Acme Mathematics 7 513. Relation between LCM and HCF of Two NumbersConsider the following example: Let 4 and 6 be two numbers. HCF of 4 and 6 = 2 and LCM of 4 and 6 = 12 Now, HCF × LCM = 2 × 12 = 24 First number × Second number = 4 × 6 = 24 Hence, HCF × LCM = First number × Second number.Solved ExampleExample 1 The LCM of two numbers is 72 and their HCF is 6. If one of the numbers is 18, what is the next number? Solution: Here, LCM = 72 and HCF = 6 First number = 18 Using: LCM × HCF = First number × Second number or, 72 × 6 = 18 × Second number or, Second number = 72 × 618 = 24 ∴ The second number is 24.Example 2 The product of two numbers is 58800 and their LCM is 840. Calculate their HCF.Solution: Here, LCM = 840Product of two numbers = 58800Using, LCM × HCF = Product of two numbers. (1st number × 2nd number)or, 840 × HCF = 58800or, HCF = 58800840 = 70.Hence, HCF of two number's is 70.Example 3 Calculate the LCM of 50 and 125 by using their HCF.Solution: Here, Two numbers are 50 and 125.Now, 50 = 2 × 5 × 5or, 125 = 5 × 5 × 5CF = 5 × 5HCF = 25
52 Acme Mathematics 7Using, LCM × HCF = First number × Second number.or, LCM × 25 = 50 × 125or, LCM = 50 × 12525or, LCM = 250Hence, LCM of 50 and 125 is 250.Exercise 2.61. For each of the following pair of numbers, show that the product of their HCF and LCM is equal to their product.(a) 24, 36 (b) 15, 35 (c) 30, 45 (d) 10, 75(e) 40, 60 (f) 72, 108 (g) 100, 120 (h) 120, 160 (i) 234, 572 (j) 119, 223 (k) 1152, 1664 (l) 525, 945 2. The product of two numbers is 20520 and their LCM is 684. Find their HCF.3. The product of two numbers is 45000 and their LCM is 300. Find their HCF.4. The product of two numbers is 1081143 and their LCM is 8253. Find their HCF.5. The product of two numbers is 1080. If their HCF is 12, find their LCM. 6. The product of two numbers is 45000. If their HCF is 150, find their LCM. 7. The product of two numbers is 22500. If their HCF is 25, find their LCM.8. The LCM and HCF of two numbers are 112 and 4 respectively. If one of the numbers is 28, what is the next number? 9. The HCF and LCM of two numbers are 16 and 96 respectively. If one of them is 32, find the other number. 10. The HCF and LCM of two numbers are 25 and 900 respectively. If one of them is 100, find the other number. 11. The HCF and LCM of two numbers are 4 and 2920 respectively. If one of them is 40, find the other number. 12. The LCM and HCF of two numbers are 240 and 16 respectively. If one number is 48, find the other number. 13. Find the LCM of 4760 and 119 using their HCF. 14. Find the LCM of 100 and 225 using their HCF.15. Find the LCM of 900 and 50 using their HCF.
Acme Mathematics 7 532.2 Integers1. Compare and put the correct sign (>, =, <) in each box.(a) 5 – 4 (b) – 6 – 6 (c) – 4 – 1 (d) – 5 – 8 (e) – 10 – 10 (f) 0 1 2. Fill in the box with correct digit:(a) 2 + = 0 (b) 1 – 4 = (c) + 7 = – 4 (d) + = 10(e) 9 – 16 = (f) + 9 = 03. Add the following using number line:(a) 2 + 5 (b) – 5 – 2 (c) 10 + (– 5)(d) 7 – 10 (e) 6 – 11 (f) 20 – 54. Write the set of the following numbers: (a) Natural numbers (b) Whole numbers (c) Negative integers (d) Positive integers 5. Complete the given number line:– 10 – 5 0 + 56. Write 'true' or 'false' for the following: (a) 0 is positive integer. (b) (– 6) is greater than (– 5). (c) 0 is greater than (– 4). (d) Numbers left to 0 are negative numbers. 7. Put the sign >, = or < . (a) (+ 7) (– 2) (b) (+ 3) (+ 6) (c) (– 3) (– 1)(d) (– 5) 0 (e) (+ 5) (+ 5) (f) (– 10) (– 10) 8. Write the integers between –3 to + 3.Warm Up Test
54 Acme Mathematics 7A RevisionYou have already studied integers in class six. Now in this class, we study more about integers. The set of natural numbers including zero and their negatives are called integers. It is denoted by Z or I. So, Z = {......., – 3, – 2, – 1, 0, 1, 2, 3, .......} The set of integers {–1, –2, –3, ...} are called negative integers and denoted by 'Z –'. So, Z – = {......–4, –3, –2, –1} The set of integers {1, 2, 3, 4, .......} are called positive integers and denoted by 'Z +'. So, Z + = {1, 2, 3, 4, .......} ‘0’ is called zero integer. It is neither a positive integer nor a negative integer. In the number line 0 is the point of reference.1. Comparison of integersWe know the following: 0 < 1 < 2 < 3 < 4 Similarly – 4 < – 3 < – 2 < – 1 < 0 The integer right to any integer is greater than that integer. i.e. 4 > 3 because 4 is one unit right to 3. (– 3) > (– 4) because (–3) is one unit right to (– 4) Similarly; the integer left to any integer is less than that integer. i.e. 3 < 4, because 3 is one unit left to 4 and (– 4) < (– 3) because (– 4) is one unit left to (– 3). Rule for comparing the integersIf 'a' and 'b' are any two integers, then only three possibilities are there; they are: (i) a > b (ii) a < b (iii) a = b a > b, if 'a' is right to 'b', a < b, if 'a' is left to 'b', a = b, if 'a' and 'b' represent the same point. These laws are called the law of trichotomy.–4 –3 –2 –1 0 1 2 3 4 50Left hand side Right hand side–4 –3 –2 –1 1 2 3 4 5
Acme Mathematics 7 552. Opposite of an integer(–5) indicates five units left from the origin '0' and (+5) indicates five units right from the origin 0. (–5) is an image of (+5). (–5) is opposite of (+5) and vice versa.If 'a' is any integer, then its opposite integer is '–a' and vice versa.3. Absolute value of the integerIn the number line, suppose A is the position of Ram's house and B is the position of Hari's house, 0 is the position of school. Ram's house is 4 km right from the school and Hari's house is 3 km left from the school. In the number line, Ram's house is (+ 4) km from 0 and Hari's house is (–3) km from 0 and their sum is (+4) km + (–3) km = 1 km which is the distance between Ram's house and Hari's house, which is not true. The distance between Ram's house and Hari's house is 7 km. Therefore (– 3) km means not negative km, it means that only direction is opposite, here ‘(–)’ve sign indicates the direction of 3 km or Hari's house which is on the left from the school. In such case, we take integers without sign which is called the absolute value of an integer. i.e. |+3| = 3 or, |–3| = 3, Therefore |+3| and |–3| indicates the absolute value of (+3) and (–3) respectively. 0 is neither positive nor negative. The absolute value of 0 is 0 i.e. |0| = 0.Classwork1. Draw a number line and answer the following questions: (a) How many integers are there between (– 5) to (+ 5) ?(b) Where do the integers smaller than zero lie?(c) Where do the integers greater than zero lie? 2. Write down ‘True’ or ‘False’ for the following statements: (a) Zero is not an integer. (b) 4 is an integer. (c) – 7 is a negative integer. (d) (– 5) × 2 is an integer. (e) The absolute value of (– 5) is 5. (f) The absolute value of + 4 is (– 4). 3. Compare the following using the sign >, =, <: (a) 5 3 (b) 7 – 7 (c) – 3 – 13 (d) – 19 – 19 (e) | + 15| | – 15| (f) | + 2 | | – 13| 4. Write the opposite of: (a) – 5 (b) 7 (c) – 8 (d) 9 (e) 100 0Origin Left Right–5 –4 –3 –2 –1 +1 +2 +3 +4 +5–3 –2 –1 0 +1 +2 +3 +4 +5B A–5 –47 km
56 Acme Mathematics 7Exercise 2.71. Write down the integers which lie 5 units right to the given integer: (a) 3 (b) – 7 (c) 5 (d) – 10 (e) 15 (f) – 182. Give the negative sense to each of the following: (a) Temperature falls by 7 °C. (b) Making a profit of Rs 700. (c) Going 10 km to the South. (d) 100 metres above the ground 3. Write down the absolute value of each of the integers: (a) + 9 (b) 0 (c) – 10 (d) + 7 (e) ± 8 4. Arrange the following integers in (i) the ascending order (ii) the descending order. (a) – 2, 9, – 7, – 8, 10, 15 (b) – 7, + 12, – 5, – 2, + 4, + 6, – 1 5. Find the distance between A and B in a number line:6. Guess the following temperature:(a) (b) (c)7. Study the given temperature and say which is more cold:(a) 3°C and 5°C (b) 0°C and 3°C(c) –10°C and 5°C (d) – 2°C and 0°C8. Guess the temperature in the following situation :(a) (b) (c)–10010 30 50 70 9020 40 60 80– 7 km 0A B+ 5 km
Acme Mathematics 7 57B Addition and Subtraction of Integer1. Addition of an IntegerConsider the following examples:Solved ExampleExample 1 Find the sum : (+ 3) + (+ 4) Solution: Here, (+ 3) + (+ 4) = (+7) Example 2 Find the sum: (– 4) + (– 3) Solution: (–4) + (–3) = (–7) To add integers of the same sign, we add their absolute values and put the common sign on to the sum. e.g. (+ a) + (+ b) = + (a + b) (– a) + (– b) = – (a + b) Example 3 Find the sum: + 7 + – 3 Solution: + 7 + – 3 = + 4 To add integers of the different signs, we find the difference between the absolute greater value and the absolute smaller value and put the sign of greater absolute value on to the difference.Laws of addition of integera. Closure property : Let +2 and –3 are integer, then +2 + – 3 = –1 where –1 is also integer. Therefore, on adding two integers, we always get an integer. If a and b are two integers then a + b is also an integer.b. Associative property :Let, +2, –3 and +4 are three integers, then, +2 + (–3 + +4) = (+2 + –3) + +4 Now, LHS = +2 + (–3 + +4) and RHS = (+2 + –3) + +4 = +2 + +1 = –1 + +4 = +3 = +3Here LHS = RHS.–2 –1 0 1 2 3 4 5 6 7 8+3 +4+7–3–3 –2 –1 0 1 2– 4 – 3–8 –7 –6 –5 –4– 70 3 4 6 7+ 4–2 –1 1 2 5 8– 3+ 7
58 Acme Mathematics 7Therefore, +2 + (–3 + +4) = (+2 + –3) + +4If a, b and c are three integers then,a + (b + c) = (a + b) + cHere, on the left hand side, b and c are added first and then the sum b + c is added to a. Whereas on the right hand side, a and b are added first and then the sum a + b is added to c. This property shows that the order of addition does not matter in integers, that is, whether you add a and b first or b and c first, the sum will always remain the same. c. Commutative property :Let +2 and –3 are two integers, then+2 + –3 = – 1–3 + +2 = – 1Hence, +2 + –3 = –3 + +2If a and b are two integers then, a + b = b + aHere on left hand side, a is added to b whereas on right hand side, b is added to aand the two sides are equal. Thus changing the order of integers in addition will not change the result. d. Additive identity property : Additive identity of integers is a number which when added to any integer, does not change the value of that integer. Therefore, additive identity of integers is '0' because for integer 'a'.a + 0 = 0 + a = aFor example:Let +2 is a integer, then +2 + 0 = +20 + +2 = +2Hence, +2 + 0 = 0 + +2 = +2e. Additive Inverse property : Additive inverse of an integer 'a' is ' – a'. Thus,a + ( – a) = (– a) + a = 0On adding any integer with its additive inverse, we get the additive identity of integers. In integers, the additive inverse of 0 is 0 itself whereas for any other integer, its additive inverse is its negative. For example, the additive inverse of 2 is – 2 and the additive inverse of – 10 is – (–10) or 10.2. Subtraction of IntegersConsider the following examples:
Acme Mathematics 7 59Solved ExampleExample 1 Find the difference: + 4 – + 3 Solution: + 4 – + 3 = + 4 + – 3 = + 1 Example 2 Find the difference: + 4 – – 3 Solution: + 4 – – 3 = + 4 + +3 = + 7 Hence, to subtract integers, we change the sign of the integer to be subtracted and do same as addition.Classwork1. Write true or false for the following statements:(a) Every positive integer is greater than negative integer.(b) Zero is less than every positive integers.(c) Zero is greater than every negative integers.(d) The quotient, when –12 is divided by –4 is positive.(e) The sum of two negative integers is a negative integers.(f) The product of two integers both positive or both negative is a positive integer.(g) Sum of any two integers is always greater than their difference.2. Simplify: (a) (+ 3) + (– 4) + (+ 5) (b) (– 60) + (+ 40) + (– 30)(c) (– 21) + (+ 32) – (– 20) (d) (– 122) – (– 234) – (+ 159)(e) –(+ 17) – (– 25) + (+ 3) + (– 14) (f) (– 20) + (+ 60) – (– 30) (g) (– 5) – (+ 2) + (+ 2) – (– 5) (h) 12 + (– 7) + (– 4) – (– 3) (i) 16 + (– 12) – (+ 5) – (– 2) (j) (– 8) – (– 5) + (– 3) – (+ 3) (k) – 15 + (– 13) – (– 7) – (+ 4) (l) 22 – (+ 12) – (– 4) + (– 6) Exercise 2.81. Demonstrate the following operations in a number line: (a) (+5) + (+ 3) (b) (+ 4) + (– 2) (c) (+ 7) – (– 5) (d) (+ 10) – (+ 5) (e) (– 4) – (– 4) (f) (+ 4) + (– 4) 4+ 4–1 0 1 5 6+ 13– 3–2 2–2 –1 0 1 2 3 4 5 6 7 8+3 +4+7
60 Acme Mathematics 72. Using number line find the sum of following:(a) (+3) + (+5) (b) (+7) + (+2) (c) (–3) + (–4) (d) (–8) + (+4)3. Using number line find the difference of following:(a) (–6) – (–3) (b) (+8) – (–3) (c) (–7) – (+2) (d) (+4) – (+5)4. Which in greater? (a) (– 8) + (– 8) or (– 8) – (– 8) (b) (+ 3) – (+ 3) or (+ 3) + (+ 3) (c) (– 9) – (– 2) or (+ 2) – (+ 2) (d) (+ 10) – (– 8) or (– 10) + (– 8) 5. Use commutative law and calculate the sum.(a) (+5) and (+8) (b) (+10) and (–3)(c) (–9) and (–4) (d) (–6) and (+2)6. Use associative law and calculate the sum.(a) (+4), (+7) and (+2) (b) (+10), (+3) and (–4)(c) (+4), (–5) and (0) (d) (–3), (–6) and (+9)7. The sum of two integers is + 15 and one of them is – 7. What is the next integer? 8. The sum of two integers is + 20 and one of them is – 14. What is the next integer? 9. The sum of two integers is – 25 and one of them is + 20. What is the next integer? 10. The difference of the two integers is + 7 and the smaller is – 4. What is the greater integer? 11. The difference of the two integers is + 15 and the smaller is – 12. What is the greater integer?12. Write a pair of integers for which the following statements are true.(a) Sum is zero. (b) Difference is zero.(c) Sum is positive number. (d) Difference is positive number.(e) Sum is smaller than both the number.(f) Difference is greater than both the numbers.(g) Sum is a negative number.(h) Difference is a negative number.13. Fill in the blanks.(a) The additive identity of integers is ............(b) Sum of an integer and its additive inverse is always a ................14. Taking the integers (+5), (– 4) and (+6), verify the associative law of addition.
Acme Mathematics 7 61C Multiplication and Division of the Integer1. MultiplicationLook at the following multiplications: (i) + 3 × + 2 From the number line, + 3 × + 2 = + 6 (ii) – 3 × + 2 From the number line, – 3 × + 2 = – 6(iii) + 2 × – 5 = – 10(iv) + 5 × – 2 = – 10 (v) – 3 × – 2 It is opposite of no. (ii)– 3 × – 2 = + 6From the above examples, we have the following sign rules:1. (+) × (+) = + 2. (+) × (–) = – 3. (–) × (–) = + [Negative of negative is positive]4. (–) × (+) = – Some properties of multiplication of integers 1. Closure property : For all integers 'a' and 'b', a × b is also an integere.g. 2 × 3 = 6, – 2 × 3 = – 6, 2 × (– 3) = – 6, (– 2) × (– 3) = 6 2. Commutative property : For all integer 'a' and 'b', a × b = b × a e.g. (– 2) × (– 3) = 6 and (– 3) × (– 2) = 6 ∴ (– 2) × (– 3) = (– 3) × (– 2) 0 3+ 3 + 3+ 6–2 –1 1 2 4 5 6 7 8 9–3 –2 –1 0 1 2–3 –3–6–6–8 –7 –5 –4–9 –8 –7 –6 –5 –4 –3 –2–2 –2 –2 –2 –2–10 –1–100 1 2 3–9 –8 –7 –6 –5 –4 –3 –2–5 –5–10 –1–100 1 2 3–2 –1 0 1 2 3 4 5 6 7363–7 –6 –5 –4 –3
62 Acme Mathematics 73. Multiplicative property : For all integers 'a', 1 × a = a × 1 = a e.g. 1 × (– 3) = – 3, (– 3) × 1 = – 3 ∴ 1 × (– 3) = (– 3) × 1 4. Associative property : For all integers 'a', 'b' and 'c', a × (b × c) = (a × b) × c e.g. (– 3) × [(– 4) × (– 5)] and [(– 3) × (– 4)] × (– 5) = (– 3) × [+20] = [+12] × (– 5) = – 60 = – 60∴ (– 3) × [(– 4) × (– 5)] = [(– 3) × (– 4)] × (– 5) 5. Distributive property : For all integers 'a', 'b' and 'c', a × (b + c) = a × b + a × c e.g. +3 × (+4 ++ 5) or, +3 × +4 + +3 × +5or, = +3 × +9 = +27 = +12 + +15∴ +3 × (+4 + +5) = +3 × +4 + +3 × +5 = +276. Multiplicative property of zero : If 'a' is a integer then a × 0 = 0 × a = 02. DivisionBecause the division operation is inverse operation to the multiplication, the rules of sign in division are same as in multiplication.Look at the following examples: (i) + 6 ÷ + 3 From the number line, + 6 ÷ + 3 = + 2 6 is divided into 2 parts(ii) – 6 ÷ + 3 From the number line, – 6 ÷ + 3 = – 2(iii) + 10 ÷ + 2 = 5 (iv) – 10 ÷ + 5 = – 20 3 3 3+ 6–2 –1 1 2 4 5 6 7 8 9–3 –2 –1 0 1 23 3–6–6–8 –7 –5 –41 2 3 4 5 6 7 85 50 9+1010 11 12 13–9 –8 –7 –6 –5 –4 –3 –22 2 2 2 2–10 –1–100 1 2 3
Acme Mathematics 7 63(v) – 10 ÷ – 5 It is opposite of no (iv)So, – 10 ÷ – 5 = 2Note: a. Since (+ 5) × (+ 3) = + 15 than (+ 15) ÷ (+ 5) = + 3 and (+ 15) ÷ (+ 3) = + 5b. Since (+ 5) × (– 3) = – 15 than (– 15) ÷ (+ 5) = – 3 and (– 15) ÷ (– 3) = + 5c. Since (– 5) × (– 3) = +15 than (+15) ÷ (– 5) = – 3 and (+15) ÷ (– 3) = – 5Classwork1. Find the value of: (a) – 10 × + 7 × – 6 (b) + 10 × – 7 × + 6 (c) – 18 × – 3 × + 7(d) – 5 × – 6 × – 7 (e) + 8 × + 9 × + 10 (f) (+ 6) × (+ 7) × (– 4) 2. Write the correct number in the blanks.(a) (+6) × ........... = +24 (b) (–8) × (+7) = ............ (c) (–5) × ........... = 0 (d) ........... ÷ (+2) = –3(e) (–12) ÷ ............ = +43. Complete the table:(a) × – 4 – 3 – 2 – 1 0 (b) × + 1 + 2 + 3 + 4 + 5 + 6– 3 – 1– 2 0– 1 + 10 + 2Exercise 2.91. Using number line find the product of: (a) + 3 × + 4 (b) + 3 × – 4 (c) – 5 × + 4 (d) – 5 × – 42. Find the product of: (a) – 10 × + 7 (b) + 7 × + 8 (c) – 9 × – 9 (d) + 15 × – 10 3. Using associative property of multiplication, multiply the followings:(a) (+4), (+5) and (–6) (b) (+12), (–8) and (–2)(c) (–10), (–7) and (–6)4. Using distributive property of multiplication simplify the followings:(a) (–4) × [(+20) – (–4)] (b) [(+7) + (–5)] × (–3)(c) (+12) × [(+8) + (–3)]
64 Acme Mathematics 75. Simplify: (a) (+ 5) × [(– 3) + (+ 6)] (b) (+ 8) × [(+ 9) + (+10)] (c) –10 × [(+7) + (–6)] (d) (– 5) × [(– 6) + (– 7)](e) [(–15) + (+12)] × (+ 6) (f) (– 4)×[(– 3) + (+15)] 6. Find the quotient of: (a) (+12) ÷ (+ 3) (b) (–18) ÷ (+ 6) (c) (–136) ÷ (–17) (d) (+ 84) ÷ (–12) (e) (– 96) ÷ (– 24) (f) (– 125) ÷ (– 25)7. Simplify :(a) [(+4) × (+6)] ÷ (–3) (b) [(–12) × (–3)] ÷ [(–9) × (–1)](c) [(+10) × (–4)] ÷ [(–2) × (+5)]8. The product of two integers is 96, one of them is –12, find the other. 9. The product of two integers is – 84. One of them is 4, find the other integer. 10. 5 times an integer is – 50, what is the integer? 11. 7 times an integer is + 77, what is the integer? 12. Find the integer which, when multiplied by – 8 gives the product 120. 13. Find the integer which, when multiplied by – 5 gives the product 75.Project WorkOn the number line do the following operations and paste on your 'Math corner'.(a) (+5) + (–3) (b) (–8) – (+4)(c) (–3) × (+4) (d) (–10) ÷ (+2)
Acme Mathematics 7 65D Simplification of Integer1. Order of OperationsWhile simplifying we follow the following order of operations: 1. B → Brackets – 2. O → Of order of brackets : ( ), { }, [ ] 3. D → Division 4. M → Multiplication5. A → Addition 6. S → Subtraction Thus, we may remember ‘BODMAS’ as a rule.2. SimplificationLook at the examples carefully and get the idea about the simplification.Solved ExampleExample 1 Simplify: 7 × 3 – 6 ÷ 3 + 2 Solution: 7 × 3 – 6 ÷ 3 + 2 = 7 × 3 – 2 + 2 [D → 6 is divided by 3] = 21 – 2 + 2 [M → 7 is multiplied by 3] = 23 – 2 [A → 21 is added to 2] = 21 [S → 2 is subtracted from 23] Example 2 Simplify: 7 + 6 ÷ 3 – (2 + 1) Solution: 7 + 6 ÷ 3 – (2 + 1) = 7 + 6 ÷ 3 – 3 [B → (2 + 1) = 3 is performed] = 7 + 2 – 3 [D → 6 is divided by 3] = 9 – 3 [A → 7 is added to 2] = 6 [S → 3 is subtracted from 9] Example 3 When the quotient of 48 and 8 is multiplied by the difference of 10 and 8, find the product. Solution: The mathematical form of above problem is (48 ÷ 8) × (10 – 8) Now, (48 ÷ 8) × (10 – 8) = 6 × 2 = 12 Thus, the product is 12.Classwork1. Simplify: (a) 5 + (– 10) – (– 3) (b) (– 3) + 7 – 12 (c) 12 × 4 ÷ 2 (d) 54 ÷ (12 × 3 ÷ 2) (e) (12 × 2) ÷ 6 + (17 – 2) (f) 9 ÷ (7 – 4) – 50
66 Acme Mathematics 72. Simplify: (a) 45 ÷ (12 × 3 ÷ 2 – 15) (b) (12 × 2) ÷ 6 + (17 – 2) ÷ 3 (c) 2 {24 + 9 ÷ (7 – 4)} – 50 (d) 3 [120 – {4 – 5 (7 – 3)}] (e) 18 ÷ {7 + 2 (7 – 6)} + 2 (f) 13 – {12 – (5 + 2 – 4 ÷ 2)} Exercise 2.101. Simplify: (a) 6 + [2 – 7 {13 – 14 ÷ (4 – 6)}] (b) [–11 ÷ {12 – (13 – 12)}] (c) 20 ÷ 2 [5 + 12 – {(4 × 2) + 4}] (d) 7 + [12 – {8 + 3 – 15}] (e) 8 – {7 + 4 – (3 + 2) – 1 + 6} (f) 50 + {18 ÷ (7 – 5) – 9 – 21} – 2 (g) 16 ÷ [{6 + (17 – 19) + 4}] + 4 (h) 35 – [6 – 6 {4 – 3 ÷ (4 – 3)}] 2. Find the product of 4 times the difference of 20 and 15 and 4 times the sum of 12 and 3. 3. One fifth of 50 is added to –6 and then added to the difference of 23 and 3. Find the result. 4. If 9 times the difference of 25 and 16 is added to 25 and then multiplied by the sum of 6 and 12, find the result. 5. Find the quotient of 16 and 8 and subtract it from the difference of 5 and 2. 6. What is the result when product of 12 and 7 is divided by 21 and – 7 is added to it? 7. One fourth of the sum of 48 and 20 is multiplied by the one third of the difference of 32 and 5. Find the result.8. A slug climbs 2 meters up a slipper piller in 1 minute and slips down by 1 meter over the next 1 minute. How many minutes will the slug take to reach the top of the 15 meters high piller. 9. I use bank account to pay the bill from mobile banking. On the month Paush –20, I pay following bills from by account.(a) Electricity bill Rs. 1175. (b) Water bill Rs. 200.(c) Mobile topup Rs. 5000. (d) Cement Bill Rs. 15713.If 10th of Paush Rs. 75000 salary deposited on my account and Rs. 20718 was already on the account. How much money are left on my account. Project WorkCollect any 10 days temperature that are used in our daily life and present classroom.Note: We do first operation under the –––– (vinculum)
Acme Mathematics 7 672.3 Rational Number1. Fill in the blanks: (a) 19 is a ................ number. (b) –10 is a ................. number. (c) 3 is a ................. number. (d) 0 is a ................. number.2. Fill 12 on the number line. 0 1 2 3 43. Identify the rational numbers : 13, 56, – 23, 34. Write True or False for the following. (a) 2 is not a number. (b) 3 is a zero integer. (c) – 4 is a Natural number. (d) 35 is a rational number. (e) All whole numbers are rational numbers.(f) There is no rational number which equals its reciprocal.(g) There is no rational number whose absolute value is negative.(h) Every rational number has a reciprocal.5. If a and b are rational numbers then fill in the blanks:(a) a + b is a ............. number.(b) a – b is a ............. number.(c) a × b is a ............. number.(d) a + b = ............................Warm Up Test
68 Acme Mathematics 7A Rational NumbersConsider the following set of numbers: The set of natural numbers, N = {1, 2, 3, 4, 5 , …} The set of whole numbers, W = {0, 1, 2, 3, 4, 5 , …} The set of integers, Z = {..., –3, –2, –1, 0, 1, 2, 3, 4, 5 , .........}Now, consider the following: Choose any two numbers from the number line. Let they are 5 and 3. Then,5 + 3 = 8, 5 – 3 = 2, 3 – 5 = – 2, 3 × 5 = 15.Here, the sum of 5 and 3, difference of 5 and 3 and product of 5 and 3 are also the integers. Now, divide 5 by 3 then we have, 5 ÷ 3 = 53or, divide 3 by 5, then we have 3 ÷ 5 = 35Are these two numbers 53 and 35 integers? Again, Suppose, we have an equation 5x – 7 = 0, and then no integer value of x can satisfy the equation. It is only the fraction x = 75 which is the solution to this equation. Similarly, the equation 5x + 7 = 0 has no solution in integers. Therefore, we need to extend our number system of integers and fractions to include numbers in which equations like this may have solutions. Let us extend the system of fractions and consider the numbers of the form pq, where p and q are any integers and q is always non zero. Thus, we consider all integers, all fractions and all ‘fraction like numbers’ (i.e. – 43 , 5– 3 etc) are called the system of rational numbers. It is denoted by Q. But the numbers like 20, – 30 are not rational numbers. Every integer (Z) is a rational number. Every fraction is a rational number. Every rational number can be represented on a number line.–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Acme Mathematics 7 69B. Properties of Rational Numbers1. Law of addition (i) Closure property If 'a' and 'b' are any two rational numbers, then a + b is also a rational number.For example: Let 12 and 23 are two rational numbers. Then,12 + 23 = 3 + 46 = 76, which is also rational number.(ii) Commutative propertyIf 'a' and 'b' are any two rational numbers, then a + b = b + aFor example: Let 12 and 23 are two rational numbers. Then,12 + 23 = 3 + 46 = 76 23 + 12 = 4 + 36 = 76 Hence, 12 + 23 = 23 + 12 (iii) Existence of additive Identity property If 'a' and '0' are any two rational numbers, then a + 0 = 0 + a = a.For example: Let 12 and 0 are two rational numbers. Then,12 + 0 = 12 0 + 12 = 12 Hence, 12 + 0 = 0 + 12 = 12
70 Acme Mathematics 7(iv) Existence of additive Inverse property Let 'a' is a rational number and (–a) is its additive inverse.Then, (a) + (– a) = (– a) + (a) = 0For example : Let 12 is a rational number and – 12 is additive inverse. Then,12 + – 12 = 12 – 12 = 0– 12 + 12 = – 12 + 12 = 0Hence, 12 + – 12 = – 12 + 12 = 02. Law of Multiplication (i) Closure propertyIf 'a' and 'b' are any two rational numbers then a × b ∈ Real number. For example: Let 12 and 23 are two rational numbers.12 × 23 = 1 × 22 × 3 = 13 Here 13 is also a rational number.(ii) Commutative propertyIf 'a' and 'b' are any two rational numbers then, a × b = b × aFor example: Let 12 and 23 are two rational numbers.12 × 23 = 1 × 22 × 3 = 13 23 × 12 = 2 × 13 × 2 = 13 Hence, 12 × 23 = 23 × 12
Acme Mathematics 7 71(iii) Associative property If 'a', 'b' and 'c' are any three rational numbers then, a × ( b × c) = (a × b) × cFor example: Let 12 , 23 and 35 are three rational numbers.12 × 23 × 35 = 12 × 2 × 33 × 5 = 12 × 25 = 1 × 22 × 5 = 15 12 × 23 × 35 = 1 × 22 × 3 × 35 = 13 × 35 = 1 × 33 × 5 = 15 Hence, 12 × 23 × 35 = 12 × 23 × 35 (iv) Existence of Unit element propertyIf 'a' is a rational number then, a × 1 = 1 × a = a, where 1 is unit element.For example: Let 12 is a rational number.12 × 1 = 1 × 12 = 12 1 × 12 = 1 × 12 = 12 Hence, 12 × 1 = 1 × 12 (v) Existence of multiplicative inverse property If 'a' is a rational numbers then, a ×1a = 1a × a = 1, where 1a is multiplicative inverse of 'a' and vice versa.For example : Let 12 is a rational number.12 × 112 = 12 × 21 = 1 × 22 × 1 = 1112 × 12 = 21 × 12 = 2 × 11 × 2 = 1Hence, 12 × 112 = 112 × 12 = 1
72 Acme Mathematics 73. Law of Distribution (i) If 'a', 'b' and 'c' are any three rational numbers then, a × (b + c) = a × b + a × c For example : Let 12 , 23 and 35 are three rational numbers.12 × 23 + 35 = 12 × 10 + 93 × 5= 12 × 1915 = 1 × 192 × 15 = 193012 × 23 + 12 × 35 = 1 × 22 × 3 = 1 × 32 × 5= 26 + 310 = 10 + 930 = 1930Hence, 12 × 23 + 35 = 12 × 23 + 12 × 35(ii) If 'a', 'b' and 'c' are any three rational numbers then, (b + c) × a = b × a + c × aFor example: Let 12 , 23 and 35 are three rational numbers.23 + 35 × 12 = 10 + 915 × 12 = 1915 × 12 = 193023 × 12 + 35 × 12 = 2 × 13 × 2 + 3 × 15 × 2 = 13 + 310 = 10 + 930 = 1930Hence, 23 + 35 × 12 = 23 × 12 + 35 × 12 Solved ExamplesExample 1 Find any two rational numbers between 14 and 12 . Solution: Let x and y be any two rational numbers between 14 and 12 . Then, x = 14 + 122 = 1 + 242 = 34 × 12 = 38 y = 38 + 122 = 3 + 482 = 78 × 12 = 716 ∴38 and 716 are two rational numbers between 14 and 12 .
Acme Mathematics 7 73Classwork1. Write ‘T’ for true statements or ‘F’ for false ones:(a) Every natural number is rational number. (b) Every fraction is rational number. (c) 10is a rational number. (d) 16 is not a rational number. (e) Every rational number is fraction.(f) Even and odd numbers are rational numbers. 2. Choose the rational numbers with reason:(a) 227 (b) 14 (c) 5125 (d) 172 00(e) – 2116 (f) 8 (g)103 (h) 0.666...(i)1625 (j) – 144 (k)949 (l) ± 256 Exercise 2.111. Convert the following rational numbers into decimal numbers: (a) 32140 (b) 481200 (c) 1517 (d) 429(e) 0.570.11 (f) 137 (g) 227 (h) 1790(i) 19 (j) 203 (k) 949 (l) 2562. Identify which of the following numbers are rational:(a) 24 (b) 67 (c) 1 (d) 2 (e) 3 (f) 4 (g) 25 (h) 0.25(i) 1.333 (j) 2.1818 (k)12 (l)1649(m)57 (n) – 36 (o) 4 – 2 5 (p) 64 × 5 (q) 75 – 11 (r)28 × 317 × 5 (s)88 + 1233 – 8 (t)16 + 960 + 4
74 Acme Mathematics 73. Find any two rational numbers between:(a) 13 and 12 (b) 17 and 15 (c) 19 and 17 (d) 15 and 14(e) 19 and 14 (f) 14 and 12 (g) 16 and 15 (h) 17 and 144. Draw the number line and represent the following rational numbers.(a) 25 (b) 611 (c) 58 (d) 510(e) 219 (f) 14 (g) 15 (h) 245. Taking 3 rational numbers 23, 34 and 12, verify the addition properties of rational number: (a) Closure property (b) Commutative property(c) Associative property (d) Identity property6. Taking 3 rational numbers 23, 34 and 12, , verify the multiplication properties of rational number: (a) Closure property (b) Commutative property(c) Associative property (d) Identity property(e) Inverse propertyC. Decimal Representation of Rational Numbers We know that every rational number, when expressed in decimal form, is expressed either by terminating or non-terminating decimal form. For example,Fraction Numbers 1 22 53 103 445 80Decimal Numbers 0.5 0.4 0.3 0.75 0.5625Fraction Numbers 1 315 362 11Decimal Numbers 0.333…... 0.41666..…. 0.181818…...If the denominator contains 2 or 5 only as prime factors, then the fraction becomes a terminating decimal. In the above examples 1 2, 2 5, 3 10 , 3 4 and 45 80 are all terminating decimals.If after the decimal point, a digit or a group of digits repeat again and again, then it is called a non-terminating or recurring or repeating decimal. In the above examples 1 3, 15 36
Acme Mathematics 7 75and 2 11 are all recurring, non-terminating or repeating decimals. In the non-terminating or recurring decimal representation 1 3, we find that the digit 3 goes on repeating. Similarly, inthe number 2 11 , the digits 1 and 8 repeat in this order infinitely many times. While writingsuch decimals, we put a bar (–) or dot (•) over the repeating part.Thus, we write, 1 3 = 0.333…, = 0.3 or 0.3.Likewise, we write 15 36 = 0.41666 … = 0.416, 2 11 = 0.181818 …. = 0.18 etc. A decimal number can be expressed in the form of a rational number. For example (i) 0.2 = 2 10 = 1 5 (ii) 0.75 = 75 100 = 5 × 5 × 3 5 × 5 × 4 = 3 4(iii) 0.3 = 1 3 (iv) 0.18 = 2 11 How ?Solved ExamplesExample 1 Convert 0.3 into rational number. Solution : Let, x = 0.3 then, or, x = 0.33………….......(i) or, 10x = 3.3 ……......…. (ii) [Multiplying both sides by 10] From (ii) and (i) on subtraction10x = 3.3– x = 0.39x = 3.0 or, x = 3 9 or, x = 1 3 ∴ 0.3 = 1 3Example 2 Convert 0.18 into rational number.Solution: Let x = 0.18 then, or, x = 0.1818 ………… (i) or, 100x = 18.18 ………........(ii) [multiplying both sides by 100] Subtracting (i) from (ii), we have, 99x = 18
76 Acme Mathematics 7 or, x = 18 99or, x = 2 11∴ 0.18 = 2 11Alternative method: (i) 0.18 = 18 – 0 99 = 18 99 = 2 11(ii) 0.183 = 183– 0 999 = 183 999 = 61 333(iii) 2.34 = 234 – 2 99 = 232 99(iv) 6.327 = 6327 – 63 990 = 6264 990 = 3132495Note 6.327 is mixed recurring number and 3 is called anti-repetend.Classwork1. Without actual division, find which of the following numbers represent a terminating decimal: (a) 3 11 (b) 17 10 (c) 7 30 (d) 41 45 (e) 1 1252. Convert the following rational numbers into decimals:(a) 17 200 (b) 15 38 (c) 2 5 (d) 135 180 (e) 3367 13000Exercise 2.121. Express as rational numbers: (a) 0.4 (b) 0.17 (c) 1.53 (d) 0.318 (e) 0.45(f) 0.3 (g) 0.8 (h) 0.5 (i) 0.12 (j) 0.15(k) 0.24 (l) 1.23 (m) 2.35 (n) 5.42 (o) 6.432. Write true or false for the following statements: (a) pq has a terminating decimal, if q is a prime. (b) Every rational number has a decimal representation. (c) Every decimal number, having a finite number of terms in the decimal part, is a rational number.
Acme Mathematics 7 773. Express each of the following decimals in the form of a rational number in lowest terms: (a) 0.34 (b) 0.34444…. (c) 1.979797… (d) 0.212121…(e) 0.363636… (f) 0.123123123….(g) 1.23452345… (h) 0.132132132...4. Insert 3 rational numbers in between: (a) 1 4 and 1 7 (b) – 3 8 and – 23 (c) – 1 2 and 1 2(d) 5 7 and – 3 9 (e) 11 6 and 13 18Project WorkRepresent the decimal number on the chart and paste in your 'Math Corner'.D. Number ChartThe following chart show the number system and their relations.Rational numberFractional number IntegersPositive integers Negative integersNatural number Whole numberZero integer
78 Acme Mathematics 7EvaluationTime: 84 minutes Full Marks: 351. Ram, Shyam and Hari are three brothers and their ages are 8 years, 16 years and 25 years. (a) What is H.C.F. ? [1](b) Find the H.C.F. of their ages. [2](c) Find out the square number and cube number from their ages. [1]2. (a) Find the square number of 27. [1] (b) Find the cube root of 343. [1](d) Find the LCM of : 75, 120 and 150 [2]3. (a) What is the HCF of any two prime numbers? [1] (b) Find the least number which when divided by 6, 8 and 18 leaves 2 as a remainder. [2] (c) Find the sum of the cube of 5 and cube root of 3375. [1] 4. (a) What is the cube number of 9? [1](b) Find the square root of 1225 by using division method . [2](c) Prove that: 25 × 36 = 25×36 [1] (d) Express as rational numbers: 0.45 [2]5. (a) Find the square number of 11. [1] (b) Find the sum of cube root of 64 and square root of 64. [1] (c) Simplify: 108 + 3 2 [2](d) Simplify: (+3) × (–16) ÷ (–4) [1] 6. Radha's father bought a square field in Dhushatar having area 8836 sq. meter.(a) Find the length of the field. [2](b) If a cubical tank is made in a side of a field writes a formula to calculatevolume of that tank. [1](c) Find the L.C.M. of 24 and 32. [2]7. (a) Simplify: [(+3) × (+2)] ÷ [(–3) × (–1)] [2](b) Multiply (+5), (+6) and (–7) by associative property using both methods. [2](c) Find the square number of –8. [1](d) Express 0.34444 decimal in the form of a rational number in lowest term. [2]
Acme Mathematics 7 792.4 Fraction and Decimal1. Compare the fractions. Use > , = or <.(a) 14 ...... 34 (b) 24 ...... 36 (c) 47 ...... 492. Complete the following: 12 = ......4 = 3...... = 4......3. Convert to the lowest term: (a) 27108 (b) 84964. Add: (a) 59 + 29 + 19 (b) 12 + 25 + 345. Subtract: (a) 34 – 14 (b) 78 – 256. Simplify: (a) 29 + 16 – 23 (b) 316 – 114 – 1127. Find the product: (a) 3 × 12 (b) 16 × 14 (c) 23 × 458. Divide: (a) 8 ÷ 45 (b) 4 ÷ 113 (c) 25 ÷ 329. Convert into the decimal number: (a) 35 (b) 72 (c) 81010. Convert into the fraction: (a) 0.2 (b) 0.35 (c) 3.2811. Add: (a) 5.473 + 8.340 (b) 0.392 + 0.12 + 0.13812. Subtract: (a) (b)13. Multiply: (a) 0.13 × 2.3 (b) 0.299 × 0.0314. Divide: (a) 1.880 ÷ 0.8 (b) 38.48 ÷ 815. Round of the following: (a) 2.62 (ones place) (b) 12.592 (tens place)16. What fraction represents the following?(a) 6 hours of a day. (b) 20 minutes of an hour.(c) 110. (d) 59 31.76 – 12.87 9.00 – 7.12Warm Up Test
80 Acme Mathematics 7A Revision on FractionThe numbers of the type pq are called the fractions, where p and q are positive integers and q ≠ 0. In fraction, the number p is called the numerator and q is called the denominator. Some examples of fractions are: 12, 23, 34, 65, 73, 94, 112 , etc.1 Types of fractions(i) Proper fraction Fraction having numerator less than the denominator is called the proper fraction. e.g. 25, 49, 1011, etc. (ii) Improper fractionFraction having numerator greater than the denominator is called the improper fraction. e.g. 53, 95, 127 , etc. (iii) Mixed fractionFraction with the whole part (number) and a proper fraction is called the mixed fractions. e.g. 112, 379, 1513, etc. (iv) Like fractions Fractions having the same denominator are called like fractions. e.g. 25, 35, 15, 125 , etc. (v) Unlike fractions Fractions having the different denominator are called unlike fractions. e.g. 12, 25, 37, 1210, etc. (vi) Equivalent fractions Fractions are said to be equivalent fractions if their reduced form are same. e.g. 14 and 28, 37 and 1228 etc. Representation of fractions Consider the fractions: 23 and 12323 = 123 = 1 + 23 = +
Acme Mathematics 7 81Hence, 23 represents 2 parts out of 3 parts. 123 represents 1 whole and 2 parts out of 3 parts. Thus, fractions represent a number of parts of the whole.2. Addition and Subtraction of FractionsIn case of like fractions, we should perform addition and subtraction by adding or subtracting the numerator and taking the denominator common.Solved ExampleExample 1 Add: 213 + 313 + 413Solution: 213 + 313 + 413= 73 + 103 + 133= 7 + 10 + 133= 10Example 2 Subtract: 1017 – 317Solution: 1017 – 317= 10 – 317= 717In case of unlike fractions, first we have to change them into like fractions, with least common denominator. After this, we have to do the same process as addition or subtraction of like fractions.Example 3 Add: 214 + 318 + 5 112Solution: 214 + 318 + 5 112= 94 + 258 + 6112= 9 × 6 + 25 × 3 + 61 × 224 [LCM of 4, 8 and 12 is 24]= 54 + 75 + 12224= 25124 = 101124
82 Acme Mathematics 7Example 4 Subtract: 1217 – 3 314Solution: 1217 – 3 314= 857 – 4514= 85 × 2 – 4514 [LCM of 7 and 14 is 14]= 170 – 4514 = 12514 = 813143. Multiplication of fractions(a) To multiply a fraction by a whole number, we have to multiply numerator of a fraction by the whole number.Example 5 Multiply: 213 and 5Solution: 213 and 5 = 2 × 513 = 1013 ∴ 213 × 5 = 1013 (b) To multiply a fraction by another fraction, we have to multiply numerator by numerator and denominator by denominator separately.Example 6 Multiply: 615 and 75 Solution: 615 × 75 = 6 × 715 × 5 = 4275 ∴ 616 × 75 = 4275 4. Division of Fraction(a) Division is the inverse process of multiplication. To divide a fraction by a whole number, we have to multiply it by the reciprocal of the whole number or the reciprocal of the divisor.Example 7 Divide: 415 by 7Solution: 415 ÷ 7 = 415 × 17 [17 is reciprocal of 7]= 4 × 115 × 7 = 4105 ∴415 ÷ 7 = 4105 (b) To divide a fraction by another fraction, we have to multiply dividend by the reciprocal of the divisor.
Acme Mathematics 7 83Example 8 Divide: 719 by 85 Solution: 719 ÷ 85 = 719 × 58 [58 is reciprocal of 85 ]= 7 × 519 × 8 = 35152 ∴ 719 ÷ 85 = 35152 5. Additive inverseConsider the fraction 23. Now, subtract 23 from it, what is the result? – 23 or 2– 3 are additive inverse of 23 . Because 23 – 23 = 0. If pq is given fraction then its additive inverse is – pq or p– q 6. Multiplicative inverseMultiply 23 and 32 . What is the result? Here 23 is the multiplicative inverse of the fraction 32 and vice versa. Because 23 × 32 = 1 If pq is a fraction, its multiplicative inverse is qp and vice versa. Note: The number 0 is called the additive identity and 1 is called the multiplicative identity.Classwork1. Represent the following fractions with figure (use circle):(a) 23 (b) 225 (c) 56 (d) 456 (e) 19 2. Which is greater?(a) 23 or 45 (b) 1517 or 911 (c) 613 or 49 (d) 37 or 123 (e) 27 or 13 3. Change the following fractions into lowest term:(a) 35100 (b) 1812 (c) 10577 (d) 1640 (e) 1239004. Find the multiplicative inverse of:(a) 79 (b) 234 (c) 7 913 (d) – 1125 (e) 55. Find the value of:(a) 12 of Rs. 500 (b) 417 of Rs. 5100 (c) 620 of 2000 kg 719 is dividend85 is divisor
84 Acme Mathematics 7Exercise 2.131. Arrange the following in ascending and descending order:(a) 17, 27, 37, 47 (b) 72, 23, 56 (c) 26, 312, 43, 7242. Add the following:(a) 27 +47 + 57 (b) 125 + 235 + 345 (c) 32 + 54 + 78(d) 138 + 1516 + 2032 (e) 139 + 2 418 + 4 736 (f) 9 110 + 10 120 + 1 1503. Subtract the following:(a) 513 – 413 (b) 2 921 – 1 521 (c) 9100 – 17(d) 219 – 1 118 (e) 81027 – 9 730 (f) 20 119 – 2154. Multiply the following:(a) 610 × 2 (b) 2 717 × 10 (c) 20 9100 × 100(d) 34 × 37 × 45 (e) 2 820 × 378 × 147 (f) 12 × 34 × 56 × 785. Divide the following:(a) 29 ÷ 74 (b) 5282 ÷ 2648 (c) 1929 ÷ 5758(d) 129 ÷ 43 (e) 100200 ÷ 75125 (f) 2/93/76. Find the value of:(a) 1938 of 1000 litre (b) 47 of 1022 m (c) 118 of 1080 hours (d) 45 of Rs. 7500 (e) 34 of Rs. 12000 (f) 49 of Rs. 81817. Find the additive inverse of:(a) 27 (b) 36 (c) 2627 (d) 7 (e) – 9118. Simplify:(a) 12 + 23 – 34 (b) 212 + 412 × 114 (c) 57 + 37 ÷ 47 (d) 12 + 3 – 13 + 16 (e) 38 – 14 + 18 – 23 (f) 334 – 112 × 4 14 ÷ 325 + 12 (g) 3 712 ÷ 912 – 234 + 178 – 98 + 912 (h) 37 + 34 + 15 ÷ 15 – 12 + 23 of 127 (i) 1 + 1 + 11 + 31(j) 2 + 2 + 22 + 32 (k) 7 + 7 + 7 + 77 + 777
Acme Mathematics 7 85B Verbal Problems on FractionsStudy the following examples:Solved ExampleExample 1 Ram had Rs. 200. He spent 14 and gave 12 of Rs. 90 to his friend. Find, how much money is left with him ? Solution: Total money = Rs. 200 Money spent = 14 of Rs. 200 = 14 × Rs. 200 = Rs. 50Money given to his friend= 12 of Rs. 90= 12 × Rs. 90= Rs. 45 Now, Money left with Ram = Total money – (Rs. 50 + Rs. 45) = Rs. 200 – Rs. 95 = Rs. 105 ∴ Rs. 105 is left with Ram. Example 2 From the product of 512 and 4, subtract the sum of 313 and 213 .Solution: The product of 512 and 4 = 512 × 4 Sum of 313 and 213 = 313 + 213 Problem in mathematical form is: 512 × 4 – 313 + 213Now, 512 × 4 – 313 + 213= 112 × 4 – 103 + 73= 22 – 173= 66 – 173 = 493 = 1613 ∴ The required result is 1613 .Alternative MethodRs 200x x x x4x = 200x = 50Rs 90x x2x = 90x = 45
86 Acme Mathematics 7Example 3 The weekly pocket money is given at the equal rate to Monika and Kundan. Monika spent 79 and Kundan spent 911 of their pocket money. Who spent more? If sum of money to each is Rs. 99, how much money is left with them all ? Solution: Monika spent 79 of her pocket money. Kundan spent 911 of his pocket money. To compare 79 and 911 we must have the same denominator, LCM of 9 and 11 is 99.Now, 79 = 7 × 119 × 11 = 7799911 = 9 × 911 × 9 = 81998199 > 7799 or,911 > 79 ∴ Kundan spent more. Again,Sum of money that Monika spent = 79 of Rs. 99= 79 × Rs. 99 = Rs. 77Money left with Monika = Rs. (99 – 77)= Rs. 22 Sum of money that Kundan spent = 911 of Rs. 99= 911 × Rs. 99 = Rs. 81 Money left with Kundan = Rs. (99 – 81)= Rs. 18 Money left with Kundan and Monika = Rs. 22 + Rs. 18= Rs. 40 Hence, Rs. 40 is left with them.
Acme Mathematics 7 87Classwork1. Fill in the blanks: (a) A fraction equivalent to 35 is .......... (b) The sum : 3 + 13 = ...................(c) The product : 13 × 185 = ............. (d) The decimal form of 85 = ............ (e) The fraction form of 0.12 =............ (f) The quotient of 2 ÷ 13 = ..............2. Solve the following word problems.(a) Mrs. Sharma bought the following foods.Maida : 212 Sugar : 1 kg 12 Dal : kg 14 Rice : 3 kg 12 kg(i) Find the total kg of her foods.(ii) Calculate the total weight of 'Dal', 'Sugar' and 'Maida'.(b) The parts of a stick is given.314518518314(i) Calculate the length of coloured part.(ii) Find the length of uncoloured part.(iii) If the black coloured part is 20 cm, calculate the length of the stick.(c) Find the sum of 34 and 12 and multiply it by 820(d) Subtract 23 from the product of 37 and 1415.(e) Divide the product of 1017 and 2840 by 117 .(f) Divide 1021 by 56 and multiply the quotient by 23 – 1 2 .
88 Acme Mathematics 71. Exercise 2.141. From a roll of 10 m long cloth, Jivan took 35 of it and Shisir took 15 of it. How long cloth is left? 2. Binod lost one half of his money. He gave 25 of the remainder to his uncle and onefourth to his brother. If he had Rs. 2000 at first, how much money is left with him? 3. Mrs. Aryal uses 14 kg of sugar per day. Now she has a stock of 10 kg. How many kg of sugar will be left over after 30 days? 4. Prem gave 13 of his property for charity. Of the rest, he gave 23 to his wife and the rest he divided equally between his son and his daughter. Find what part of his property was given to the son and the daughter. 5. One-tenth of a class is absent. If the total attendance of the day is 126, find: (a) number of absent students (b) total number of students 6. Rajesh gave 12 of his provident fund to his wife, 14 to his son and the remainder was to be divided equally between his two brothers so that each brother got Rs. 10000. What was the amount of provident fund?7. The total income of a family is Rs. 20000. One twentieth is given to a milkman, one-fifth is spent on food and one-tenth is spent on miscellaneous. How much money is left? 8. Kiran's house is 710 km from the school. Sulu's house is 45 km from the school and Ritesh's house is 12 km from the school. Who is the farthest from the school ? How far is Kiran's house from Sulu's house if they are in the same direction? 9. If the product of 312 of a number and – 413 is 7, find the number. 10. Binaya spent 910 of his pocket money. He has Rs. 120 left now. Find: (a) How much money did he spend? (b) How much money did he have? 11. A rope is 2212 m long. How many pieces of each of 112 m long can be cut from it? How much long piece is left? 12. A number multiplied by its 13 gives the product 9075, what is the number?13. Today is Kushal's birthday. Kushal's friends took a cake and tried to divide it by 14 part for Pashang, 13 for Kiran and 12 for Kushal. Is it possible?14. In a class there are 700 students. Out of them 37 like Mathematics, 15 like English, 110 like Nepali and rest like Science. Calculate the number of students in each subject.
Acme Mathematics 7 89Project WorkDivide your study time ( 6 hours) in the following subjects:a. Mathsb. Nepalic. Englishd. ScienceNow express the time of each subject in fraction.C Revision on DecimalThe fractions 12 , 34 , 125 , 87100 and 5510 can be written as 0.5, 0.75, 2.4, 0.87 and 5.5 respectively, which are the decimal number or decimal fraction. Thus, the fraction can be changed into the decimal and decimal into the fraction.1. Addition of DecimalIn case of addition, there must be the same line with decimal point in the addend. The given example gives the clear idea about the addition.Example 1 Add: 2.34 + 3.456 + 0.12Solution:Note: There is no change in the decimal number when zeros on the right of decimal point are added or subtracted. e.g. 2.3450 = 2.345 = 2.34500002. Subtraction of DecimalsSubtraction is the similar to the addition. Example 1 Subtract 2.34 from 3.456Solution:3. Multiplication of DecimalsExample 1 Multiply: 2.345 by 6.78Solution: Numbers after removing decimals are, 2345 and 678. 2.340 3.456 + 0.120 5.916 3.456 – 2.340 1.116
90 Acme Mathematics 7Now,Here, 2.345 × 6.78 = 15.89910To understand this problem look at the following diagram carefullyand write your conclusion.4. Division of DecimalsIn case of division, shift the decimal of divisor and make it like natural number and shift the decimal of dividend same as divisor and divide in a normal way. Example 1 Divide: 0.56 by 0.5 Solution: 0.56 ÷ 0.5 = 0.560.5= 5.65= 1.12Classwork1. Change into decimal:(a) 4510 (b) 37100 (c) 192 (d) 458 (e) 6114 2. Change into fraction:(a) 0.25 (c) 0.123 (c) 2.732 (d) 5.03 (e) 10.0033. Add:(a) 3.65 + 4.28 (b) 70.31 + 47.5(c) 9.01 + 1.243 (d) 1.051 + 40.891 + 5.17(e) 67.581 + 9.012 + 5.6 (f) 1.001 + 50.12 + 3.00174. Simplify: (a) 0.25 + 2.56 – 1.89 (b) 2.34 – 4.567 + 5.89 – 1.307 (c) 77.83 – 26.82 – 155.832 + 408.206 (d) 6.7 – 0.7 × 2.3 (e) (5.68 + 1.87) × 4.01 (f) (4.264 × 3.2) ÷ 1.62 3 4 5× 6 7 81 8 7 6 01 6 4 1 5 0+ 1 4 0 7 0 0 01 5 8 9 9 1 0 2.345 × 6.78 = 15.89910 3 decimal places2 decimal places5 decimal places5 3 + 2 = 5Decimal placesDecimal point is shifted to one place right
Acme Mathematics 7 91Exercise 2.151. Add:(a) 4.17 + 7.043 + 3001.5 (b) 343.456 + 239.764 + 467.349(c) 1.152 + 2.1 + 13.89 + 1.732 (d) 11.52 + 42.183 + 5.25 + 18.182. Subtract:(a) 4.031 – 2.157 (b) 8.54 – 2.739 (c) 573.5 – 509.307(d) 93.35 – 70.882 (e) 7 – 2.7392 (f) 17 – 0.9323. Multiply:(a) 2.345 × 2.9 (b) 0.346 × 2.23 (c) 567.897 × 0.123(d) 90.4 × 1.2732 (e) 3700.1234 × 2.5 (f) (7.046) (2.5)(g) (4.004) (2.4) (h) (46.025) (1.25)(i) 3575.432 × 0.12344. Divide:(a) 5.85 ÷ 0.02 (b) 4.136 ÷ 0.02 (c) 38.745 ÷ 14.35(d) 72.512 ÷ 45.32 (e) 0.09225 ÷ 1.23 (f) 19.095 ÷ 0.15 (g) (80.35) ÷ 3.214 (h) (16.576) ÷ ( 0.004) (i) 561.6 ÷ 0.0312 5. Simplify: (a) (2.4227 × 10.42) ÷ 0.02 (b) (7.5)2 – (1.5)2(b) {(1.2)2 + (0.9)2} ÷ 1000 (c) 6.3 ÷ (0.3)2(d) 4.7 + {3.14 × 2.3 + 6.2} (e) (38.72 – 20.38) ÷ (20.35 – 0.35) (f) (8.16 –6.0632) ÷ (1.6 × 2.532 + 2.5013) (g) 45 ÷ (0.25 – 0.5) + 3 (h) 23 – 12 × (1.5 + 5.1) ÷ 0.4 (i) 12 × 0.5 + 13 × 0.63 + 23 × 1.23(j) 1.2 × 1.3 – 0.7 × 0.8 3.2 – 2.8 (k) 2.4 × 4.32 – 0.068 1.2 + 6.26. If the length and breadth of a rectangular field are 55.75 m and 40.15 m respectively find: (a) The perimeter of the field. (b) The area of the field.
92 Acme Mathematics 77. If the area of a rectangular garden is 8610.45 m2 and its length is 125.7 m find, (a) Breadth of the garden (b) Perimeter of the garden. 8. The circumference of a wheel of motorcycle is 1.88 m, find how many times it moves when the motorcycle travels 14.10 km ?9. Himalaya bought 2.5 kg of potatoes for Rs. 45.50, onion for Rs. 9.75, vegetables for Rs. 16.35 and orange for Rs. 40.50. How much money does he spend? 10. Hareram bought 2.4 m cloth for shirt, 1.15 m for pant and 2.90 m for coat. Find the total length of the cloth that he bought. 11. Krishna bought 13 pens for Rs. 304.85. How much does it cost for 6 pens? 12. Surendra runs 3.455 km at 6 o'clock, 4.650 km at 8 o'clock and 5.305 km at 11 o'clock by cycle. Find how long he runs altogether?13. If the length of base of a triangle is 30.75 ft. and its height 10.25 ft. Calculate its area.[Note : Area of triangle = 12 × base × height]Take a book and a copy from your bag. Measure the length and breadth of book and copy.Now, Answer the following questions.(a) How long is your book?(b) How broad is your copy?(c) Compare the area of book and copy.(d) Is there any difference between length of book and copy?Project Work
Acme Mathematics 7 932.5 Percentage, Ratio and Proportion1. Change into percentage:(a) 25100 (b) 5050 (c) 325 (d) 4102. Change into fraction: (a) 16% (b) 75% (c) 90% (d) 1.25% 3. Calculate the value of the following: (a) 10% of 200 (b) 5% of Rs. 100 (c) 2% of 800 4. Ram spends 80% of his income per month and save Rs. 2000. How much is his income per month. 5. Convert to the lowest term. (a) 20 : 30 (b) 115 : 230 (c) 2 24 : 126. Fill in the blanks: (a) Ratio of x to y is written as ............... (b) Ratio of 3 to 7 is written as ...............(c) Ratio of Rs. 3 to Rs. 9 is written as .............. (d) If a : b = 2 : 4 then b is double of ...................... (e) The ratios 1 : 2 and 2 : 4 are in .................... (f) If x10 = 65 then, x =.................... 7. Convert to the fractions:(a) 3 : 9 (b) 6 : 8 (c) 10 : 20 (d) 50 : 1508. Convert to the decimal:(a) 19% (b) 110% (c) 98% (d) 2.25%Warm Up Test
94 Acme Mathematics 7A Revision on PercentageA fraction with denominator 100 is called a percent. Percent is an abbreviation for the Latin word 'per centum' that means per hundred or hundredths. It is denoted by '%'. Thus, 15100 = 15 × 1100 = 15% 32100 = 32 × 1100 = 32% 450 = 4 × 250 × 2 = 8100 = 8 × 1100 = 8% etc.The percentage is a form of fraction; we can express it as fractions or decimals and vice versa. We express it as in the following diagram. (a) Conversion of fraction or decimal into percentageStudy the following examples carefully.Solved ExampleExample 1 Express as percentage: (a) 35 (b) 0.23Solution: (a) Here, 35= 3 × 205 × 20 = 60100= 60 × 1100 = 60%(b) Here, 0.23= 023100 = 23 × 1100 = 23%(b) Conversion of percentage into fraction or decimalStudy the following examples carefully.Solved ExampleExample 1 Express 12% as fraction. Solution: Here, 12%= 12 × 1100= 12100 =2 × 2 × 32 × 2 × 25 = 325Hence, 12% = 325Fraction or Decimal Percentage× 100%÷ 10060 of 100 is 60%% is replaced by 1100
Acme Mathematics 7 95Example 2 Express 145% as decimalSolution: Here, 145%= 145 × 1100 = 145100 = 1.45Hence, 145% = 1.45(c) To find the value of the given percentage of a given quantity.Study the following examples carefully.Example 1 Find the value of 25% of Rs. 125. Solution: Here, 25% of Rs. 125 = 25100 × Rs. 125= Rs. 31.25 Example 2 Toya spends 75% of his monthly income. If his income is Rs. 5100, find his monthly saving. Solution: Toya's monthly income = Rs. 5100 His expenditure = 75 % of Rs. 5100 = 75100 of Rs. 5100 = Rs. 75100 × 5100 ∴ His expenditure = Rs. 3825 Now, his saving per month = Total income – Total expenditure = Rs. 5100 – Rs. 3825 = Rs. 1275 Hence, monthly saving of Toya is Rs. 1275. Example 3 Increase 800 kg by 40%. Solution: Here, 40% of 800 kg = 40100 × 800 kg= 320 kg Now, 800 kg + 320 kg = 1120 kg Hence, 800 kg increased by 40% = 1120 kg. 100) 145 ( 1.45 –100 450 –400 500 –500 0
96 Acme Mathematics 7(d) To find the quantity whose value of certain percentage is givenStudy the following example carefully.Solved ExampleExample 1 Find the sum of money whose 10% is Rs. 15. Solution: Here, 10% of the sum of money is Rs. 15. Let, Sum of money = Rs. x, then 10% of x = Rs. 15 or, 10100 × x = Rs. 15 or, 110 × x = Rs. 15 or, x = Rs. 15 × 10 or, x = Rs. 150 Thus, sum of money is Rs. 150.Classwork1. Change into percentage: (a) 12 (b) 125 (c) 0.6 (d) 1.25 (e) 0.655 2. Change into decimals and fractions: (a) 55% (b) 6014% (c) 3712% (d) 3215% (e) 125% 3. Find the value of: (a) 20% of Rs. 150 (b) 5% of 1 kg(c) 12.5% of 2 hours (d) 3313 % of 21m.(e) 23% of 300 kg (f) 0.35% of 2250 litres 4. There are 1400 students in a school out of which 780 are girls, find the, (a) Number of boys in the school. (b) Percentage of boys in the school. (c) Percentage of girls in the school.Alternative Method:10%Rs 1510%Rs 15Here, 1 room = Rs. 15 10 rooms = Rs. 15 × 10 = Rs. 150
Acme Mathematics 7 97Exercise 2.161. What percent of: (a) 570 kg is 190 kg (b) Rs. 555 is Rs. 105 (c) 1115 litres is 115 litres (d) 2005 m is 205 m (e) 3333 km is 333 km (f) 20 hrs. is 30 minutes2. Increase the quantity as given percentage: (a) 72 by 25% (b) 125 by 20% (c) 16 by 112% (d) Rs. 100 by 200% (e) 700 kg by 8012 % (f) 300 m by 30%3. Decrease the quantity as required: (a) 100 by 25% (b) 700 kg by 20% (c) 75 by 15% (d) Rs. 24 by 12.5% (e) 90 by 1623 % (f) 125 m by 3% 4. (a) Ram spends 60% of his income and saves Rs. 2000. How much is his income? (b) What is the sum, if 20% of it is Rs. 20? 5. If there are 35% girls and 325 boys in a group, calculate the total member of the group and number of girls. 6. (a) Hari earns Rs. 4300 per month. He spends 80% from his income. How much amount does he save in a year? (b) The annual income of Amir is Rs. 50,000. If he spends 20% from this and deposits the remaining in a bank, find his expenditure amount and the amount deposited in the bank. (c) In an election, 400 votes were cast. The winning candidate received 60% votes. How many more votes did winning candidate get? 7. The price of kerosene was increased by Rs. 24 per litre to Rs. 28 per litre. Find how much percent is increased in the price. 8. Calculate the difference between 25% and 20% of Rs. 1250. 9. If one sixth of 300 oranges are rotten, what percentage is good? 10. A customer gets discount of 8%. How much should he pay for a shirt marked Rs. 450 ?
98 Acme Mathematics 711. (a) There were 700 pigs in a village. 5% died of disease and 20% of the remaining were sent for meat supplier. How many pigs were left? (b) Nikhil has Rs. 80,000. From this he gives 30% to his wife and 20% to his father. (i) Find the amount received by wife and father.(ii) How much money has he now? (c) Jems has monthly income of Rs. 12500. If he donated 4% of his income for poor people and deposited 70% of remainder in a bank, find the amount of donation and deposited in the bank.B Ratioa. IntroductionWe know that ratios are used to compare quantities of the same kind. Suppose, in a class, there are 35 boys and 25 girls. To compare the number of boys with the number of girls, we write the ratio 35:25 and read as 35 to 25 or 35 is to 25. In the ratio, 35 is called antecedent and 25 is called the consequent. The simplest form of ratio 35:25 is 7 : 5, it means that in each 12 (7+5) students there are 7 boys and 5 girls. Now study the given examples carefully.Solved ExampleExample 1 Find the ratio of Re 1 and 80 paisa and convert to the lowest term. Solution: Here, Re 1 = 100 paisa Now, Ratio of 100 paisa and 80 paisa = 100 : 80 = 10080= 108= 54 = 5 : 4 Here, ratio in the lowest term is 5 : 4.
Acme Mathematics 7 99Example 2 Compare the ratios 1:2 and 3:4 Solution: Here, 1 : 2 = 12 and 3 : 4 = 34Now, Compare the fractions 12 and 34LCM of 2 and 4 = 4 So, 12 = 1 × 22 × 2 = 2424 < 34Here, 1 : 2 < 3 : 4 Example 3 There are 48 students in the class. The ratio of the number of boys to the number of girls in the class is 5:3. How many of them are boys ? How many of them are girls? Solution: Total number of students = 48 Ratio of boys to girls = 5 : 3 = 535 : 3 means, for every 5 boys there are 3 girls or 58 are boys. Therefore, out of 8 students, 5 are boys and 3 are girls.∴ Number of boys = 58 × 48 = 30 Number of girls = 38 × 48 = 18 Therefore, there are 30 boys and 18 girls. Example 4 The ratio of the age of father and his son is 5:2 and the age of son is 20 years now. Find the age of father. Solution: Here, age of son = 20 years Ratio of their age = 5 : 2 = 52Now, Age of fatherAge of son = 52or, Age of father20 years = 52or, Age of father = 52 × 20 years = 50 years Therefore, the age of father is 50 years.
100 Acme Mathematics 7b. Relation Between Ratio and FractionStudy the given figure.Its 2 parts out of 5 parts are shaded.So, its fraction is 25. Non-shaded part is 35Ratio between shaded and non-shaded part = 2 : 3, where, 2 and 3 represents the parts of fraction.Classwork1. Express the shaded part and non-shaded part in the ratio.(a) (b) (c)(d) (e) (f)2. Convert the following into the lowest term: (a) 12 : 18 (b) 25 : 70 (c) 49 : 245 (d) 234 : 4563. Fill in the blanks. (a) Find the ratio of 7 kg to 14 kg. It is ................ (b) Write the ratio of 50 cm to 1 m. It is................ (c) If x : 4 = 5 : y, then the value of x = ................ and y = ............... (d) If a5 = 210 , then the value of a =.................... 4. The ratio 30y in equal to 6 : 4. Find the value of y. 5. 3, 5 and 6 x are in equal ratio. Find the value of x.