Arithmetic 51 CHAPTER 3 Integers Lesson Topics Pages 3.1 Review on Integers 52 3.2 Addition and Subtraction of Integers 57 3.3 Multiplication and Division of Integers 64 3.4 Simplifications of Integers 71 What is the initial number of the ruler ? Count the scales of the ruler in cm and inches. What is the depth of the above swimming pool ? How can we represent the depth of the swimming pool ? What represents the water level of the swimming pool ? What is the maximum temperature on the Earth ? What is the minimum temperature on the Earth ? WARM-UP 6 feet
52 The Leading Mathematics - 7 3.1 Review on Integers At the end of this topic, the students will be able to: ¾ introduce the integers. Learning Objectives Now, we have an enlarged set consisting of three distinct sets of numbers: 1. The set of positive integers, Z+ : +1, +2, +3, ... ... 2. The integer zero: 0 3. The negative integers, Z− : ... ..., −3, − 2, −1. This new set of numbers is called the set of integers. We can define addition and multiplication for the set of integers as in the case of whole numbers. Such a set then becomes a system: the system of integers. Number line for integers The set of integers can be beautifully represented by means of the points on a straight line. Now, we see how a number line for integers can be constructed. We begin with a horizontal straight line. On this line, a point O is fixed. The direction to the right of O is taken to be positive and that to the left is taken to be negative. A point P in the positive direction is then chosen. The distance or length OP is taken as a unit of length. 0 O –1 N –2 M –3 L 1 P 2 Q 3 R – + Points are laid off at equal intervals of length OP on either side of O. Then, we assign the numbers 1, 2, 3, … respectively to the points P, Q, R,… on the positive or right side; and the numbers –1, –2, –3, … to the points N, M, L, … to the negative or left of O. The point O itself is called the origin. In this way, we have set up a one-to-one correspondence between the set of numbers and the set of points so far laid off on the number line (i.e., to each such point on the line there is an integer and to each integer there is a point). This line is a number line for the integers: ... ... –3 –2 –1 0 1 2 3 ... ...
Arithmetic 53 Ordering of integers and inequality Consider two points on a number line. One is always on the left of the other or one is always right of the other. We agree to accept that a number representing a point on the left is less than the number representing the point on the right. We may also say that a number representing a point on the right side is greater than the number representing a point on the left. Left 0 1 2 3 4 Right – + In the number line 2 is left of 3 or 3 is right of 2. Thus, we have 2 is less than 3 or 3 is greater than 2. In symbols, we write 2 < 3 for 2 is less than 3 and 3 > 2 for 3 is greater than 2. In such situation, the two numbers are said to be ordered and not in equality or unequal. The symbols < and > are called signs of inequality. The two symbols are shown in the following table: Symbol Meaning < > Less than Greater than Two numbers related by any of the above two signs form an inequality. Looking at the number line for the integers, we observe that i) Every positive integer is greater than zero, e.g., 3 > 0 or 0 < 3 ii) Every negative integer is less than zero, e.g., –3 < 0 or 0 > –3 iii) Every positive integer is greater than every negative integer, e.g., 1 > –5 or –5 < 1, and iv) The greater is the whole number the smaller is its negative, e.g., 5 > 3 but –5 < –3, and –5 < –7 but 7 > 5.
54 The Leading Mathematics - 7 On the whole, we may conclude that “ Given two integers a and b, then only one of the following is true: a < b or a = b or a > b ” By definition, an integer is a whole number or zero or the negative of a whole number. An integer without the negative sign is called its absolute value. For instances, The absolute value of 3 is 3, 0 is 0, –3 is 3. To denote an absolute value of an integer, we draw two vertical lines, one on either side of the integer. For instances, we write the absolute value of +3 as |+3| and |+3| = 3. 0 as |0| and |0| = 0. –3 as |–3| and |–3| = 3. EXERCISE 3.1 Your mastery depends on practice. Practice like you play. 1. Answer the following questions on the basis of the number line. (a) On which side of the point of reference do the numbers greater than 0 lie? (b) On which side of the point of reference do the numbers less than 0 lie? (c) On which side of a given number does a number 2 units less than the given number lie? (d) On which side of a given number does a number 1 unit greater than the given number lie? (e) Of the two numbers – 4 and –1, which is greater? (f) How many integers are there between – 4 and 4? 2. Write a number 4 units to the right on the basis of a number line. (a) 1 (b) –3 (c) – 6 (d) 1 (e) 0 3. Write a number 2 units to the left on the basis of a numbers line. (a) 3 (b) 2 (c) – 2 (d) – 5 (e) 0 –3 3 units 3 units 6 units –2 –1 0 1 2 3
Arithmetic 55 4. Write the opposite integer of each of the following given integers. (a) – 10 (b) + 6 (c) – 2 (d) – 3 (e) + 4 (f) + 0 5. Making the number line from –8 to 8 find the value of the following. (a) 2 units right and –3 units left to 0. (b) –3 units right and 4 units left 2. (c) 2 units right and 3 units left to –1. 6. List the elements of the sets of the integers. (a) A = {integers between + 2 and –3} (b) B = {integers between – 4 and +2} (c) C = {integers from –4 to +3} (d) D = {integers from +2 to –4 } 7. Put the symbols ‘>’ or ‘<’ or, ‘=’ between the following two numbers: (a) + 7 –2 (b) +3 + 4 (c) – 3 – 4 (d) +2 +2 (d) – 5 –6 (d) –5 –5 (e) – 5 + 5 (f) +4 – 5 8. Arrange the following integers in ascending order: (a) – 3, + 8, + 10, 0, –4, + 6, –7 (b) 0, –4, + 3, –6, – 16, + 12, –24, +27 (c) – 8, + 9, – 10, + 21, + 32, – 36, +26, –25 (d) – 14, + 15, – 5, + 9, + 13, – 10, – 12, +14 9. Arrange the following integers in descending order: (a) +6, –2, 0, +6, +5, –5, +21, –2, +32 (b) –6, –5, +8, –8, – 14, + 18, –34, +37 (c) – 18, + 19, – 13, + 31, + 42, – 46, +34, –15 (d) – 34, + 55, – 65, + 19, + 53, – 16, – 10, +24 10. Write the value of: (a) |– 8| (b) |+ 9| (c) |+ 6| (d) |+ 4| (e) |0| (f) |– 12| (g) |+ 7| (h) |– 24|
56 The Leading Mathematics - 7 11. Write the absolute value of each of the following given integer. (a) – 2 (b) +4 (c) – 7 (d) 0 (e) – 9 (f) + 5 12. Observe the given situation and write the answers of the following questions. (a) A dolphin was 1 m below the water level of a pond. If the dolphin jumped 3 m above the water level then how many meters has the dolphin jumped? (b) A man was standing 2 meters above the water surface of a swimming pool. He jumped into the pool and covered 5 meters from the place of standing. How much depth did he reach in the water? 13. Jesus is at a place 4 km East and Shiva is at a place 2 km west of a statue. (a) Using integers show this information on a number line. (b) What is the distance between Shiva and Jesus? Find on the basis of absolute value of integers. ANSWERS 1. to 5. Consult with your teacher. 6. (a) {– 2, – 1, 0, + 1} (b) {– 3, – 2, – 1, 0, + 1} (c) {– 4, – 3, – 2, – 1, 0, + 1 + 2, + 3} (d) {+2, + 1, 0, – 1, – 2, – 3, – 4} 7. to 9. Consult with your teacher. 10. (a) 8 (b) 9 (c) 6 (d) 4 (e) 0 (f) 12 (g) 7 (h) 24 11. (a) 2 (b) 4 (c) 7 (d) 0 (e) 9 (f) 5 12. (a) + 4 m (b) + 3 m 13. (a) Consult with your teacher. (b) 6 km
Arithmetic 57 3.2 Addition and Subtraction of Integers At the end of this topic, the students will be able to: ¾ add and subtract the integers. Learning Objectives Addition of Integers Adding two whole numbers is easy. We have already done so. But adding a whole number and a negative whole number or two negative whole numbers is somewhat different. We need more rule or rules. We can make such rules from our daily life experiences. A pictorial way of doing so is the use of the number line and set model. Adding physically means putting together, we add integers using either of the models a set model and a measurement model. Set Model While adding by set model we put the disjoint sets together as in the following examples: Here, denotes positive integer and denotes negative integer. (+3) + (+1) (+3) + (–1) (–3) + (+1) (–3) + (–1) +4 +2 –2 – 4 Measurement Model To begin with, we consider a number line for the integers. In the number line, the positive integers are written towards the right side of 0 and the negative integers towards the left of 0. Now, we take up concrete examples to illustrate various possible cases: Case 1: Addition of two positive integers (+a) and (+b), i.e., to find (+a) + (+b) To add +3 and +1 is to find (+3) + (+1). We start at ‘0’ move 3 units to the right. We reach at the point marked 3. We further For addition ‘+’, moving right side or East direction.
58 The Leading Mathematics - 7 move 1 unit to the right of the point marked 3 and arrive at the point marked +4. –3 –2 –1 0 +1 +2 +3 +4 +4 +3 1 +5 +6 +7 +8 Thus, we have (+3) + (+1) = +4. We can easily see that (+1) + (+3) = +4. Case 2: Addition of a positive integer (+ a) and a negative integer (–b). i.e., to find (+a) + (–b), where a > b To add (+3) and (–1) is to find (+3) + (–1). We start at ‘0’ move 3 units to the right. We arrive at the point marked +3. Then, we move 1 to the left of the point marked +3 and arrive at the point marked +2. –3 –2 –1 0 +1 +2 +3 +4 +2 3 1 +5 +6 +7 +8 Thus, we have (+3) + (–1) = +2. We can easily see that (–1) + (+3) = +2. Case 3: Addition of a negative integer (–a) and a positive integer (–b), i.e., to find (–a) + (+b), where a > b To add (–3) and (+1) is to find (–3) + (+1). We start at ‘0’ move 3 units to the left from the origin. We arrive at the point marked –3. We move 1 to the right of the point marked –3 and arrive at the point marked – 2. –3 –2 –1 0 +1 +2 +3 –2 –3 1 Thus, we have (–3) + (+1) = –2. We can easily see that (+1) + (–3) = –2. Case 4: Addition of two negative integers (– a) and (– b) , i.e., to find (– a) + (– b) To add (–3) and (–1) is to find (–3) + (–1). We start at ‘0’ move 3 units to the left. We arrive at the point marked –3. We then move 1 further to the left of the point marked –3 and arrive at the point marked – 4.
Arithmetic 59 –5 –4 –3 –2 –1 0 +1 +2 –4 1 3 Thus, we have (–3) + (–1) = –4. We can easily see that (–4) + (–1) = – 4. Case 5: Addition of a positive or negative integer and the number zero i.e., +a or – a and 0 , i.e., to find (+ a) + 0 or (– a) + 0 To add +3 and 0 is to find (+3) + 0. We start at ‘0’ move 3 units to the right. We arrive at the point marked +3. Then, we move 0 further to the right of the point marked +3 (i.e., no movement) and so remain at the same point marked + 3. Thus, we have (+3) + 0 = +3. We can easily see that (–3) + 0 = – 3. –3 –2 –1 0 +1 +2 +3 +4 3 On the basis of the above observations, we can draw certain conclusions. They are called the properties of addition of integers. We denote the set of integers by Z and the positive integer Z+ and negative integer Z– . For any two integers a and b, there is a unique integer called their sum. The sum of a and b is denoted by a + b 1. Closure Property: If a, b ∈ Z then (a + b) ∈ Z. 2. Commutative Property: If a, b ∈ Z, then a + b = b + a. 3. Associative Property: If a, b, c ∈ Z, then (a + b) + c = a + (b + c). 4. Additive Identity: If a ∈ Z, there is a unique 0 ∈ Z such that a + 0 = 0 + a. 5. Additive Inverse: If a ∈ Z, then a + (– a) = 0 = (– a) + a. Subtraction of Integers The positive and the negative numbers form an unlimited number of pairs of opposite integers. For instance, 1 and –1, 2 and –2, 3 and –3, 4 and – 4 are some such pairs. Clearly, 1 + (–1) = 0, 2 + (–2) = 0, 3 + (–3) = 0, 4 + (–4) = 0, …. In such a case, one integer of each pair is known as the additive inverse of the other. Subtraction of integers can be performed in many ways. Look at the following way:
60 The Leading Mathematics - 7 Here, denotes positive integer and denotes negative integer. (+4) + (–3) +1 (–3) + (–1) –4 (–3) – (–1) –2 Now, we can give a meaning of subtraction of an integer b from another integer a. By subtraction of a number b from another integer b, we mean the addition of the negative of the whole number b to the whole number a, i.e., the sum of a and (−b). In symbols, we write the sum : a + (−b) as a − b. Subtraction can also be easily understood with the help of a number line. For instances, 1. 4 – 3 = 4 + (– 3) = 1 may be interpreted as movement from the point marked zero (0) by 4 units to the right followed by a movement of 3 units to the left from the point marked 4. For subtraction ‘–’, moving left side or West direction. –3 –2 –1 0 +1 +2 +3 +4 1 4 3 +5 +6 +7 +8 2. Since (– (– 3)) = opposite of (– 3) = + 3, 4 – (–3) = 4 + (– (– 3)) = 4 + 3 = 7 may be interpreted as movement from the zero mark (0) by 4 units to the right followed by a movement of 3 units to the right from the point marked 4. Rules on Addition ‘+’ + ‘+’ = ‘+’ (Add) ‘–’ + ‘–’ = ‘+’ (Add) ‘+’ + ‘–’ = ‘–’ (Subtract) ‘–’ + ‘+’ = ‘+’ (Subtract)
Arithmetic 61 CLASSWORK EXAMPLES Example 1 Add: (a) (+7) + (+2) (b) (–7) + (–2) (c) (+7) + (–2) (d) (–7) + (+2) Solution: Here, (a) (+7) + (+2) = +(7 + 2) = + 9 (b) (–7) + (–2) = –(7 + 2) = – 9 (c) (+7) + (–2) = + (7 – 2) = + 5 (d) (–7) + (+2) = – (7 – 2) = – 5 Example 2 Subtract: (a) (+9) – (+4) (b) (–9) – (–4) (c) (+9) – (–4) (d) (–9) – (+4) Solution: Here, (a) (+9) – (+4) = (+9) + (–4) = +(9 – 4) = + 5 (b) (–9) – (–4) = –(9 – 4) = – 5 (c) (+9) – (–4) = + (9 + 4) = + 13 (d) (–9) – (+4) = – (9 + 4) = – 13 Example 3 Use the number line to verify that 4+ + 4– = 0. Solution: Draw a number line and move 4 units to the left from the zero 0. We arrive at the point marked + 4. Again, move 4 to the left of the point marked +4 and arrive at the same origin. –3 –2 –1 0 +1 +2 +3 +4 +5 4 4 ∴ 4+ + 4– = (+4) + (–4) = 0. Proved. If a, b ∈ Z & a> b then, (+a) + (+b) = +(a + b) (–a) + (–b) = – (a + b) (+a) + (–b) = +(a – b) (–a) + (+b) = – (a – b) If a, b ∈ Z & a> b then, (+a) – (+b) = + (a – b) (–a) – (–b) = – (a – b) (+a) + (–b) = + (a – b) (–a) + (+b) = – (a – b)
62 The Leading Mathematics - 7 EXERCISE 3.2 Your mastery depends on practice. Practice like you play. 1. Identify whether the following statements about integers are true or false. (a) Zero is a positive number. (b) The integers on the right of the origin are positive. (c) Of the numbers on the left the origin, the number nearer to the origin is greater than the further one. (d) 0 is the smallest integer. (e) The sum of two integers is always an integer. (f) The difference of two integers is always an integer. (g) 0 is called the identity element of addition. (h) (– a) – (+ b) = – (+ b) + (– a) is true. 2. Add the following integers: (a) (+ 5) + (+ 6) (b) (+ 10) + (+ 3) (c) (+12) + (– 6) (d) (+ 6) + (– 4) (e) (– 12) + (+ 4) (f) (– 24) + (+ 13) (g) (– 8) + (– 3) (h) (– 18) + (– 9) (i) (+ 25) + (– 8) 3. Subtract the following integers: (a) (+ 8) – (+ 3) (b) (+ 8) – (+ 1) (c) (+ 15) – (– 3) (d) (+ 17) – (– 4) (e) (– 23) – (+ 14) (f) (– 34) – (+ 17) (g) (– 39) – (– 3) (h) (– 10) – (– 16) (i) (– 5) – (– 8) 4. Show the following operations of integers on a number line. (a) (+ 4) + (+ 3) (b) (+ 8) + (– 4) (c) (– 5) – (+ 3) (d) (– 6) + (– 4) (e) (– 2) – (– 4) (f) (+ 4) + (– 4) (g) (+ 8) – (+ 3) (h) (+ 10) + (+ 6) (i) (+ 5) – (– 8) 6. Fill in the blanks with suitable integers and state the direction. (a) East 3 + West 4 = .................. (b) West 2 + West 5 = ................... (c) West 3 + East 5 = .................. (d) West 2 + East 3 = ................... (e) East 2 + ............... = East 5 (f) East 2 + ............... = East 8
Arithmetic 63 (g) Below 2 + Above 5 = ............. (h) Below 4 + Above 3 = ............ (i) Right 5 + left 4 = ................ (j) ............ + left 4 = Right 2 5. Fill in the table below with addition and subtraction. (a) (b) + –2 –1 0 +1 +2 +3 – –2 –1 0 +1 +2 +3 –2 –2 –1 –1 0 0 +1 +1 +2 +2 +3 +3 7. An snail moved 25cm towards East from the origin and then 14 cm West. (a) Show this information on a number line. (b) How far has the snail reached from the origin? 8. (a) If the sum of two integers is – 87 and if the greater integer is 127, then what is the smaller integer? (b) The difference of two integers is – 115. If one is + 152 then what is the other? 9. The minimum temperature of Kathmandu on 3rd Magh, 2073 is – 2o c and maximum, 17o c. (a) What is the difference between the maximum and minimum temperature? (b) If the minimum temperature forecast for that day is expected to be 2o , then what is the difference in the minimum temperatures? ANSWERS 2. (a) + 11 (b) + 13 (c) + 6 (d) + 2 (e) – 8 (f) – 11 (g) – 11 (h) – 27 (i) + 17 3. (a) + 5 (b) + 7 (c) + 18 (d) + 21 (e) – 37 (f) – 51 (g) – 36 (h) + 6 (i) + 3 7. (a) Consult with your teacher. (b) 11 cm 8. (a) – 214 (b) + 267 9. (a) 19° C (b) 15° C
64 The Leading Mathematics - 7 3.3 Multiplication and Division of Integers At the end of this topic, the students will be able to: ¾ multiply and divide the integers. Learning Objectives Multiplication of Integers Multiplication is understood as a repeated addition. Given a number, by repeated addition we mean addition of the same number several times to the number zero. In case of integers, we meet several cases. Case 1: Given the whole number 4, we write 3 times repeated addition of 4 to 0 = 0 + 4 + 4 + 4 = 4 + 4 + 4 = 12 = 3 × 4; Repeated addition of 4 to 0 three times = 0 + 4 + 4 + 4 = 4 + 4 + 4 = 12 = 4 × 3. Thus, 3 × 4 = 12 = 4 × 3. In general, given two whole numbers a and b, we have ( + whole number a) × ( + whole number b) = + ( a × b ) = + ( b × a). or, ( +) × ( +) = + . Case 2: Given the negative integer (– 4), 3 times repeated addition of (– 4) to 0 = 0 + (– 4) + (– 4) + (– 4) = – 4 – 4 – 4 = –12 = 3 × (– 4); Repeated addition of (– 4) to 0 three times = 0 + (– 4) + (– 4) + (– 4) = – 4 – 4 – 4 = –12 = (– 4) × 3. Thus, 3 × (– 4) = –12 = (– 4) × 3. In general, given a negative whole numbers – a and a positive whole number b, we have (– whole number a) × ( + whole number b) = – ( a × b ) = – ( b × a). or, ( –) × ( +) = – = ( +) × ( –).
Arithmetic 65 Case 3: Suppose the negative integer (– 4) is given. Observe the following: Type A: In (– 4) × 1, × 1 or × (+1) stands for 1 time addition of (– 4). In (– 4) × 2, × 2 or × (+2) stands for 2 times addition of (– 4). In (– 4) × 3, × 3 or × (+3) stands for 3 times addition of (– 4). Type B: In (–4) × (–1), × (–1) stands for 1 time subtraction of (– 4). In (– 4) × (–2), × (–2) stands for 2 times subtraction of (– 4). In (– 4) × (–3), × (–3) stands for 3 times subtraction of (– 4). Type C: 3 times repeated subtraction of – 4 from 0 = 0 – (–4) – (–4) – (–4) or, (– 3) × (–4) = 4 + 4 + 4 = 12 (–4) repeatedly subtracted three times from 0 = 0 – (–4) – (–4) – (–4) or, (–4) × (–3) = 4 + 4 + 4 = 12 Thus, (–3) × (–4) = 12 = (–4) × (–3). In general, given two negative whole numbers – a and –b, we have (– whole number a) × ( – whole number b) = (– a) × (– b ) = a × b (– whole number b) × ( – whole number a) = (– b) × (– a ) = b × a. We know a × b = b × a. Thus, ( –) × ( –) = + . On the basis of the above observations, we can draw certain conclusions. They are called the properties of multiplication of integers: 1. Closure Property: Given any two integers, their product is also n integer. In symbols, if a, b ∈ Z, then a × b ∈ Z 2. Commutative Property: Multiplication of two integers is commutative. In symbols, if a, b ∈ Z, then a × b = b × a. 3. Associative Property: Multiplication of integers is associative. In symbols, if a, b, c ∈ Z, then (a × b) × c = a × (b × c). 4. Distributive Property Over Addition: Multiplication of integers is distributive over addition. In symbols, if a, b, c ∈ Z, then a × (b + c) = a × b + a × c. Sign of Multiplication × + – + + – – – +
66 The Leading Mathematics - 7 5. Multiplicative Identity Property: Multiplication of an integer and 1 is itself integer. In symbols, if a, 1∈ Z, then a × 1 = a = 1 × a. So, 1 is multiplicative identity. 6. Multiplicative Null Property: Multiplication of integers and 0 is also zero. In symbols, if a, 0∈ Z, then a × 0 = 0 = 0 × a. So, 0 is multiplicative null. Fun Facts: ZERO is the only number which has so many names such as nought, naught, nil, zilch and zip. CLASSWORK EXAMPLES Example 1 Multiply : (a) + 3 × + 2 (b) (+ 2) × (– 4) (c) – 2 × + 5 (d) (– 5) × (– 4) Solution: (a) + 3 × + 2 ⇒ + 3 [ + 3 and + 2 have the same sign] × + 2 + 6 (b) (+ 2) × (– 4) ⇒ + 2 [ + 3 and – 4 have opposite sign] × – 4 – 8 (c) – 2 + 5 ⇒ – 2 [ – 2 and + 5 have opposite sign] × + 5 – 10 (d) (– 5) × (– 4) ⇒ (– 5) [ – 5 and – 4 have the same sign] × – 4 + 20 Example 2 Multiply the following and show them in the number line: (a) (+ 3) × (+ 4) (b) + 3 × (– 4) Solution: (a) + 3 × + 4 ⇒ + 3 [ + 3 and + 4 have the same sign] × + 4 + 12 –1 0 +1 +2 +3 +4 +5 +6 +3 +3 +3 +3 +7 +8 +9 +10 +11 +12 +12
Arithmetic 67 (b) + 3 × (– 4) = + 3 [ + 3 and – 4 have opposite sign] × – 4 – 12 +10–1–2–3–4–5–6 –3 –3 –3 –3 –12 –11 –7–8–9–10 –12 Division of Integers We have already seen that a whole number divided by another whole number may or may not be a whole number. For examples, 6 divided by 2 is 3 (i.e., 6 ÷ 2 = 6 2= 3) but 6 divided by 5 is not a whole number (i.e., 6 ÷ 5 = 6 5 ≠ a whole number). Consequently, division as such may not be meaningful in the set of all integers. We shall therefore start with the multiplication of two integers; and then define division as the inverse of multiplication. This can be done in the following ways: If the product of two non–zero integers a and b is another integer c, then c is said to be divisible by a and c. The result of division of c by a is denoted by c a and is equal to b. In the same way, the result of division of c by b is denoted by c b and is equal to a. The number c to be divided is called the dividend and the number which divides it is called the divisor. The result of division, i.e., c a and c b are called quotients. The sign of the quotient is easily determined by the rule or rules of signs for product. 1. If the product of two integers (or dividend) is positive, then the quotient is positive if the divisor is positive, the quotient is negative if the divisor is negative 2. If the product of two integers (or dividend) is negative, then the quotient is negative if the divisor is positive, the quotient is positive if the divisor is negative. In symbols and signs, (+) ÷ (+) = (+), (+) ÷ ( −) = (−), and (−) ÷ ( +) = (−), (−) ÷ ( −) = (+), Some important special cases of divisions are: a ÷ a = 1, (a 0); a ÷ 1 = a; a (−1) = − a; a ÷ (−a) = −1 (a ≠ 0); (−a) ÷ a = −1. Sign of Division ÷ + – + + – – – +
68 The Leading Mathematics - 7 CLASSWORK EXAMPLES Example 1 Find the quotient of: (a) (+ 12) ÷ (+ 4) (b) (– 12) ÷ (– 3) (c) (+ 16) ÷ (– 8) (d) (– 18) ÷ (+ 6) Solution: Here, (a) (+12) ÷ (+4) = + 3 [ +12 and +4 have the same sign and 4 × 3 = 12] (b) (–12) ÷ (–3) = + 4 [ –12 and –3 have the same sign and 4 × 3 = 12] (c) (+16) ÷ (–8) = – 2 [ +16 and –8 have different signs and 8 × 2 = 16] (d) (–18) ÷ (+6) = – 3 [ –18 and +6 have the same sign and 6 × 3 = 18] Example 2 Simplify: (a) + 4 × (– 16 ÷ + 4) (b) (+ 9 × – 4) ÷ – 6 Solution: Here, (a) + 4 × (– 16 ÷ + 4) = + 4 × – 8 = – 32. [Work within brackets first] (b) (+ 9 × – 4) ÷ – 6 = – 36 ÷ – 6 = + 6. [Work within brackets first] EXERCISE 3.3 Your mastery depends on practice. Practice like you play. 1. Multiply and demonstrate the product in the number line: (a) + 2 × + 3 (b) + 3 × + 4 (c) + 5 × + 3 (d) + 2 × – 3 (e) (+ 3) × (– 4) (f) (+ 5) × (– 3) (g) (– 2) × (+ 3) (h) (– 3) × (+ 4) (i) – 5 × + 3 (j) – 3 × – 2 (k) – 3 × – 4 (l) – 5 × – 3 2. Find the product of: (a) + 2 × – 3 (b) – 3 × + 4 (c) – 4 × + 5 (d) – 5 × + 2 (e) (– 3) × (+ 8) (f) (+ 6) × (– 9) (g) (– 6) × (– 7) (h) (– 7) × (– 8) (i) – 9 × – 4 (j) – 14 × – 2 (k) – 16 × + 3 (l) + 17 × – 6 3. Divide: (a) + 6 ÷ + 3 (b) + 8 ÷ + 4 (c) + 12 ÷ + 6 (d) – 6 ÷ + 3 (e) (– 8) ÷ (+ 4) (f) (– 12) ÷ (+ 6) (g) (+ 6) ÷ (– 3) (h) (+ 8) ÷ (– 4) (i) + 12 ÷ – 6 (j) – 6 ÷ – 3 (k) – 8 ÷ – 4 (l) – 12 ÷ – 6
Arithmetic 69 4. Complete the following multiplication table: × –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 –6 +36 –5 –4 +12 –3 –15 –2 0 –1 0 +1 +2 –8 +8 +3 +4 +5 +6 5. Complete the following division table: ÷ +2 +4 +6 +8 +10 +12 –12 –10 –8 –6 –4 –2 –2 –1 –3 –4 –5 –6 +6 +1 +5 +4 +3 +2
70 The Leading Mathematics - 7 6. (a) The product of two integers is +24. If one of them is –4, what is the other? (b) The product of two integers is –42. If one of them is –6, what is the other? (c) By what quantity –12 should be multiply to make the product + 24? (d) By what quantity +18 should be multiply to make the product – 54? ANSWERS 1. (a) + 6 (b) + 12 (c) + 15 (d) – 6 (e) – 12 (f) – 15 (g) – 6 (h) – 12 (i) – 15 (j) + 6 (k) + 12 (l) + 15 2. (a) – 6 (b) – 12 (c) – 20 (d) – 10 (e) – 24 (f) – 54 (g) + 42 (h) + 56 (i) + 36 (j) + 28 (k) – 48 (l) – 102 3. (a) + 2 (b) + 2 (c) + 2 (d) – 2 (e) – 2 (f) – 2 (g) – 2 (h) – 2 (i) – 2 (j) + (k) + 2 (l) + 2 6. (a) – 6 (b) + 7 (c) – 2 (d) – 3 Project Work 3.3 In a quiz competition of three groups A, B and C, the rules are followed as (+10) points for correct answer, (–5) points for incorrect answer and (0) point for left answer. (a) If the group A gives 5 correct answers, 4 false answers and 3 questions are left, how many points does the group A get? (b) If the group B gives 4 correct answers, 5 false answers and 3 questions are left, how many points does the group B get? (c) If the group C gives 6 correct answers, 5 false answers and 1 question is left, how many points does the group C get? (d) Who is winner in this quiz competition ?
Arithmetic 71 3.4 Simplifications of Integers At the end of this topic, the students will be able to: ¾ simplify the integers using brackets. Learning Objectives Order of Operations and Use of Brackets So far we have confined ourselves mostly to simple numerical expressions with one or two operations and brackets. Now, we try to simplify numerical expressions involving several fundamental operations. Suppose we calculate the expressions (i) 6 + 9 – 3 (ii) 6 + 9 × 3 (iii) 6 – 9 × 3 (iv) 6 + 9 ÷ 3 Now, we operate them as follows: (i) In 6 + 9 – 3, at first add, the result is 12 and at first subtract, the result is 12. In the both cases, the results are the same. (ii) In 6 + 9 × 3, at first add, the result is 45 and at first multiply, the result is 33. In the both cases, the results are not the same. It is problem somewhere. (iii) In 6 – 9 × 3, at first subtract, the result is –9 and at first multiply, the result is –21. In the both ways, there are different results. It is problem somewhere. (iv) In 6 + 9 ÷ 3, at first add, the result is 5 and at first divide, the result is 9. In the both ways, the results are not the same. It is problem somewhere. To eliminate these confusions of getting different results, we follow the usual procedure of performing the operation from left to right in the order: division, multiplication, addition and subtraction. In many cases, we use brackets such as; — Horizontal bar or Vinculum or simply bar, ( ) Parentheses or small brackets, { } Braces or curly brackets, [ ] Large or square brackets,
72 The Leading Mathematics - 7 to indicate which of the operations is to be performed first. In case of fundamental operations and brackets, we follow the familiar BODMAS rule. Its full form is given below: Brackets Of, Divide, Multiplication, Addition and Subtraction CLASSWORK EXAMPLES Example 1 Simplify: 36 + 12 ÷ 6 – 1 × 5 Solution: Here, 36 + 12 ÷ 6 – 1 5 = 36 + 2 – 1 × 5 [ Dividing] = 36 + 2 – 5 [ Multiplying] = 38 – 5 [ Adding] = 33. [ Subtracting] Example 2 Simplify: 15 – 8 + 9 ÷ (2 + 1) × 2 Solution: Here, 15 – 8 + 9 ÷ (2 + 1) × 2 = 15 – 8 + 9 ÷ 3 × 2 = 15 – 8 + 3 × 2 = 15 – 8 + 6 = 7 + 6 = 13. Example 3 Simplify: 4 - {(5 – 2) × 5} Solution: Here, 4 – {(5 – 2) × 5} = 4 – {3 × 5} = 4 – 15 = – 9 Example 4 Simplify: 4×11 ÷[-11 ÷ {12 – (13 – 12)}] Solution: Here, 4 × 11 ÷[– 11 ÷ {12 – (13 – 12)}] = 4 × 11 ÷[– 11 ÷ {12 – 1}] = 4 × 11 ÷[– 11 ÷ 11] = 4 × 11 ÷ [– 1] = 4 × (– 11) = – 44
Arithmetic 73 EXERCISE 3.4 Your mastery depends on practice. Practice like you play. 1. Simplify: (a) (+12) + (+4) – (+6) (b) (+17) + (+12) – (+23) (c) (+12) – (+8) – (+2) (d) (+12) + (–10) – (+3) (e) + 14 – + 12 – + 6 (f) – 7 + – 4 + + 13 (g) – 22 + + 12 – + 8 (h) – 15 – + 14 – + 9 2. Simplify: (a) (+2) × (+12) ÷ (+6) (b) (–3) × (+16) ÷ (+2) (c) (+12) ÷ (+4) × (–2) (d) (+18) ÷ (–9) × (–1) (e) (+7) × (+14) ÷ (+2) (f) (–28) ÷ (+4) × (+6) 3. Simplify: (a) (+2) × (–3) + (+4) (b) (+3) – (+6) × (+2) (c) (+14) ÷ (+7) + (–2) (d) (–1) × (–9) + (+5) (e) (+7) × (+2) – (–4) + (+2) (f) (+27) ÷ (+3) + (–4) – (+2) (g) (–18) ÷ (+2) + (+8) – (–7) × (+2) (h) (+8) × (+24) ÷ (–4) – (+2) + (+8) 4. Simplify: (a) + 5 + (+ 4 – – 2) (b) + 7 – (+ 8 – + 2) (c) + 8 + (– 5 – – 2) (d) + 3 – (– 8 + – 2) (e) + 12 + (– 8 + + 2) (f) + 14 + (– 9 – + 7) (g) + 13 – (+ 4 + – 1) (h) + 23 + (– 16 + + 14) (i) + 25 + (+ 13 – – 8) 5. Simplify: (a) + 3 × (– 3 × – 4) (b) (+ 4 × – 5) × + 2 (c) (– 6 × – 2) × – 3 (d) + 4 × (+ 5 × – 2) (e) (+ 3 ÷ – 2) × + 6 (f) ( + 12 ÷ – 2) × + 3 (g) ( + 14 ÷ – 2) × + 4) (h) (+ 4 × – 6) ÷ + 2 (i) (+ 24 ÷ – 6) × + 2 6. Simplify: (a) 20 – {8 + (5 – 2)} + 15 (b) 28 – 3 {24 ÷ (18 ÷ 6)} (c) 48 – 7{42 ÷ 56 ÷ 8)} (d) 39 – 4{16 ÷ (7 – 3)} – 23 (e) (200 ÷ 5 – 10) ÷ {2(7 – 4) – 1 } (f) 16 ÷ [3 + 4 {2 + 4 ÷ (4 – 2)}] (g) 5 + [2{18÷ (7 –5)} × 2 – 25] (h) [{(40 + 80) ÷ 10} + 18} 5
74 The Leading Mathematics - 7 7. Write the following word problems into mathematical expressions and simplify: (a) 45 divided by 5 minus 7. (b) 6 subtracted from the quotient of 36 divided by 6. (c) 60 divided by 15 times the difference of 18 and 19. (d) 8 added to the quotient of 56 divided by 7. (e) 81 divided by the sum of 4 and 5. (f) The quotient of the product of 7 and 3 divided by 21. (g) The sum of 5 and 6 divided by 11. (h) 6 times the difference of 10 and 7 divided by 9. (i) The difference of 30 and 2 divided by the product of 7 and 2. (j) The product of 7 and 6 plus 80 divided by 10. (k) 81 divided by 9 times 3 added to 18 minus 19. (l) The sum of 5 and 10 divided by the difference of 18 and 13. (m) 40 minus from the product of 5 and 16 minus 48 divided by 4. ANSWERS 1. (a) + 10 (b) + 6 (c) + 2 (d) – 1 (e) – 4 (f) + 2 (g) – 18 (h) – 38 2. (a) + 4 (b) – 24 (c) – 6 (d) + 2 (e) + 49 (f) – 42 3. (a) – 2 (b) – 9 (c) 0 (d) + 14 (e) + 20 (f) + 3 (g) + 13 (h) – 42 4. (a) + 11 (b) + 1 (c) + 5 (d) + 13 (e) + 6 (f) – 2 (g) + 10 (h) + 21 (i) + 46 5. (a) + 36 (b) – 40 (c) – 36 (d) – 40 (e) – 9 (f) – 18 (g) – 28 (h) – 12 (i) – 8 6. (a) 24 (b) 4 (c) 6 (d) 0 (e) 6 (f) 16 19 (g) 16 (h) 150 7. (a) 2 (b) 0 (c) – 4 (d) 16 (e) 9 (f) 1 (g) 1 (h) 2 (i) 2 (j) 50 (k) 11 (l) 3 (m) 28 Project Work 3.4 Present the any five examples of simplification of addition, subtraction, multiplication and division that are used in our daily life in your classroom.
Arithmetic 75 CHAPTER 4 Rational Numbers Lesson Topics Pages 4.1 Rational Numbers 76 1. How many athletics participates in the above race? 2. How much distance is traveled by them and how much time ? 3. Who is the first, second and third in position ? 4. What would be changed in heart beat when we are running ? 5. How much money is there in the purse ? 6. How much money is spent and how much saved ? 7. How many pieces is a pizza divided into ? WARM-UP
76 The Leading Mathematics - 7 4.1 Rational Numbers At the end of this topic, the students will be able to: ¾ introduce the rational numbers. Learning Objectives We use number lines to give the geometrical representation of integers. We define a unit length on the number line, then the other integers are represented by the points which are at a distance from the origin to the multiples of the unit length. –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 If the integer is positive it lies to the right side of the number zero. If the integer is negative, it lies to the left side of the number zero. Now question arises. If the number line is divided into number of unit length, what happens if the unit length is divided by the integer itself? Suppose for example, what happens when + 1 is divided by + 2 ? i.e., where does + 1 2 lie on the number line? This means to divide the unit length 0 to 1 into two equal parts and one part to represent 1 2 (called half). –2 –2 –1 –2 –2 0 +1 +2 1 3 1 2 + + Hence + 1 2 lies exactly half way between 0 and 1. Similarly, + 1 3 lies one third distance from 0 between 0 and + 1. In this way, while working with integers, the product of two integers can be represented by a point on the number line. Consider for example, (+ 1 + + 3) × - 2 = + 4 × (- 2) = - 8 – 8 can be represented by a point 8 unit away to the left from 0. Let’s think, what happens when the sum of two integers is divided by an integer? Can it be represented by a point on the number line? Consider for example + 1 + + 2 + 2 , i.e., the sum of + 1 and + 2 divided by + 2 is + 1 + + 2 + 2 = + 2 + + 1 + 2 = + 2 + 2 = + 1 + 2 = 1 + + 1 + 1 = + 1 1 2
Arithmetic 77 –2 –2 –1 –2 –2 0 +1 +2 1 2 +1 The number + 11 2 is one and a half unit away to the right side from the origin (0). Actually , + 1 + + 2 is an integer sum +3 , hence + 3 divided by + 2 and this is the best question of dividing an integer by an integer. A number obtained by dividing an integer by a non-zero integer is a rational number. Symbolically, Q = {n: n = 1 b, where a, b are integers and b ≠ 0 } is the set of rational number (fraction). Note: (a) Every integer is a rational number because it has denominator 1. i.e., 4 = 4 1, –3 = – 3 1, 0 = 0 1, etc. So, every natural number and whole numbers are also rational numbers. (b) Decimal numbers are also rational numbers. i.e., 0.4 = 4 10 = 2 25 , 0.467 = 467 1000 , etc. Rational number is a terminating decimal number. Let’s take a rational number as 3 8. Now, convert the rational number into decimal number, 3 8 = 0.375, which has no more digits after 5 except 0. Therefore, it is a terminating decimal number. Rational number is a terminating decimal number. Let’s take a rational number as 25 11. Now, convert the rational number into decimal number, 25 11 = 2.272727..... = 2.27, which has many more 27 after 27. Therefore, it is a repeating decimal number. 8 ) 3 0 ( 0.375 – 28 2 0 – 16 4 0 – 40 0 11 ) 2 5 ( 2.2727 – 22 3 0 – 22 8 0 – 77 3 0 – 22 8 0 – 27 3
78 The Leading Mathematics - 7 Non-terminating and non-repeating decimal number is not rational number. Let’s take a number as 15. Now, evaluate the square root of 15, 15 = 3.8729......... , which has many more digits after 9 but no repeated. Therefore, it is a non-repeating and nonterminating decimal number. CLASSWORK EXAMPLES Example 1 Locate 3 2 on the number line. Solution: Construct a number line. Construct two parallel lines l 1 and l 2 from 0 and respectively. Q1 Q2 P1 Q3 P2 Q4 P3 Q5 P4 E P5 l2 l1 0 +1 +2 A B C D Construct 5 equal division with a pencil compass along l1 and l2. 3 3 15 – 9 3.8729 68 + 8 600 544 767 + 7 5600 5369 7742 + 2 23100 15484 77449 + 9 761600 637851 77458 63749
Arithmetic 79 Let P1, P2, P3, P4 and P5 be the points on l2 dividing into 5 equal parts and Q1, Q2, Q3, Q4 and Q5 be the points dividing l1 into 5 equal parts. Now join 0 to Q1, P1 to Q2, P2 to Q3, P3 to Q4, P4 to Q5, and P5 to +1. Now A, B, C, D, E divide the length of 1 unit between 0 and 1 into 5 equal parts. The point C is at 3 5 distance from 0. Hence, C is the required point. Similarly, to locate 2 3 5 on the number line repeat the process as above constructing two parallel lines from 2 and 3. 0 +1 +2 +3 Q1 Q2 P1 Q3 P2 Q4 P3 Q5 P4 E P5 l2 l1 X is the point indicating 2 3 5 . Example 2 Insert rational number between 2 and 3. Solution: One of the rational numbers that lies half way between 2 and 3 is the arithmetic average of 2 and 3. Therefore, one of the rational numbers is 1 2 (2 + 3) = 5 2 = 21 2 . The other rational number may lie half way between 2 and 21 2 ; which is the arithmetic average of 2 and 21 2 . We have, 21 2 2 + 21 2 = 1 2 2 + 1 2 = 1 2 × 9 4 = 9 4 = 21 4 . And the next rational number may be lying half way between 21 2 and 3; which is again the arithmetic average of 21 2 and 3 Now, 1 2 2 1 2 + 3 = 1 2 5 2 +3 = 1 2 × 11 2 = 11 4 = 23 4 . Hence, the required rational numbers between 2 and 3 are 9 4 , 5 2 and 11 4 as 21 4 , 21 4 and 23 4 .
80 The Leading Mathematics - 7 0 1 2 3 4 2 1 4 2 1 2 2 3 4 In fact in between two numbers there lie infinite rational numbers. Therefore, there is not unique solution for this type of problem. Example 3 Find the 5 rational numbers between 11 4 and 11 3 . Solution: Here, the given two rational numbers are 11 4 and 11 3 . The LCM of the denominator 4 and 3 is 12. Convert the denominator to 12, 11 4 = 11 × 3 4 × 3 = 33 12 and 11 4 = 11 × 3 3 × 4 = 44 12. ∴ The 5 rational numbers between 11 4 and 11 3 are; 34 12, 36 12, 39 12, 42 12 and 43 12. EXERCISE 4.1 Your mastery depends on practice. Practice like you play. 1. Which of the following are rational numbers? (a) 8 9 (b) –31 2 (c) 5 (d) 0 (e) 0.32 (f) – 0.337 (g) 9 (h) – 1 3 2. State whether the following statements are true or false: (a) Every natural number is rational number. (b) Each number of a set of integers is not rational number. (c) Whole numbers are rational numbers. (d) Zero is rational number. (e) Fractional numbers are not rational numbers. (f) Terminating decimal numbers are rational numbers. (g) Repeating decimal numbers are not rational numbers.
Arithmetic 81 3. Locate the following rational numbers in the number line. (a) 1 2 (b) 3 1 2 (c) 1 1 2 (d) 3 3 4 (e) 1 8 (f) 2 3 8 (g) 5 5 8 (h) 2 1 4 4. Locate the following rational numbers by the method of construction. (a) 2 5 (b) 3 3 5 (c) 2 4 5 (d) 4 1 7 (e) 2 3 7 (f) 5 4 9 (g) 8 7 10 (h) 5 9 10 5. Insert any three rational numbers in between the following pairs of numbers. (a) 2 and 3 (b) 2 1 2 and 3 (c) 3 1 2 and 3 3 4 (d) 5 and 6 (e) 4 1 8 and 4 7 8 (f) 5 4 7 and 5 6 7 6. Find 5 rational numbers in between the following pairs of numbers. (a) -2 and -1 (b) 1 2 and 1 (c) 3 1 4 and 3 1 3 (d) 3 1 5 and 3 7 4 (e) 5 9 and 7 12 (f) 5 8 and 5 12 7. How many rational numbers are there between? (a) 1 and 2 (b) 1 2 and 1 4 (c) 3 6 and 7 10 (d) 6 and 6 1 2 (e) Any two numbers a and b. 8. Convert the following into fractions: (a) 0.2 (b) 0.75 (c) 1.125 (d) 0.3 (e) 1.25 (f) – 2.25 (g) 3.125 (h) 2.225 ANSWERS Consult with your teacher. Project Work 4.1 Show the relation among natural number, whole number, integer and rational number in a chart paper. Present it in your classroom.
82 The Leading Mathematics - 7 CHAPTER 5 Fraction and Decimal Lesson Topics Pages 5.1 Review on Fraction 83 5.2 Simplification of Fractions 89 5.3 Word Problems on Fractions 96 5.4 Review on Decimals 99 5.5 Simplifications on Decimals 107 5.6 Verbal Problems on Decimals 116 How many parts of the cake are remaining ? How many parts of the cake have been taken ? If a boy took the one-one piece, how many parts of pizza will remain? What would be the rate of petrol per litre? What is the quantity of petrol filled for the scooter? How much total amount is paid for the petrol by the conductor? WARM-UP Petrol price is 169.75/ltr. Rs. 339.50 0 2 0 0.
Arithmetic 83 5.1 Review on Fraction At the end of this topic, the students will be able to: introduce the fraction. identify the types of fraction. Learning Objectives Man created counting numbers in ancient time. But, it is believed that even the primitive man had some idea about halves, thirds, quarters and so on. In other words, fraction appeared as a portion of a whole long before other numbers were created. With the development of the process of division or division rule of numbers, fraction became more meaningful and useful. A fraction formed by dividing a number by 10 or 100 or 1000 or the like is found to be of special importance. Such fractions are called decimal fractions. Fraction We have seen division may or may not have meaning in the system of integers. But we need division operation. To make division meaningful, it became necessary to introduce or invent new numbers. Reciprocal Number Now, we use the notion of multiplication of integers to construct a new kind of number. Suppose we have a non–zero integer a. Then, we agree to construct a number y such that the product of a and y is 1, i.e., a × y = 1. (a ≠ 1) This new number y is called the reciprocal of a. It is denoted by 1 a ; and is read as “ fraction or ratio of 1 to a ”. Based upon the above construction, the reciprocals of the non-zero integers … ,– 101, –100, …, –12, –11, … – 2, – 1, 1, 2, …, 11, 12, …, 111, 112, … would look like ..., – 1 101, – 1 100,..., – 1 12, – 1 11, ..., – 1 2, – 1 1, 1 1, 1 2, ..., 1 11, 1 12, ..., 1 111, 1 112... In particular, consider the non-zero integer 3. We construct a new number denoted by 1 3 such that 3 × 1 3 = 1. We call the number 1 3 the reciprocal of the whole number 3. This number is read “1 upon 3 ” or “ 1 over 3 ” or “ 1 is to 3 ”.
84 The Leading Mathematics - 7 We also call it a fraction or ratio with numerator 1 and denominator 3. In each case, we say that 1 is divided by 3. In case, we have; a × y = b, where b is an integer and a ≠ 0, we denote the new number y by b a . It is also called a fraction or ratio with b as numerator and a as denominator. It will be read as “ b over a ” . In such a case, we say that b is divided by a. We can then write; a × b a = b , a ≠ 0. We are now in a position to give one of the most important definitions in arithmetic. A rational number (or fraction or ratio) is a number that can be expressed in the form p q , where p and q are integers and q ≠ 0. We know how to add and multiply the natural numbers and integers. The operations of addition and multiplication enjoyed many properties. Likewise, we can add, subtract, multiply and divide the rational numbers. Here also, we have the commutative, associative and distributive laws. We can therefore say that the set of rational numbers also constitutes a system under addition and multiplication. The detailed description of the properties of addition and multiplication is similar to the ones already discussed. So, it is not necessary to repeat them. A number line for the set of rational numbers also can be constructed as before. Equivalent Fractions Equivalent fractions are different fractions that name the same number, even though they may look different. 1 2 2 4 4 8 The fractions 1 2, 2 4 and 4 8 are equivalent since each represents the same value.
Arithmetic 85 Here is why those fractions are really the same: 1 2 2 4 4 8 = × 2 × 2 × 2 × 2 = If the different numbers multiply a fraction, then the fractions so obtained are all equivalent fractions. Fraction in its Lowest Term Here are some more equivalent fractions, this time by dividing: 18 36 9 18 3 6 1 2 = ÷ 2 ÷ 2 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = = Choose the number you divide by carefully, so that the results (both top and bottom) stay whole numbers. If we keep dividing until we cannot go any further, then we have simplified the fraction (made it as simple as possible). That means cancel out the common factors from the numerator and denominator of the fraction. This is the lowest term of the given fraction. Types of Fractions A fraction has two parts: numerator and denominator. So, we may meet cases in which a) Numerator is smaller than the denominator, b) Numerator is equal to the denominator, and c) Numerator is greater than the denominator. In the first case, we say that the fraction is a proper fraction, eg, 1 2, 1 3, 2 3, 1 4, 2 4, 3 4. A fraction in which numerator is smaller than the denominator is called proper fraction. In the second and the third cases, we say that the fraction is an improper fraction, eg, 2 2, 3 3, 3 2 , 5 4 , 6 4 , 7 4 , 8 4.
86 The Leading Mathematics - 7 A fraction in which numerator is greater than the denominator is called improper fraction. An improper fraction always consists of a whole number part and a proper fraction part. For instance, the improper fraction 3 2 consists of 1 full part and 1 half part as shown below: 3 2 1 2 1 = 2 2 2 ) 3 ( 1 – 2 1 3 2 = 11 2 We observe that 1 + 1 2 = 3 2. In practice the + sign is deleted and the fraction is written as 11 2. A fraction having a whole number part and a proper fraction part is called a mixed fraction. Note: Mixed Fraction = Quotient Remainder Denominator Improper Fraction = Quotient × Denominator + Remainder Denominator In all cases, fractions having the same denominator are said to be like fractions. For instances, the fractions 1 4, 2 4, 3 4 , 4 4, 5 4 , 6 4 , 7 4 and 8 4 are all like fractions. Fractions having different denominators are called unlike fractions. The fractions 1 4 and 2 3 are unlike. Like fractions can be easily compared. In such cases, a fraction with a greater numerator is greater than that with lesser numerator. In the same way, a fraction with a smaller numerator is smaller than that with larger numerator. Comparison of Fractions Consider two fractions with the same denominator as 1 4 and 3 4. Since 1< 3 so 1 4 < 3 4 or 3 4 > 1 4.
Arithmetic 87 Again, consider two fractions with different denominator as 3 4 and 5 6. Which one of them is greater? For this at first, we change the denominator into the same by taking the LCM of their denominators and then compare their numerators. CLASSWORK EXAMPLES Example 1 Arrange the given fraction into ascending order: 5 6, 7 9 , 2 3, 11 12 Solution: Here, the LCM of the denominators 6, 9, 3, and 12 is 36. 5 6 = 5 × 6 6 × 6 = 30 36; 7 9 = 7 × 4 9 × 4 = 28 36; 2 3 = 2 × 12 3 × 12 = 24 36; 11 12 = 11 × 3 12 × 3 = 33 36. Clearly, 24 36 < 28 36 < 30 36 < 33 36. Arranging the given fractions in ascending order as 2 3, 7 9 , 5 6, 11 12. EXERCISE 5.1 Your mastery depends on practice. Practice like you play. 1. Identify the following fractions as proper fraction or improper fraction or mixed fraction: (a) 1 4 (b) 1 8 (c) 2 7 8 (d) – 11 12 (e) 26 17 (f) – 4 1 9 2. Write the five equivalent fractions of the following fractions: (a) 1 3 (b) 2 5 (c) 7 8 (d) 11 13 (e) 13 15 (f) 19 20 3. Write the following fractions in their lowest terms: (a) 4 6 (b) 16 24 (c) 32 36 (d) 34 51 (e) 26 104 (f) 100 210
88 The Leading Mathematics - 7 (g) 114 126 (h) 106 204 (i) 325 625 (j) 834 951 (k) 2605 2505 (l) 3423 4362 4. Write down the following mixed fractions into improper fractions: (a) 2 1 3 (b) 3 2 5 (c) 4 7 8 (d) 1 11 13 (e) 3 13 15 (f) 5 19 20 5. Write down the following improper fractions into mixed fractions: (a) 4 3 (b) 12 5 (c) 37 8 (d) 101 13 (e) 137 15 (f) 193 20 6. Write the correct sign (< , > or =) in the gap: (a) 1 3 ..... 2 3 (b) 2 5 ..... 2 3 (c) 7 9 ..... 7 9 (d) 11 13 ..... 15 17 (e) 10 13 ..... 12 17 (f) 13 25 ..... 19 35 (g) 7 28 ..... 9 42 (h) 30 88 ..... 15 44 7. Arrange the following fractions into ascending order: (a) 5 9, 7 3, 2 6 (b) 7 12, 3 8, 5 6 (c) 8 21, 2 14, 27 35, 17 28 (d) 13 20, 7 15, 23 60, 19 40 8. Arrange the following fractions into descending order: (a) 2 3, 7 12, 5 9 (b) 7 18, 13 24, 1 6 (c) 25 36, 16 27, 29 45, 11 9 (d) 7 40, 7 30, 13 60, 29 50 ANSWERS 3. (a) 2 3 (b) 2 3 (c) 8 9 (d) 2 3 (e) 1 4 (f) 10 21 (g) 19 21 (h) 53 102 (i) 13 25 (j) 278 317 (k) 521 501 (l) 1141 1454 4. (a) 7 3 (b) 17 5 (c) 39 8 (d) 24 13 (e) 58 15 (f) 119 20 5. (a) 11 3 (b) 22 5 (c) 45 8 (d) 710 13 (e) 9 2 15 (f) 913 20 7. (a) 2 6 , 5 9 , 7 3 (b) 3 8 , 7 12, 5 6 (c) 2 14, 8 21, 17 28, 27 35 (d) 23 60, 7 15, 19 40, 13 20 8. (a) 2 3 , 7 12, 5 9 (b) 13 24, 7 18, 1 6 (c) 11 9 , 25 36, 29 45, 16 27 (d) 29 50, 7 30, 13 60, 7 40
Arithmetic 89 5.2 Simplification of Fractions At the end of this topic, the students will be able to: ¾ simplify the fraction. Learning Objectives Four Fundamental Operations on Fractions Addition and multiplication are meaningful in the set of whole numbers. Subtraction is possible in the set of integers. Division is possible only when we have rational or fractional numbers. That is, all four simple rules are applicable in the set of fractional or rational numbers. To handle fractions, we must accept the following principles: (a) Fundamental principle of equivalent fractions “A fraction remains unchanged if its numerator and denominator are multiplied or divided by the same non-zero number.” For examples, 1) 1 3 = 1 × 2 3 × 2 = 1 × 3 3 × 3 = 1 × 5 3 × 5 = 1 × 30 3 × 30 = 30 90. 2) 30 90 = 30 ÷ 30 90 ÷ 30 = 30 ÷ 15 90 ÷ 15 = 30 ÷ 10 90 ÷ 10 = 30 ÷ 12 90 ÷ 12 = 1 3. We may state the above principle in the following way also: (b) Fundamental principle of cancellation: “A fraction remains unchanged if factors common to the numerator and denominator are cancelled.” For examples, 1 3 = 1 × 2 3 × 2 = 1 × 3 3 × 3 = 1 × 5 3 × 5 = 1 × 30 3 × 30 = 30 90. In problems of fractions, we often apply this principle to put the fraction in the simplest form or reduce it to the lowest term. (c) Fundamental principle of addition and subtraction of fractions: When add or subtract one fraction from another fraction follow the following three steps:
90 The Leading Mathematics - 7 Step 1 : Reduce all fractions into like fractions Step 2 : Add or subtract the numerator keeping the denominator intact Step 3 : Reduce the result to the lowest term. For examples, 1) 1 5 + 2 5 = 1 + 2 5 = 3 5. 2) 1 + 1 5 = 1 × 5 1 × 5 + 2 5 = 5 5 + 2 5 = 5 + 2 5 = 7 5. 3) 1 2 + 2 3 = 1 × 3 2 × 3 + 2 × 2 3 × 2 = 3 6 + 4 6 = 3 + 4 6 = 7 6 4) 1 2 – 2 3 + 1 4 = 1 × 6 2 × 6 – 2 ×4 3 × 4 + 1 × 3 4 × 3 (Taking the product of the denominators as the new denominator) = 6 12 – 8 12 + 3 12 = 6 – 8 + 3 12 = 1 12 A shorter and commonly followed technique is to take the LCM of the denominators as the new denominator. For example, 1 2 – 5 6 + 3 4 = 1 × 6 2 × 6 – 5 × 2 6 × 2 + 3 × 3 4 × 3 (Taking the LCM 12 of the denominators as the new denominator) = 6 12 – 10 12 + 9 12 = 6 – 10 + 9 12 = 5 12 The third basic principle is that of the multiplication of two fractions. (d) Fundamental principle of multiplication of fractions “The product of two fractions is a fraction whose numerator is the product of the numerators of the given fractions and denominator is the product of the denominators of the given fractions.”
Arithmetic 91 For examples, 1) 1 2 × 3 4 = 1 × 3 2 × 4 = 3 8. 2) 2 × 3 5 = 2 1 × 3 5 = 2 × 3 1 × 5 = 6 5. 3) 1 1 2 × 3 4 = 3 2 × 3 4 = 3 × 3 2 × 4 = 9 8. 4) 2 1 2 × 3 5 = 5 2 × 3 5 = 1 × 3 2 × 1 = 3 2 (d) Fundamental principle of division of fractions: “The quotient of a fraction divided by another is the product of the first fraction and the reciprocal of the second fraction.” For examples, (1) 1 2 ÷ 3 5 = 1 2 × 5 3 = 1 × 5 2 × 3 = 5 6. (2) 2 ÷ 3 4 = 2 × 4 3 = 2 × 4 1 × 3 = 8 3. (3) 2 1 2 ÷ 5 = 5 2 × 1 5 = 1 × 1 2 × 1 = 1 2. (4) 3 4 ÷ 21 2 = 3 5 ÷ 5 2 = 3 5 × 2 5 = 3 × 2 5 × 5 = 6 25. (e) Order of operations under grouping signs: Grouping of fractions may be done in various ways. Some of the brackets used for grouping are: (a) a horizontal bar —, known as vinculum, drawn above the numbers, (b) small brackets or parentheses ( ), (c) curly brackets or braces { } and (d) large or square brackets [ ]. (1) These brackets are removed inside out, that is, in the order from the inner one first and then the next and so on. While removing the brackets, the signs of the terms remain the same if the sign just before the bracket removed is positive. But all signs are changed if the sign before the bracket removed is negative. Reciprocal means the product of two numbers is equal to 1. e.g. 3 5 × 5 3 = 1
92 The Leading Mathematics - 7 (2) In case, the word ‘of’, the division sign ‘ ÷ ’ and the multiplication sign × in between a number or a group of numbers, the order in which they should be performed should be in the order indicated by the word BODMAS. B for Bracket O for Of or Order D for Division M for Multiplication A for Addition S for Subtraction. CLASSWORK EXAMPLES Example 2 Simplify: 8 15 + 1 2 – 7 12. Solution: Here, 8 15 + 1 2 – 7 12 = 8 × 4 15 × 4 + 1 × 30 2 × 30 – 7 × 5 12 × 5 = 32 60 + 30 60 – 35 60 = 32 + 30 – 35 60 = 27 60 = 9 20 Example 3 Simplify: 2 5 8 ÷ 2 1 12 × 5 70 Solution: Here, 2 5 8÷ 2 1 12 × 5 70 = 21 8 ÷ 25 12 × 5 70 = 21 8 × 12 25 × 5 7 = 9 10 2 15, 2, 12 3 15, 1, 6 5, 1, 2 ∴ LCM of 15, 2 and 12 = 2 × 3 × 5 × 1 × 2 = 60
Arithmetic 93 Example 4 Simplify: 2 9 + 1 2 of 7 8 + 3 4 – 5 12 ÷ 1 2 Solution: Here, = 2 9 + 1 2 of 7 8 + 3 4 – 5 12 ÷ 1 2 = 2 9 + 1 2 of 7 8 + 3 × 3 4 × 3 – 5 12 ÷ 1 2 = 2 9 + 1 2 of 7 8 + 9 12 – 5 12 ÷ 1 2 = 2 9 + 1 2 of 7 8 + 4 12 ÷ 1 2 = 2 9 + 1 2 of 7 8 + 1 3 ÷ 1 2 = 2 9 + 1 2 × 21 + 8 24 ÷ 1 2 = 2 9 + 1 2 × 29 24 × 2 1 = 2 9 + 29 24 = 2 × 8 + 29 × 3 72 = 16 + 87 72 = 103 72 = 1 31 72 EXERCISE 5.2 Your mastery depends on practice. Practice like you play. 1. Add: (a) 5 7 + 3 14 (b) 3 8 + 5 6 + 7 12 (c) 2 4 9 + 1 7 12 (d) 3 13 15 + 2 7 25 +3 1 20
94 The Leading Mathematics - 7 1. (a) 13 14 (b) 119 24 (c) 4 1 36 (d) 9 59 300 2. (a) 1 9 (b) 17 30 (c) 31 54 (d) 179 210 3. (a) 17 24 (b) – 11 36 (c) 20 27 (d) 1 7 12 (e) 59 90 (f) 3 7 12 4. (a) 1 6 (b) 4 15 (c) 11 9 (d) 8137 800 5. (a) 13 1 2 (b) 31 2 (c) 25 86 (d) 11 2 6. (a) 1 25 (b) 81 500 (c) 3 4 (d) 15 9 (e) 31 88 (f) 53 84 7. (a) 61 72 (b) – 1571 2205 (c) – 13 25 (d) 1 10 (e) 1 1 24 (f) 23 90 (g) 7244 2100 (h) 7 63 128 (i) 1242 375 (j) 3 45 112 8. (a) – 2 15 (b) 3 15 64 (c) 7113 224 (d) – 53 72 (e) 28 1215 (f) 6109 162 (g) – 12 1 3 (h) 3 26 45 (i) 8 33 (j) – 257 70 2. Subtract: (a) 2 3 – 5 9 (b) 7 10 – 2 15 (c) 2 8 27 – 1 5 36 – 7 12 (d) 4 9 35 – 2 4 21 – 1 3 14 3. Simplify: (a) 3 4 + 5 6 – 7 8 (b) 4 9 + 5 12 – 7 6 (c) 8 27 – 1 3 + 7 9 (d) 2 9 10 – 314 30 + 2 3 20 (e) 5 2 9 – 3 4 15 – 1 3 10 (f) 4 3 6 – 2 4 15 + 1 7 20 4. Multiply: (a) 2 9 × 3 4 (b) 9 10 × 8 27 (c) 14 15 by 25 21 (d) 4 3 25 by 2 15 32 5. Divide: (a) 8 ÷ 16 27 (b) 4 ÷ 8 7 (c) 1 9 16 by 227 8 (d) 3 4 9 by 2 8 27 6. Simplify: (a) 2 9 × 3 8 ÷ 25 12 (b) 7 8 × 6 35 ÷ 25 27 (c) 8 9 ÷ 2 3 × 9 16 (d) 2 1 10 ÷ 4 1 2 × 3 1 3 (e) 3 4 9 ÷ 18 3 ÷ 22 3 (f) 4 5 12 ÷ 3 1 9 ÷ 2 1 4 7. Simplify: (a) 2 3 + 5 6 × 3 4 – 4 9 (b) 4 5 × 6 7 × 2 21 – 7 9 (c) 9 10 + 2 25 – 4 9 × 27 8 (d) 6 25 ÷ 9 10 – 7 8 × 4 21 (e) 21 16 of 4 14 + 7 8 of 16 21 (f) 2 3 of 5 6 + 2 5 – 7 10
Arithmetic 95 (g) 1 4 15 of 2 9 10 ÷ 7 8 – 3 4 (h) 2 3 8 + 3 1 3 of 7 8 ÷ 2 3 (i) 3 4 9 – 2 6 25 ÷ 5 6 + 8 9 (j) 2 4 7 + 1 2 14 – 7 8 ÷ 2 4 5 8. Simplify: (a) 2 5 of 2 5 – 2 5 – 2 15 (b) 3 – 3 4 of 3 16 – 4 8 (c) 2 3 4 of 7 8 + 1 3 4 + 2 7 (d) 5 6 + 2 3 of 4 9 – 5 12 – 7 9 (e) 2 5 of 2 3 + 1 6 – 4 9 ÷ 27 4 (f) 4 9 ÷ 4 5 × 4 25 – 7 10 + 1 2 (g) 2 5 + 5 6 ÷ 2 3 of 6 10 – 3 4 (h) 1 + 2 2 3 of 4 5 + 1 2 – 1 3 (i) 1 – 2 3 ÷ 2 3 + 2 3 – 1 4 × 1 2 + 1 2 (j) 31 5 + 2 3 7 ÷ 3 5 8 – 15 8 × 2 5 – 2 4 5 ANSWERS 1. (a) 13 14 (b) 119 24 (c) 4 1 36 (d) 9 59 300 2. (a) 1 9 (b) 17 30 (c) 31 54 (d) 179 210 3. (a) 17 24 (b) – 11 36 (c) 20 27 (d) 1 7 12 (e) 59 90 (f) 3 7 12 4. (a) 1 6 (b) 4 15 (c) 11 9 (d) 8137 800 5. (a) 13 1 2 (b) 31 2 (c) 25 86 (d) 11 2 6. (a) 1 25 (b) 81 500 (c) 3 4 (d) 15 9 (e) 31 88 (f) 53 84 7. (a) 61 72 (b) – 1571 2205 (c) – 13 25 (d) 1 10 (e) 1 1 24 (f) 23 90 (g) 7244 2100 (h) 7 63 128 (i) 1242 375 (j) 3 45 112 8. (a) – 2 15 (b) 3 15 64 (c) 7113 224 (d) – 53 72 (e) 28 1215 (f) 6109 162 (g) – 12 1 3 (h) 3 26 45 (i) 8 33 (j) – 257 70
96 The Leading Mathematics - 7 5.3 Word Problems on Fractions At the end of this topic, the students will be able to: ¾ solve the word problems related to fraction. Learning Objectives To solve the word problems on fractions first express the given statement in expression of fraction then simplify using the operations of fractions. CLASSWORK EXAMPLES Example 1 Martha spent 5 9 of her money on shopping. What fraction of her money has she left? Solution: Here, Money with Martha = 1 Spent money on shopping = 5 9 ∴ Remaining money with her = 1 – 5 9 = 9 – 5 9 = 4 9 Thus, She has 4 9 of her money left. Example 2 3 5 of a group of children were girls. If there were 24 girls, how many children were there in the group? Solution: Here, Fractional part of girls = 3 5 Number of girls = 24 ∴ Total number of children = 24 ÷ 3 5 = 24 × 5 3 = 40. Alternatively 3 5 of Total children = 24 or, 3 5 × Total children = 24 or, Total children = 24 3 5 = 24 ÷ 3 5 = 24 × 5 3 = 40 Thus, there were 40 children in the group.
Arithmetic 97 Example 3 Sam had 120 teddy bears in his toy store. He sold 2 3 of them at Rs.120 each. How much money did he receive? Solution: Here, total number of teddy bears = 120 Selling of teddy bears = 2 3 of 120 = 2 3 × 120 = 80 He sold 80 teddy bears. Rate of selling teddy bears = Rs. 120 ∴ Selling amount of 80 teddy bears = Rs. 120 × 80 = Rs. 9600 Thus, he received Rs. 9600. EXERCISE 5.3 Your mastery depends on practice. Practice like you play. 1. (a) A drum full of rice weighs 40 1 6 kg. If the empty drum weighs 13 3 4 kg, find the weight of rice in the drum. (b) From an 11 m long rope, two pieces of lengths 13 5 m and 33 10 m are cut off. i. Find the length of cutting parts of rope. ii. What is the length of the remaining rope ? 2. (a) A basket contains three types of fruits having weight 19 1 3 kg in all. 81 9 kg of these are apples, 31 6 kg oranges and the rest pears. i. Find the total weight of apples and oranges. ii. What is the weight of the pears in the basket? (b) On one day a rickshaw puller earned Rs. 80. Out of his earnings, he spent Rs. 68 5 on tea and snacks, Rs. 51 2 on food and Rs. 22 5 on repairing of the rickshaw. i. Find his spending money. ii. How much did he save on that day? 3. (a) One liter of petrol costs Rs. 187 4 . What is the cost of 35 liters of petrol? (b) Find the cost of 3 2 5 meters of cloth at Rs. 36 3 4 per meter. 4. (a) A car is moving at an average speed of 200 3 km/hr. How much distance will it cover in 15 2 hours? (b) An airplane covers 1020 km in an hour. How much distance will it cover in 25 6 hours?
98 The Leading Mathematics - 7 5. (a) Find the area of a rectangular park which is 183 5 m long and 50 3 m broad. (b) Find the area of a square plot of land whose each side measures 17 2 meters. 6. (a) The cost of 3 1 2 meters of cloth is Rs. 573 4 . What is the cost of one meter of cloth? (b) A cord of length 143 2 m has been cut into 26 pieces of equal length. What is the length of each piece? 7. (a) The area of a room is 65 1 4 m2 . If its breadth is 5 7 16 meters, what is its length? (b) The product of two rational numbers is 9 3 5. If one of the rational number is 93 7 , find the other rational number. 8. (a) Rita had Rs. 300. She spent 1 3 of her money on notebooks and 1 4 of the remainder on other stationery items. i. How much money did she spend? ii. How much money is left with her? (b) Anoj earns Rs. 16000 per month. He spends 1 4 of his income on food; 3 10 of the remainder on house rent and 5 21 of the remainder on the education of children. i. How much money did he spend in total? ii. How much money is still left with him? ANSWERS 1. (a) 26 5 12 kg (b) 5 9 10 m, 5 1 10 m 2. (a) 11 5 18 kg, 8 1 18 kg (b) Rs. 431 2, Rs. 361 2 3. (a) Rs. 16361 4 (b) Rs.124 19 20 4. (a) 500 km (b) 4250 km 5. (a) 610 m2 (b) 721 4 m2 6. (a) Rs. 161 2 (b) 2 3 4 m 7. (a) 12 m (b) 1 1 55 8. (a) Rs. 175, Rs. 125 (b) Rs. 12609 11 21, Rs. 3390 10 21 Project Work 5.3 How much times do you spend on daily different activities (refreshment, taking meal, writing, reading, gaming, watching TV/mobile, etc.), on Saturday (6 am to 6 pm) ? Write them in fractions and present it in the classroom.
Arithmetic 99 5.4 Review on Decimals At the end of this topic, the students will be able to: introduce the decimal numbers. identify the types of decimal numbers. Learning Objectives Fractions with numbers such as 10 or 100 or 1000 or the like were given special attention some 1200 years ago. The Arabs are known to have used such them from those days. The modern dot (.) notation is accepted to be of Chinese origin. The westerners have started to use them nearly 800 years ago. Fractions like; 1 10, 2 10, 3 10, ....... are denoted by.1, .2, .3, ... . Here the dot before the numerator digit is called the decimal point; and the numbers .1, .2, .3, … are called decimals. These decimals are read as “ point 1 ”, “ point 2 ”, “ point 3 ” and so on. Fractions with 10, 100, 1000, … may be proper or improper. In all such cases, there is a special mechanical rule of writing them as decimals, i.e. putting a dot or point before or in between the digits of the numerator. It is as follows: Start from the unit digit of the numerator. Put the decimal point after counting the same number of digits as the number of zeros in the denominator. Thus, we have Fraction notation Decimal notation Read as 1 100 0.01 point zero one ... ... ... 11 100 0.11 point one one ... ... ... 1 1000 0.001 point zero zero one 12 1000 0.012 point zero one two ... … …
100 The Leading Mathematics - 7 Fraction notation Decimal notation Read as 1 10000 0.0001 point zero zero zero one ... … … 1234 10000 0.1234, point one two three four ... … … 1234 10 123.4 one twenty three point four 1234 1000 1.234 one point two three four. In such cases, the number before the decimal point represents the whole number or integral part and the number after decimal point, pure decimal. In particular, Improper Fraction Mixed Decimal Number Integral Part Pure Decimal Part Proper Fraction 1234 10 123.4 123 .4 4 10 = 2 5 1234 100 12.34 12 .34 34 100 = 17 50 1234 1000 1.234 1 .234 234 1000 = 117 500 An important point to be noted is that we can add as many zeroes as we like after the last digit following the decimal point. For instances, .4 = 4 10 = 40 100 = .40 = 400 1000 = .400 = .4000 and so on. Sometimes, we call decimals having the same number of decimal places as like decimals. For examples, the decimals 0.125 and 1.123 are like decimals. In the same way, decimals having different number of decimal places are called unlike decimals. For examples, the decimals 0.210 and 0.215 are unlike decimals. Unlike decimals can be converted into like decimals by just adding necessary number of zeros at the end of decimals having lesser number of decimal places. For instance, adding 1 zero at the end of 0.21 we can change the decimals 0.21 and 0.215 into like decimals. Comparison, addition and subtraction of decimals become easily understandable when converted into like decimals.