 5 + 5 4 × 4 × 2 – 3 3 × 3 × 2 – 6 3 × 3 × 5 = 2 5 + 5 × 4 2 – 3 × 3 2 – 6 × 3 5 = 2 5 + 20 2 – 9 2 – 18 5 = 20 2 – 9 2 + 2 5 – 18 5 = (20 – 9) 2 + (2 – 18) 5 = 11 2 – 16 5 Exercise 2.2.2 1. State whether the following statements are true or false. a) Every natural number is a rational number. b) Every irrational number is a real number. c) Every real number is a rational number. d) The product of two irrational numbers is always an irrational number. e) Set of Integers ⊂ Set of Irrational Numbers ⊂ Set of Real Numbers. f) 5 is a rational number. g) 9 is an irrational number.
34 Perfect Mathematics Class 8 2. Add the following : a) 3 2 + 5 2 b) 2 6 + 6 6 c) 5 7 + 2 7 + 3 7 d) 3 5 + 2 5 + 5 3. Subtract the following : a) 8 2 – 3 2 b) 12 5 – 6 5 c) 12 11 – 8 11 d) 12 15 – 8 15 4. Simplify the following : a) 8 3 – 3 3 + 3 3 b) 6 20 – 6 45 + 3 80 c) 13 3 + 9 2 – 6 3 – 5 2 d) 9 5 + 7 3 + 6 5 – 3 3 5. Show the given numbers on a number line. a) 5 b) 6 c) 7 d) 8 6. a) Show the relation between rational and irrational numbers in a Venndiagram. b) Write any two differences between rational and irrational numbers. 1. A packet of balloon contains 48 balloons. A teacher bought 8 packets of balloons for his 56 students. One fourth of the students are eligible to get the balloons. The teacher distributed all the balloons among the eligible students equally. How many balloons did each of the eligible students get? 2. Take a sheet of square shaped chart paper and prepare a graph assuming 2 cm length as 1 unit. Then, show the numbers 2 and 3 on the number line. Also, present it in the classroom. 3. Take any 5 rational numbers and reduce them to decimal form. Identify whether they are terminating or non–terminating recurring decimals. Project Work Exercise 2.2.2 Irrational Numbers 1. a) True 1. b) True 1. c) False 1. d) False 1. e) False 1. f) False 1. g) False 2. a) 8 2 2. b) 8 6 2. c) 10 7 2. d) 6 5 3. a) 5 2 3. b) 6 5 3. c) 4 11 3. d) 4 15 4. a) 8 3 4. b) 6 5 4. c) 7 3 + 4 2 4. d) 15 5 + 4 3 5. Show to your teacher
Irrational Numbers 35 2.3 Scientific Notation of Numbers Introduction A scientific notation, also known as a exponential notation, is a way of representing numbers that are either very large or very small. It is a standard notation used in scientific fields such as physics, chemistry, astronomy, and engineering. Importance of using scientific notation: (i) It makes large and small numbers easier to read and understand : Scientific notation simplifies the representation of large and small numbers. For example, instead of writing 0.0000000000234, you can write 2.34×10 – 11. This makes the number easier to read and understand. (ii) It helps avoid errors in calculations : When working with large or small numbers, it's easy to make mistakes. Scientific notation reduces the number of digits that need to be calculated, making it less likely that errors will occur. (iii) It saves space : Scientific notation takes up less space than writing out the entire number. This is especially important when dealing with large datasets or when writing equations. (iv) It enables easy comparison of numbers : When numbers are written in scientific notation, it's easy to compare their relative magnitudes. For example, it's clear that 2.34×10–11 is smaller than 1.23×10–8 . Overall, scientific notation is an important tool for anyone who needs to work with large or small numbers in scientific or technical fields. It simplifies calculations, saves space, and helps to ensure clear communication. Complete the following table of some numbers in the power of 10. Power of 10 Factorized form Number Number Name 10 2 10 × 10 100 1 hundred 10 3 10 × 10 × 10 1000 105 10 × 10 × 10 × 10 × 10 100000 100 1 1 10 –1 1 ÷ 10 0.1 1 tenth 10–2 1 ÷ (10 × 10) 0.01 10–4 0.0001 1 ten-thousandth 10 –6 0.000001 Activity - 1
36 Perfect Mathematics Class 8 The weight of the earth = 6,600,000,000,000,000,000,000 tonnes The weight of the electron = 0.00,000,000,000,000,000,000,000,000,000,091gm Sometimes, we have to work with huger numbers than the numbers given above. Such huge numbers can be written as the product of a number and a power of 10. For example : 6,600,000,000,000,000,000,000 = 66 × 100,000,000,000,000,000,000 = 66 × 1020 = 6.6 × 1021 Similarly, 0.00,000,000,000,000,000,000,000,000,000,091 = 91 × 10– 32 = 9.1 × 10–31 and 150,000,000,000 = 15 × 10 10 = 1.5 × 1011 Here are the distances between planets from the sun, in order from the closest to the farthest : Average Distance of Planets from the sun Planet Average Distance (Km) Scientific Notation Mercury 57,950,000 5.795×107 Venus 108,200,000 1.082×108 Earth 149,600,000 1.496×108 Mars 227,900,000 2.279×108 Jupiter 778,300,000 7.783×108 Saturn 1,430,000,000 1.43×109 Uranus 2,870,000,000 2.87×109 Neptune 4,500,000,000 4.5×109 Note that these distances can vary depending on the position of each planet in its elliptical orbit.
Irrational Numbers 37 This system of writing a number in a short form is called Scientific Notation or Standard Form. In Scientific Notation, numbers are expressed in the form of a × 10n , where, 1 ≤ a ≤ 10. For example, 283000 = 2.83 × 10 5 (but 28.3 × 10 4 is incorrect scientific notation) 0.00386 = 3.86 × 10 – 3 (but 38.6 × 10–4 is incorrect scientific notation) In case of a large number, separate whole number part and decimal part by using a decimal point after one digit. Count the number of digits after the decimal point and express as the product of the same number of power of 10. For example : There are 5 digits after the decimal point. So the power of 10 is 5. 275000 = 2.75 × 105 Whole number part Decimal part Insert decimal point 5 digits In case of a very small number, separate the whole number part and decimal part by using a decimal point after one significant digit. Count the number of digits before the decimal point and express as the product of the same number of negative power of 10. For example, The decimal point has shifted 4 digits right from its initial position. So, the power of 10 is – 4. 0.0002435 = 2.435 × 10– 4 Whole number part Decimal part Insert decimal point 4 digits Complete the following table of an equivalent decimal number and a scientific notation. Standard Decimal Number Scientific Notation 10 1 × 101 1000 1 × 10 3 1000000 1 × 10 6 1.2 × 104 32460 – 2500 –2.5 × 103 –1824 0.2 2 × 10 – 1 0.003 1.2 × 10 – 3 5.6 × 10– 5 Activity - 2
38 Perfect Mathematics Class 8 Example 1: The distance of the moon from the earth is 380000000m. Express it in a scientific notation. Solution: Since, 380000000m = 38 × 10000000m = 38 × 10 7 m = 3.8 × 10 × 10 7m = 3.8 × 10 8m Hence, the distance of the moon from the earth is 3.8 × 108 m. Example 2: Write the following in the standard decimal number. a) 2.04 × 109 b) 0.38 × 10–4 Solution: a) 2.04 × 10 9 = 2.04 × 1000000000 = 204 100 × 1000000000 = 204 × 10000000 = 2040000000 \ 2.04 × 10 9 = 204000000 b) 0.38 × 10–4 = 0.38 × 1 104 = 038 100 × 1 104 = 38 100 × 10000 = 38 1000000 = 0.000038 \ 0.38 × 10– 4 = 0.000038h Example 3: Multiply : (4 × 105 ) and (6 × 107 ) Solution: Here, (4 × 105 ) × (6 × 107 ) = 4 × 10 5 × 6 × 107 = 4 × 6 × 10 5+7 = 24 × 10 12 = 2.4 × 10 13
Irrational Numbers 39 Example 4: Divide : (9 × 106 ) by (4 × 104 ) Solution: Here, (9 × 106 ) ÷ (4 × 104 ) = 9 × 106 4 × 104 = 9 4 × 106 –4 = 2.25 × 102 Example 5: Find the difference : 4.8 × 106 – 2.3 × 104 Solution: Here, 4.8 × 106 – 2.3 × 104 = 4.8 × 104+2 – 2.3 × 104 = 4.8 × 104 × 102 – 2.3 × 104 = 104 (4.8 × 102 – 2.3) = 104 (480 – 2.3) = 104 × 477.7 = 104 × 4.777 × 102 = 4.777 × 104+2 = 4.777 × 106 Example 6: Simplify : (2 × 109 ) × (3.6 × 10 –3 ) 1.2 × 10 –2 Solution: Here, (2 × 109 ) × (3.6 × 10–3 ) 1.2 × 10–2 = 2 × 3.6 × 109 × 10–3 1.2 × 10–2 = 7.2 × 109–3 1.2 × 10–2 = 72 × 106 12 × 10–2 = 72 12 × 10 6 10–2 = 6 × 106+2 = 6 × 108
40 Perfect Mathematics Class 8 Exercise 2.3 1. Fill in the blanks in the following conversions. a) 2250000 = 2.25 × ... b) 752300 = 7.523 × ... c) 31050000 = 3.105 × ... d) 0.2 = 2 × ... e) 0.0052 = 5.2 × ... f) 0.0000302 = 3.02 × ... 2. a) Speed of light is 1080000000 kilometer per hour. Express it in a scientific notation. b) The radius of the moon is 1,737.4 kilometer. Express it in a scientific notation. c) The distance between the moon and the earth is 384,400km. Express it in a scientific notation. 3. Write the following numbers in a scientific notation. a) 170000 b) 152300 c) 7050000 d) 56000000 e) 50000000 f) 15700000 g) 10000000000 h) 2.43 i) 0.073 j) 0.000005 k) 0.02047 l) 0.0004305 4. Write the following numbers in the standard decimal number. a) 2.3 × 103 b) 5.02 × 105 c) 2.3 × 102 d) 0.356 × 1010 e) 0.32 × 102 f) 5.6 × 107 g) 25 × 108 h) 126 × 105 i) 588 × 104 5. Multiply the following. a) (3 × 103 ) and (4 × 103 ) b) (6 × 103 ) and (7 × 108 ) c) (4.3 × 103 ) × (2 × 105 ) d) (1.2 × 105 ) × (2.6 × 103 ) 6. Divide the following. a) (5 × 105 ) by (2 × 103 ) b) (6 × 108 ) by (5 × 106 ) c) (4 × 107 ) ÷ (5 × 102 ) d) (1.2 × 107 ) ÷ (3 × 103 ) 7. Find the sum or difference. a) (2.4 × 107 ) + (1.3 × 105 ) b) (3.8 × 106 ) + (2.3 × 108 ) c) (3.7 × 107 ) – (2.3 × 105 ) d) (6.3 × 106 ) – (1.2 × 103 ) 8. Simplify and express in a scientific notation. a) (2.8 × 10–5 ) × (5.2 × 102 ) b) (175 × 104 ) ÷ (25 × 108 ) c) (3 × 1010) × (4 × 1015) d) (1.2 × 109 ) × (6.2 × 10–3 ) 6 × 10–2 e) (6 × 107 ) × (3 × 10–8 ) 2 8.2 × 10–20 f) (2 × 105 ) × (1.24 × 10–2 ) (2 × 10–3 ) × (2.7 × 108 )
Irrational Numbers 41 9. Umesh and Usha are husband and wife. In 5 years' time, Umesh earned Rs. 1.086 × 10 7 and Usha earned Rs.8.145 × 106 . a) Find the sum of money earned by both of them together. b) How much more money did Umesh earn than Usha? c) How much less in percentage did Usha earn than Umesh? 10. There are two fuel reservoirs at an airport. If the capacity of the first reservoir is 6.03 × 105 litres and that of the second is 8.503 × 106 litres, a) What is the total capacity of both the reservoirs? b) How much more fuel can the second reservoir contain than the first reservoir? Exercise 2.3 Scientific Notation 3. a) 1.7 × 105 3. b) 1.523 × 105 3. c) 7.05 × 106 3. d) 5.6 × 107 3. e) 5 × 107 3. f) 1.57 × 107 3. g) 1 × 1010 3. h) 2.43 × 100 3. i) 7.3 × 10–2 3. j) 5 × 10–6 3. k) 2.047 × 10–2 3. l) 4.305 × 10 –4 4. a) 2300 4. b) 502000 4. c) 230 4. d) 3560000000 4. e) 32 4. f) 56000000 4. g) 2500000000 4. h) 12600000 4. i) 5880000 5. a) 1.2 × 107 5. b) 4.2 × 1012 5. c) 8.6 × 108 5. d) 3.12 × 108 6. a) 2.5 × 102 6. b) 1.2 × 102 6. c) 8 × 104 6. d) 4 × 103 7. a) 2.413 × 107 7. b) 2.338 × 108 7. c) 3.677 × 107 7. d) 6.2988 × 106 8. a) 1.456 × 10– 2 8. b) 7 × 10–4 8. c) 1.2 × 1026 8. d) 1.24 × 108 8. e) 2.585 × 1011 8. f) 4.5925 × 10–3 9. a) Rs. 1.9005 × 10 7 9. b) Rs.2.774×106 9. c) 25% 10. a) 9.106 × 106 litres 10. b) 7.9 ×106 litres Mixed Exercise B Whole number, Rational Number, Scientific Notation 1. The length of Araniko Highway is 112.8 ,kilometer. a) Which of the followings is in the scientific notation of given length? (i) 11.28 × 10 (ii) 1.128 × 10 2 (iii) 0.1128 × 103 (iv) 112.8 × 10–1 b) Write the number 1128 in the quinary number system. c) If the symbol denotes 1 and denotes 0, Color the following blocks to represent 35 in binary. 2. A farmer produced 23 quintal of paddy. a) Express kilogram measure of 23 quintal in a scientific notation. b) Convert 0.6 into fraction. c) Write the binary number expressed as 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 .
42 Perfect Mathematics Class 8 3. The length of Mahendra Highway from Kakadbhitta in the east to Gaddachauki of Mahendranagar in west is 1027.67 kilometer. a) Which of the following denotes the scientific notation of that length? (i) 1.02767 × 106 m (ii) 10.2767 × 106 m (iii) 102.767 × 106 m (iv) 0.102767 × 10 6 m b) Write 1028 in quinary number system. c) If and denote 1 and 0 respectively, which of the following blocks should be shaded to denote 35 in binary number system? Show by shading. a b c d e f 4. An ocean ship travels at the speed of 1.2 × 104 miles per hour. a) Write the distance travelled by the ship per hour in the usual form. b) Write the distance travelled by the ship in 1 hour into the quinary number system. c) Convert 1. 22 into fraction. 5. An ocean ship travels at the speed of 1.5 × 10 miles per hour. a) Write the distance travelled by the ship per hour in the usual form. b) Write the distance travelled by the ship in 1 hour into the quinary number system. c) Convert 1. 45 into fraction. 6. In a shop, the price of articles is an attached in binary number. a) If the price of a pen is written Rs.100112, then write the price of the pen in decimal number system. b) What is the price of the pen in the quinary number system? c) Convert 0.3 into fraction. 7. A space ship's engine consumes 3.5 × 103 litres of fuel per hour. a) If the ship has 1.4 × 105 litres of fuel, how many hours can it fly? b) Convert 40 into binary and quinary number systems. c) Express 4.4 into fraction. 8. The capacities of two tanks, A and B are 4.35 × 105 litres and 8.70 × 10 3 litres respectively. a) Write the capacity of tank A in the usual form. b) How much less is the capacity of tank B than the capacity of tank A? c) Convert 348 into the quinary number system. 9. The capacities of two tanks, A and B are 7.5 × 104 litres and 2.5 × 10 3 litres respectively. a) Write the capacity of tank A in the usual form. b) How much less is the capacity of tank B than the capacity of tank A? c) Convert 725 into the quinary number system.
Irrational Numbers 43 10. One student of Grade 8 wrote his monthly fee in the expanded form as follows : 4 × 54 + 2 × 53 + 1 × 52 + 0 × 51 + 0 × 50 Answer the following questions a) Write the short form of his fee into the quinary number system. b) Write the monthly fees in decimal number system and then write in scientific notation. c) Find his fee for one year.) d) Simplify : 2 8 + 3 2 11. There are two numbers 110112 and 4325. a) Which one is greater between these? b) Write the larger number in a scientific notation. c) If cost of 5 pens is Rs. 85, find the cost of 2 pens. 12. Gita wants to simplify the following problem to get the answers in different forms. 11102 + 105 + 4 × 103 2 × 102 a) Calculate the answer of Gita in quinary number. b) Find the answer of Gita in a scientific notation. c) What is the answer of Gita in binary number system? 1. Search the distance of the Moon and the mars from the earth in the Google. Convert the distances into a scientific notation form and then find the farthest object among the Moon or Mars by how much percent. Also, find their total distance. 2. Search the weights of electron, newton and proton of an atom. Convert their weights in scientific notations and then sum of the atom. Which is more heavier and which is lighter? Calculate and present them in your classroom. Project Work Mixed Exercise A Whole number, Rational number, Scientific notation 1. a) ii 1. b) 140035 1. c) 2. a) 2.3 × 104 kg. 2. b) 2 3 2. c) 10102 3. a) a 3. b) 131035 3. c) a, e, f 4. a) 12,000 miles 4. b) 3410005 miles 4. c) 11 9 5. a) 1,500 miles 5. b) 220005 miles 5. c) 16 11 6. a) Rs. 19 6. b) Rs. 345 6. c) 1 3 7. a) 40 hours 7. b) 1010002, 1305 7. c) 40 9 8. a) 43500l 8. b) 34800l 8. c) 23435 9. a) 75000l 9. b) 72500l 9. c) 10400 10. a) 421005 10. b) 2775; 2.775 × 103 10. c) Rs.33300 10. d) 72 11. a) 4325 11. b) 1.17 × 102 11. c) Rs.34 12. a) 745 12. b) 0.39 × 102 12. c) 1001112
44 Perfect Mathematics Class 8 2.4 Ratio, Proportion and Percentage 2.4.1 Ratio The weight of two students, Ronish and Sunita of class 8 is 50 kg and 40kg respectively. We can compare the weight of these two students in two different ways. (i) In terms of difference : Difference in weight = Weight of Ronish – Weight of Sunita = 50kg – 40 kg = 10 kg This is the comparison of two quantities by differences. (ii) In terms of division : Weight of Ronish Weight of Sunita = 50kg 40 kg = 50 40 = 5 4 i.e., Weight of Ronish = 5 4 times the weight of Sunita. The ratio of the weight of Ronish to that of Sunita is 5 4 . We read it as '5 is to 4' and it is written as '5 : 4'. Similarly, Weight of Sunita Weight of Ronish = 40 kg 50 kg = 40 50 = 4 5 . The weight of a pen-drive, a book and a laptop is given in the following table. Particular Weight Weight in gm (same unit) Pen-drive 25gm 25 gm Book 350gm 350 gm Laptop 2kg 2000 gm Find the following ratios. (i) Weight of laptop Weight of book = = (ii) Weight of pen-drive Weight of book = = Activity - 1
Irrational Numbers 45 Ratio in the Simplest Form The ratio (a : b) is said to be in the simplest form if HCF of a and b is 1. So, the ratio of 200 and 125 is 200 125 = 8 × 25 5 × 25 = 8 5 Thus, the ratio of two quantities of the same kind and same unit is the fraction of the two quantities in a simplest form. The ratio of a and b is the fraction a b , written as a : b. Example 1: Find the ratio of each of the following in the simplest form. a) Rs.30 to Rs.600 b) 3m to 75 cm Solution: a) The ratio Rs.30 to Rs.600 = Rs.30 Rs. 600 = 1 × 30 20 × 30 = 1 20 b) For the ratio of 3m and 75cm, Here, m and cm are different in units. Those units must be changed into the same to take their ratio. = 3m 75 cm = 300 cm 75 cm = 4 × 75 1 × 75 = 4 1 Example 2: If a : b = 2 : 3 and b : c = 4 : 5, find a) Express a in terms of b? b) Express c in terms of b? c) Using the values of a and c, find a : c. d) Also find a : b : c. Solution: a) Here a : b = 2 : 3 i.e. a b = 2 3 or, 3a = 2b or, a = 2b 3 b) Here b : c = 4 : 5 i.e. b c = 4 5 or, 4c = 5b or, c = 5b 4 c) Now, a = 2b 3 and c = 5b 4
46 Perfect Mathematics Class 8 Hence, a c = 2b 3 5b 4 = 2b 3 × 4 5b = 8 15 \ a c = 8 15 . d) Here, in the first ratio b = 3 and in the second ratio b = 4. The LCM of 3 and 4 is 12. Now, a b = 2 3 = 2 × 4 3 × 4 = 8 12 and b c = 4 5 = 4 × 3 5 × 3 = 12 15 Hence, a : b : c = 8 : 12 : 15. Example 3: The ratio of the number of boys and girls in a school is 4 : 3. If there are 270 girls, find the number of boys. Solution: Let the number of boys be x, then according to the question : x 270 = 4 3 or, 3x = 4 × 270 or, x = 4 × 270 3 = 360 So, the required number of boys is 360. Example 4: There are 540 students in a school. If the ratio of the number of boys and girls is 4 : 5, find the number of girls in the school. Solution: Total number of students = 540 The ratio of the number of boys and girls is 4 : 5. Let the number of boys be 4x and the number of girls 5x. Hence, number of boys + number of girls = total number of students or, 4x + 5x = 540 or, 9x = 540 or, x = 540 9 = 60 \ The number of girls is 5x = 5 × 60 = 300.
Irrational Numbers 47 Example 5: The ratio of present ages of a father and his son is 7 : 2. Five years later, the ratio of their age will be 8 : 3. Find their present age. Solution: Let the present age of the father be 7x years and that of the son be 2x years. 5 years later, the age of the father = (7x + 5) years 5 years later, the age of the son = (2x + 5) years. According to the question : 7x + 5 2x + 5 = 8 3 or, 3(7x + 5) = 8(2x + 5) or, 21x + 15 = 16x + 40 or, 21x – 16x = 40 – 15 or, 5x = 25 \ x = 5 The present age of the father = 7x = 7 × 5 = 35 years. The present age of the son = 2x = 2 × 5 = 10 years. Example 6: The ratio of the length and breadth of a rectangular hall is 3 : 2. If the length is 4m longer than its breadth, find the perimeter and the area of the hall. Solution: The ratio of length and breadth of a hall is 3 : 2. The length is 4m longer than the breadth. Suppose the breadth of the hall is 2xm. Then, its length will be (2x + 4)m. Now, by question, Length Breadth = 3 2 or, 2x + 4 2x = 3 2 or, 6x = 4x + 8 or, 6x – 4x = 8 or, 2x = 8 or, x = 8 2 = 4 Hence, the length of the hall is (2x + 4)m = (2×4 + 4)m = 12m.
48 Perfect Mathematics Class 8 The breadth of the hall is 2x = 2 × 4m = 8m. The perimeter of the hall (P) = 2(l + b) or, P = 2(12m + 8m) or, P = 2 × 20m or, P = 40m Therefore, the perimeter of the hall is 40m. Example 7: Three persons bought a bus in partnership sharing the investments in the ratio 3 : 5 : 7. If the bus costs Rs.15,00,000, what is the share of each person? Solution: The ratio of investment of three persons is 3 : 5 : 7. Let their share be Rs.3x, Rs. 5x and Rs.7x. Then, Rs.3x + Rs.5x + Rs. 7x = Rs. 1500000 or, 15x = Rs. 1500000 or, x = Rs.1500000 15 or, x = Rs. 100000. The shares of three persons is Rs.3x = Rs. 3 × 100000 = Rs.300000, Rs.5x = Rs.5 ×100000 = Rs.500000 and Rs.7x = Rs.7 ×100000 = Rs.700000. \ The shares of partners are Rs.300000, Rs.500000 and Rs.700000 respectively. Example 8: 6000 students appeared at a Basic Level Examination. In their result, the ratio of the number of students in grade A and grade B was 3 : 5. 1200 students had grade C and 800 had grade D. a) How many students got the result in grade A and grade B each? b) Find the ratio of the number of students with grade A, grade B and grade C. Solution: Number of students appeared in examination = 6000 Ratio of students in grade A and grade B = 3 : 5 Number of students in grade C = 1200 and Number of students in grade D = 800. Here, the total number of students in grade A and grade B = Total number of students – Number of students in grade C – Number of students in grade D
Irrational Numbers 49 = 6000 – 1200 – 800 = 6000 – 2000 = 4000 Let the number of students in grade A be 3x and the number of students in grade B be 5x. Then, we get 3x + 5x = 4000 or, 8x = 4000 \ x = 4000 8 = 500 a) The number of students in grade A and grade B is 3×500 and 5×500, i.e., 1500 and 2500. b) Find the ratio of the number of students with grade A, grade B and grade C. = 1500 : 2500 : 1200 = 15 : 25 : 12 Exercise 2.4.1 1. Find the ratio of the following and reduce it in the simplest form. a) 36 to 54 b) 1225 to 725 c) 3kg to 7.5 kg d) 1m 20 cm to 2m e) 2m to 70 cm f) Rs.2.10 to 75p 2. In a school, there are 1200 students. Out of them 500 are girls. Find the following. a) The ratio of boys to girls b) The ratio of girls to total students c) The ratio of boys to total students 3. a) If a : b = 1 : 2 and b : c = 2 : 3, then (i) find the value of a in terms of b. (ii) find the value of c in terms of b. (iii) find the ratio a : c (iv) find the ratio a : b : c. b) If p : q = 2 : 3 and q : r = 2 : 5, find p : r and p : q : r. 4. a) The ratio of the number of boys and girls in a school is 3 : 4. If there are 210 boys, find the number of girls. b) The ratio of the number of boys and girls in a school is 5 : 7. If there are 280 girls, find the number of boys. 5. a) 150 staff of an office drink either tea or coffee at noon. The number of staff drinking tea and coffee is in the ratio 7 : 8. Find the number of staff who drink coffee. b) Two numbers are in the ratio 2 : 3 and their sum is 35. Find the numbers. c) If the ratio of two numbers is 5 : 7 and their difference is 8, find the numbers.
50 Perfect Mathematics Class 8 6. a) The ratio of the present ages of a father and his son is 4 : 1. Five years later, the ratio of their ages will be 3 : 1. Find their present ages. b) The age of father and his son is in the ratio of 4 : 1. Five years ago, the ratio of their age was 7 : 1. Find their present age. c) Two numbers are in the ratio 4 : 5. If 10 is added to each, the new ratio becomes 9 : 11. Find the numbers. d) The ratio of two numbers is 4 : 5. If 9 is subtracted from each, the ratio becomes 7 : 9. Find the numbers. 7. a) The ratio of the length and the breadth of a rectangular room is 5 : 3. If the length is 4m longer than its breadth, find the perimeter and the area of the hall. b) The ratio of the length and the breadth of a rectangular playground is 3 : 2. If the perimeter of the playground is 50m, find the area of the playground. 8. a) Three partners started their business with a total amount of Rs. 54,00,000. If their shares are in the ratio 5 : 6 : 7, find the sum of money invested by each partner. b) Weight of Rojeena, Dinesh and Suresh is 40kg, 45kg and 50kg respectively. If a sum of Rs. 27000 is divided among them in the ratio of their weight, how much money does each receive? 9. 450 students appeared at a Basic Level Examination. In their result, the ratio of the number of students in grade A and grade B was 2 : 3. 120 students had grade C and 80 had grade D. a) How many students got the result in grade A and grade B each? b) Find the ratio of the number of students with grade A, grade B and grade C. Exercise 2.4.1 Ratio 1. a) 2 : 3 1. b) 49 : 29 1. c) 2 : 5 1. d) 3 : 5 1. e) 20 : 7 1. f) 14 : 5 2. a) 7 : 5 2. b) 5 : 12 2. c) 7 : 12 3. a) (i) b 2 3. a) (ii) 3b 2 3. a) (iii) 1 3 3. a) (iv) 1 : 2 : 3 3. b) 4 : 15, 4 : 6 : 15 4. a) 280 4. b) 200 5. a) 80 5. b) 14, 21 5. c) 20, 28 6. a) 40 yrs, 10 yrs 6. b) 40yrs, 10 yrs 6. c) 80, 100 6. d) 72, 90 7. a) 32m, 60m2 7. b) 150m2 8. a) Rs.1500000, Rs.1800000, Rs.2100000 8. b) Rs. 8000, Rs. 9000, Rs. 10000 9. a) 100, 150 9. b) 10: 15 : 12
Proportion 51 2.4.2 Proportion Suppose Bina and Nisha bought 20kg and 30kg rice respectively at Rs. 25 per kg. Then, Bina paid Rs. 500 and Nisha paid Rs. 750. The ratio between the quantities of rice purchased is 20 : 30 = 2 : 3 and the ratio of the amount paid by them is 500 : 750 = 2 : 3. Both the ratios of quantities of rice and the amount paid are equal. Such four quantities 20kg, 30kg and Rs. 500, Rs.750, in which the ratio of the first two quantities is equal to the ratio of the next two quantities are said to be in proportion. If a, b, c and d are in proportion, then we write, a b = c d or a : b = c : d. The first and the fourth terms i.e. a and d are called the extremes and the second and the third terms i.e. b and c are called the means. a : b means extremes Proportionals Proportionals 1st 3rd 2nd 4th c : d From a b = c d, we get ad = bc. Therefore, the product of extremes = the product of means. Continued Proportion Three quantities are said to be in a continued proportion if the ratio of the first two quantities is equal to the ratio of the second and the third. If x, y and z are in continued proportion, then x y = y z . The second quantity, y is called the mean proportional. The third quantity z is called the third proportional to x and y. The three quantities 6, 12 and 24 are in a continued proportion as 6 12 = 12 24. Here 12 is called the mean proportional between 6 and 24 and 24 is called the third proportional to 6 and 12. Example 1: Find the value of x : a) 3 : 4 = x : 48 b) (x + 2) : x = 6 : 5 Solution: a) Here, 3 : 4 = x : 48 or, 3 4 = x 48 or, 4x = 3 × 48 or, x = 3 × 48 4 \ x = 36 b) Here, (x + 2) : x = 6 : 5 or, x + 2 x = 6 5 or, 6x = 5x + 10 or, 6x – 5x = 10 \ x = 10
52 Perfect Mathematics Class 8 Example 2: If 4, 6, x and 24 are in proportion, find the value of x. Solution: Since, 4, 6, x and 24 are in proportion, then or, 4 6 = x 24 or, 6x = 4 × 24 or, x = 4 × 24 6 or, x = 16 \ The value of x = 16. Example 3: A map is drawn to the scale 1 : 1200. If the distance between two places in the map is 8.5 cm, what is the actual distance? Solution: Here, let the actual distance between two places be x cm. The ratio of scale of map = 1 : 1200 Distance in Map Actual distance 1 1200 8.5 cm x Now, 1 8.5 cm = 1200 x or, x = 8.5cm × 1200 = 10200 cm = 10200 100 m = 102m Hence, the actual distance between two places is 10200cm or 102m. Example 4: Find the fourth proportional to 3, 4 and 9. Solution: Let the fourth proportional of 3, 4 and 9 be x. Then, 3 4 = 9 x or, 3x = 36 or, x = 12 \ The fourth proportional to 3, 4 and 9 is 12.
Proportion 53 Example 5: What number must be subtracted from each of the numbers 17, 27, 32 and 52 to make them proportional? Solution: Here, the four given numbers are 17, 27, 32 and 52. Let x must be subtracted from each of 17, 27, 32 and 52 to make them proportional. Then, 17 – x, 27 – x, 32 – x and 52 – x are in proportion. i.e., 17 – x 27 – x = 32 – x 52 – x or, (17 – x) (52 – x) = (32 – x) (27 – x) or, 884 – 17x – 52x + x2 = 864 – 32x – 27x + x2 or, –69x + 59x = 864 – 884 or, –10x = – 20 or, x = 2 Hence, the required number to be subtracted is 2. Example 6: Find the mean proportional between 6 and 54. Solution: Let x be the mean proportional between 6 and 24. So, 6, x and 24 are in continued proportion. i.e., 6 x = x 24 or, x2 = 6 × 24 or, x2 = 6 × 4 × 6 or, x2 = 122 or, x = 12. Hence, the mean proportional between 6 and 54 is 12. Example 7: Find the third proportional to 9 and 15. Solution: Let x be the third proportional of 9 and 15. Then, 9, 15 and x be in continued proportion. i.e., 9 15 = 15 x or, 9x = 15 × 15 or, x = 15 × 15 9 = 5 × 5 = 25 \ The third proportion to 9 and 15 is 25. or, 9x = 15 × 15 or, x = 15 × 15 9 or, x = 5 × 5 = 25 \ The third proportional to 9 and 15 is 25.
54 Perfect Mathematics Class 8 Example 8: What number must be subtracted from each of the numbers 8, 20 and 56 so that the remainders may be in a continued proportion? Solution: Let x must be subtracted from each of 8, 20 and 56 to make them in a continued proportion. Then 8 – x, 20 – x and 56 – x are in a continued proportion. i.e., 8 – x 20 – x = 20 – x 56 – x or, (20 – x)2 = (8 – x) (56 – x) or, 400 – 40x + x2 = 448 – 8x – 56x + x2 or, –40x + 8x + 56x = 448 – 400 or, 24x = 48 or, x = 2 Hence, the required number is 2. Exercise 2.4.2 1. State which of the following numbers are in proportion. a) 2, 4, 6 and 12 b) 10, 15, 26 and 39 c) 10, 12, 14 and 16 d) 4, 12, 6 and 15 e) 16, 24, 32 and 48 f) 14, 35, 10 and 20 2. Find the value of x. a) x : 2 = 15 : 30 b) 2 : x = 14 : 21 c) 5 : 8 = 20 : x d) (x + 5) : x = 16 : 6 e) 2x : (x – 3) = 10 : 7 f) 8 : (x – 3) = 16 : x 3. a) If 2, 3, x and 30 are in proportion, find the value of x. b) If x, 7, 15 and 21 are in proportion, find the value of x. c) If x, 10, 2x – 4 and 12 are in proportion, find the value of x. d) If 2, x + 3, 3 and 2x + 1 are in proportion, find the value of x. 4. a) A map is drawn to the scale 1 : 1200. If the distance between two places in the map is 6cm, what is the actual distance between the two places? b) The scale of a map is 1cm: 2500m. The distance between two places in the real field is 15km. What is the distance between the places in the map? 5. Find the fourth proportional for each of the following. a) 9, 15, 30 b) 3.5, 7, 10 c) 8, 12, 16 d) 4.5, 6, 15 6. a) What number must be added to each of the numbers 6, 15, 20 and 43 to make the four numbers in proportion? b) What number must be subtracted from each of the numbers 8, 13, 25 and 47 to make the four numbers in proportion?
Proportion 55 7. Find the mean proportional for each of the following. a) 16 and 25 b) 3 and 12 c) 4 and 16 d) 9 and 16 8. Find the third proportional of the following. a) 5, 15 b) 10, 20 c) 6, 18 d) 2 2 3 , 4 9. a) What number must be subtracted each of the numbers 8, 20 and 56 from so that the remainders may be in continued proportion? b) What number must be added to each of the numbers 6, 22 and 70 so that the sums may be in a continued proportion? c) Shanta has some pens, books and notebooks. The ratio of the number of pens to the number of books is the same as the ratio of books to the number of notebooks. She has 18 pens and 2 notebooks. How many books does she have? 10. a) To prepare tea in a party, milk and water is mixed in the ratio 3 : 4. If there is 3.5 litres of tea in a kettle, find the quantity of water in the tea. b) Anup and Ishan are two brothers. In a particular month, the ratio of their incomes was 7 : 5. If the earning of Ishan was Rs.46000, then what was the earning of Anup? c) In an examination, Samjhana got 69, 94, 51 and x marks in English, Maths, Nepali and Science respectively. If these four marks are in proportion, then what is the value of x? List out the number of boys and girls from class 1 to 10. Find out the ratios of boys and girls, as well as that of girls and boys. Identify which classes have the same ratios. What are they called? Prepare a report and present in your classroom. Project Work Exercise 2.4.2 Proportion 1. (a), (b), (e) 2. a) 1 2. b) 3 2. c) 32 2. d) 3 2. e) – 15 2 2. f) 6 3. a) 20 3. b) 5 3. c) 5 3. d) 7 4. a) 72m 4. b) 6cm 5. a) 50 5. b) 20 5. c) 24 5. d) 20 6. a) 3 6. b) 3 7. a) 20 7. b) 6 7. c) 8 7. d) 12 8. a) 45 8. b) 40 8. c) 54 8. d) 6 9. a) 2 9. b) 2 9. c) 6 10. a) 2 litres 10. b) Rs.64400 10. c) 68
56 Perfect Mathematics Class 8 2.4.3 Percentage Introduction The word percentage means the number of parts taken out of 100 parts. Hence, 25 percent (25%) means 25 parts out of 100 or 25% or 25 100. Conversion of Percentage into Fraction We can convert each percentage into an equivalent fraction. Complete the following table with figures, equivalent percentage and fraction. Figure Percentage 20% 10% Fraction 1 5 Activity - 1 To convert a percentage into a fraction, we divide it by 100 and remove the % sign. Example 1: Convert the following percentage into fraction. a) 15% b) 37.5% Solution: a) 15% = 15 100 = 3 20 b) 37.5% = 37.5 100 = 375 1000 = 15 × 25 40 × 25 = 15 40 = 3 8 Conversion of Fraction or Decimal into Percentage We can convert any fraction or decimal into percentage just multiplying it by 100. For example, 1 5 = 1 5 × 20 20 = 20 100 = 20% 0.24 = 0.24 × 100% = 24% and so on. Example 2: Convert the following into percentage. a) 2 3 b) 0.32
Percentage 57 Solution: a) 2 3 = 2 3 × 100% = 200 3 % = 662 3% b) 0.32 = 0.32 × 100% = 32% Example 3: Find 18% of Rs. 1500. Solution: Here, we have, 18% of Rs.1500 = Rs. 18 100 × 1500 = Rs. 27000 100 = Rs. 270 Example 4: Find the number whose 12% is 30. Solution: Let the required number be x, then 12% of x = 30 or, 12 100 × x = 30 or, 12x = 3000 or, x = 3000 12 = 250 Hence, the required number is 250. Exercise 2.4.3 1. Convert the percentage into fraction. a) 10% b) 28% c) 40% d) 67.5% 2. Convert the following fractions into percentage. a) 1 2 b) 3 4 c) 1 5 d) 3 2 3. Convert the following decimals into percentage. a) 0.2 b) 0.12 c) 0.735 d) 2.38 4. Find the value of : a) 15% of Rs.1500 b) 12% of 1600 litres c) 20% of 70 kg d) 6 1 5% of 32 hours 5. a) Find the number whose 20% is 60. b) Find the number whose 42% is 105. c) If 10% of the population of a town is 300, find the total population. d) If 16% of the monthly salary of a man is Rs.2400, find his monthly income.
58 Perfect Mathematics Class 8 400 students of a school participated in the voting to select one out of two students Anisha or Rukesh for speech competition. a) If Anisha got three votes for every vote Rukesh got, how many votes did each get? b) If it was necessary to get more than 60% of the total votes to be selected, how many more votes than 60% did Anisha get? Project Work Exercise 2.4.3 Percentage 1. a) 1 10 1. b) 7 25 1. c) 2 5 1. d) 27 40 2. a) 50% 2. b) 75% 2. c) 20% 2. d) 150% 3. a) 20% 3. b) 12% 3. c) 73.5% 3. d) 238% 4. a) Rs. 225 4. b) 192 litres 4. c) 14kg 4. d) 1.984 hours 5. a) 300 5. b) 250 5. c) 3000 5. d) Rs.15000 Mixed Exercise B Ratio, Proportion and Percentage 1. Two student,s Krish and Purna studying at grade 8 have got new rulers. Their friend Sonam asked how much price they paid for it. The prices paid by them were as follows : Krish : Rs.101002, Purna : Rs.1005 a) Sonam could not understand the numbers they told. Give your idea to express the numbers in the proper form so that everyone can understand the prices easily. b) If Sonam also needs to buy a ruler, which shop is better for him? c) Express prices paid by Krish and Purna into fraction. d) How much more percent did Purna paid than Krish? Find out. 2. a) Convert the number 2.45 × 108 in a scientific notation into a decimal number. b) Two numbers are in the ratio of 3: 4. If their sum is 133, find the numbers. c) For what value of P, the quinary number 10P45 represents 134? d) Samir spent 25% of his income Rs.6000 and Sandeep spent 20% of his income Rs. 8000. Who spent more and by how much? 3. a) Convert the number 8.416 × 10–5 in a scientific notation into a decimal number. b) Convert 45810 into quinary number system. c) The age of Sajan, Rajan and Ramesh is 5, 7 and 8 years respectively. Divide Rs. 50,000 in the ratio of their age. d) What percent of total amount did Ramesh get?
Percentage 59 4. a) Write 14200000 × 2 in a scientific notation. b) Convert 0.3 into fraction. c) If a : b = 3: 4, b: c = 2: 5 then find a : c. d) If 35 sheep give 10 kg wool, how much wool is given by 63 sheep? 5. a) Divide 30 mangoes to Manisha and Nisha in the ratio of 2: 3. b) If 1112 is taken out from 100112, what is the remainder? c) On Saturday, the number of visitors in a zoo was 8400. On Monday, the number of visitors is decreased to 4200. By what percentage was it decreased? d) Simplify : 18 + 8 6. a) Write 437.65 in a scientific notation. b) How much will be the 30% of Rs.600? c) 12 workers can do a piece of work in 20 days. How many workers should be added to complete the work in 16 days? d) Which common number should be subtracted from each of the numerator and the denominator in the ratio 19: 16 to make the ratio 5: 4. 7. a) Find the value of x, if x, 3, 5 and 15 are in proportional. b) If there are 25 boys in a class of 40 students, find the percentage of girls in the class. c) If 9 pumps can pumps 425 litres of water in 10 minutes, how many liters of water will be pumped by 6 pumps in 12 minutes? d) If x = 2 + 3 then find the value of x – 1 x . 8. a) Write the number 235000 in a scientific notation. b) If the cost 10kg of apples is Rs. 2750, what will be the cost of 15kg apples? c) The ratio of the present ages of Reshma and Rekha is 2 : 3. After how many years, the ratio of their ages will be 5 : 7? d) Convert the quinary number 1235 into decimal number system. Mixed Exercise B Ratio, Proportion and Percentage 1. a) Rs. 20, Rs.25 1. b) The shop from which Krish had bought as price is cheaper by Rs.5. 1. c) 4 : 5, 1. c) 25% 2. a) 245000000 2. b) 57, 76 2. c) 1 2. d) Rs. 100 more by Sandeep 3. a) 0.00008416 3. b) 33135 3. c) Rs.12,500, Rs.17,500, Rs. 20,000 3. d) 40% 4. a) 2.84 × 107 4. b) 1 3 4. c) 3 : 10 4. d) 18 kg 5. a) 12, 18 5. b) 1100 5. c) 50% 5. d) 5 2 6. a) 4.3765 × 102 6. b) Rs.180 6. c) 3 men 6. d) 4 7. a) 1 7. b) 37.5% 7. c) 340 ltr. 7. d) 2 3 8. a) 2.35 × 105 8. b) Rs.4125 8. c) 5 years 8. d) 38
60 Perfect Mathematics Class 8 2.5 Profit, Loss and Discount 2.5.1 Profit and Loss Ramesh buys a bicycle for Rs. 4,000 and sells it for Rs.4200. He makes a profit of Rs. 200. If he sells it for Rs. 3700, he will get a loss of Rs.300. Here, Rs. 4000 is the cost price. Rs.4200 and Rs.3700 are the selling prices in two situations. The cost price is denoted by (CP) and the selling price is denoted by (SP). Santosh started a business of electronic gadgets. Some events of his business are displayed in the following table. Fill in the blank cells having the question mark. Gadget Name Cost Price Selling Price Profit Loss Mobile Rs. 20000 Rs.22000 Rs.2000 – Laptop Rs.58000 ? Rs.3000 – Tablet Rs. 45000 – Rs.2000 Headphone Rs.3200 Rs.3100 – ? Activity - 1 Cost Price (CP) : The price at which an article is purchased is called its cost price, written as CP. Selling Pirce (SP) : The price at which an article is sold is called its selling price, written as SP. Profit or Gain : When the selling price (SP) of an article is more than its cost price (CP), a profit is made. Profit is the difference between the selling price and the cost price. Loss : When the selling price of an article is less than its cost price, it leads to a loss. Loss also is the difference between the cost price and the selling price.
Percentage 61 Cost price (CP), selling price (SP), profit and loss have the following relations. Profit Loss If SP > CP, then there is profit. If SP < CP, then there is loss. 1. Profit = SP – CP 1. Loss = CP – SP 2. Profit = Profit% of CP 2. Loss = Loss% of CP 3. Profit percent = Profit Cost Price × 100% 3. Loss percent = Loss Cost Price × 100% 4. SP = 100 + P% 100 × CP 4. SP = 100 – L% 100 × CP 5. CP = 100 100 + P% × SP 5. CP = 100 100 – L% × SP Example 1: A shopkeeper sold 70 books at Rs. 65 each, which he had bought at Rs. 58 each. Find his profit amount. Solution: Here, Cost price of a book = Rs.58 Cost price of 70 books = Rs. 58 × 70 = Rs. 4060 Selling price of a book = Rs. 65 Selling price of 70 books = Rs. 65 × 70 = Rs.4550 Profit = SP – CP = Rs. 4550 – Rs.4060 = Rs. 490 Example 2: Krishna bought a laptop for Rs. 42000. He sold it at a loss of 15% after two years. What was its selling price? Solution: Given : Cost price of a laptop (CP) = Rs.42000 Loss percent (%) = 15% To find : Selling price of the laptop (SP) By formula, SP = 100 – L% 100 × CP = 100 – 15 100 × 42000 = 85 100 × 42000
62 Perfect Mathematics Class 8 = 85 × 420 = 35700 \ The selling price of the laptop was Rs.35700. Example 3: A shopkeeper bought a watch for Rs. 1,250 and sold it for Rs. 1,500. a) Did the shopkeeper make a profit of loss? Give reason. b) How much profit or loss did he make? c) Express profit or loss in percent. Solution: a) The shopkeeper made profit. Because his selling price is more than the cost price. b) Here, CP = Rs. 1,250, SP = Rs.1,500 Profit = SP – CP = Rs. 1,500 – Rs. 1,250 = Rs. 250 Hence, the required profit is Rs. 250. c) Again, profit percent = Profit C.P × 100% = 250 1250 × 100% = 20% Hence, the required profit percent is 20% Example 4: A man bought 480 glasses at Rs.300 per dozen. 20 glasses were broken and he sold the remaining glasses at Rs. 32 each. Find his profit or loss. Solution: Here, Cost price of 480 glasses = Rs.300 per dozen. Number of broken glasses = 20 Selling price of each glass = Rs. 32 Profit or loss = ? The cost price of 12 glasses = Rs.300 The cost price of 1 glass = Rs. 300 12 = Rs. 25 The cost price of 480 glasses = Rs.(480 × 25) = Rs. 12000 Number of broken glasses = 20.
Percentage 63 Number of fresh glasses = 480 – 20 = 460 Selling price of a glass = Rs. 30 Selling price of 460 glasses = Rs.(460 × 30) = Rs. 13800 Since selling price > cost price, there is profit. \ Profit = SP – CP = Rs. 13800 – Rs.12000 = Rs. 1800 Example 5: By selling a table for Rs.2700, a man makes a loss of 10%. What percent would he have gained if he had sold it for Rs. 3150? Solution: For first condition, Selling price of the table (SP) = Rs. 2700 Loss percent = 10% Here, SP CP Alternately, SP = 100 – L% 100 × CP or, 2700 = 100 – 10 100 × CP or, CP = Rs. 3000 Rs.90 Rs.100 Rs.2700 ? If selling price is Rs. 90, its cost price = Rs.100. If selling price is Re.1, its cost price = Rs. 100 90 . If selling price is Rs.2700, its cost price = Rs.100 90 × 2700 = Rs. 3000 For second condition, Selling price (SP) = Rs.3150, Cost price = Rs.3000 Profit amount = SP – CP = Rs. 3150 – Rs. 3000 = Rs. 150 Profit percent = Profit Cost Price × 100% = Rs.150 Rs.3000 × 100% = 5% Therefore, he would have gained 5%. Example 6: Sunita bought a mobile phone and sold to Sailendra at 10% loss. Sailendra again sold it for Rs. 6,750 at 25% profit.