202110721-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G08-FY_Optimized

13. Direct and Inverse Proportions Learning Outcomes At the end of this chapter, you will be able to: • Understand proportionality between two quantities. • Differentiate between direct and inverse proportions. • Find the missing values using direct and inverse proportions. Concept Map ������������ = ������������ Key Points We know that in direct Proportion x/y= k, x=ky • We come across many such situations in our So we can find value of k from known values, and day-to-day life, where we need to see variation then use the formula to calculate the unknown in one quantity bringing in variation in the other values. quantity. • Inverse proportion For example: Two quantities x and y are said to be in inverse (i) As the speed of a vehicle increases, the time proportion: if an increase in x causes a proportional decrease taken to cover the same distance decreases. in y (and vice-versa) in such a manner that the (ii) More apples cost more money. product of their corresponding values remains (iii) More interest earned for more money constant. That is, if xy = k= Constant deposited. Then x and y are said to vary inversely. (iv) More distance to travel, more petrol needed. In this case if y1, y2 are the values of y corresponding to the values x1 , x2 of x • Direct Proportion respectively then Two quantities x and y are said to be in direct proportion: x1 u y1 x2 u y2 If they increase (decrease) together in such a We know that in inverse Proportion manner that the ratio of their corresponding xy= k,x=k/y values remains constant. So we can find value of k from known values, and That is if x/y=k= [k is a positive number] = then use the formula to calculate the unknown Constant values. Then x and y are said to vary directly. In such a case if y1, y2 are the values of y corresponding to the values x1 , x2 of x respectively then x1 = x2 y1 y2 87

13. Direct and Inverse Proportions Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEETS PS – 1 Introduction • Examples on direct and inverse proportion PS – 2 • Differentiating direct and inverse PS – 3 Proportion   Direct Proportion • Direct Proportion PS – 4 • Word problems on direct proportion Self-Evaluation Sheet Indirect Proportion • Indirect Proportion • Word problems on indirect Proportions   • Textual problems Worksheet for “Direct and Inverse Proportions” Evaluation with Self Check or Peer ---- Check* 88

PRACTICE SHEET - 1 (PS-1) 1. Identify as direct or inverse proportion (i) Time taken by train to cover a fixed distance to the speed of the train. (ii) Cost of pen and number of pen. (iii) Number of people in a party and amount of food. (iv) Amount of fuel to the distance travelled. (v) Area of land to its cost. (vi) Number of people working to the quantity of work. (vii) Age of a tree and height of the tree. (viii) Increase in cost of pant to the number of pants bought if the total amount spent is same. (ix) Sales tax and amount of the bill. (x) Compound interest on a sum of money and rate of interest. PRACTICE SHEET - 2 (PS-2) 1. If x varies directly as y, if x= 80 , y= 240 , find y if x = 100. 2. If x and y are directly proportional, find the missing values. X 3 13 B C y 15 A 90 105 3. If the area occupied by 15 carpets is 600 m2, find the area covered by 120 carpets. 4. Harish travels 50 m taking 75� steps, how many kms will he cover by taking 375 steps. 5. If 14 kg of rice cost Rs 441 , find the cost of 22 kg of rice. 6. 3 cans of paint is used to paint a rectangular board of length 10 m and breadth 9 m. Find the number of cans required to paint an area of 300� m2. 7. Hari types 108 words in 6 minutes, how many words can he type in half an hour? 8. If a deposit of Rs 3000 earns a interest of Rs 600 in 3 years. How much should be the sum invested to earn a interest of Rs 10800 in 3 years at the same rate of interest? 9. Mass of a steel rod varies directly with its length, if 16 cm long rod weighs 256 gms. Find the mass of 80 cm long rod. 10. A pole of length 36 m casts a shadow of 54 m on the ground. Find the height of an abject which casts a shadow of 120 m on the ground. 89

PRACTICE SHEET - 3 (PS-3) 1. If x and y varies inversely, if x= 12,� y= 24 . Find the value of y when x = 20. 2. 30 people can reap a field in 17 days. How many more people will be required to reap the same field in 10 days? 3. A packet of sweet was distributed among 10 children and each child receives 4 sweets. If it is distributed among 8 children, how many sweets will each child receive? 4. A car takes 4 hours to reach a place at a speed of 60 km/hr. How long will it take to reach the same place if the speed is increased to 80 km/hr? 5. Food in the hostel is sufficient for 24 days for 20 people. If 6 more people come to the hostel, how many days will the food last? 6. 12 men can dig a pond in 8� days. How many men can dig the same pond in 6� days? 7. A factory works in 3 shifts of 6 hours each, how long will each shift be if the number of shifts is increased to 4 keeping the total number of hours same. 8. A factory requires 50 machines to produce a given number of articles in 75 days. How many machines would be required to produce the same number of articles in 30 days? 9. A contractor estimates to finish a work in 10 days using 4 laborers. If he uses 2 persons more, in how many days will he be able to finish the same work? 10. If 8 taps flowing at same speed fills a tank in 27 minutes, how long will it take for taps to fill the tank if 2 taps went out of order? 11. 1200 soldiers in a fort had enough food for 28 days. After 4 days, some soldiers were transferred to other fort and thus the food lasted for 32 days. Find the number of soldiers that left. 12. A group of 3 friends staying together consume 54 kg of wheat in a month. Some more friends join the group and they find that the wheat lasts for 18 days. Find the number of friends who joined the group? 13. 420 men can finish a piece of work in 9 months. How many extra men should be employed to finish the same piece of work in 7 months? 14. A car can finish a journey in 10 hours at a speed of 48 Km/hr. By how much should the speed be increased so as to cover the same distance in 8 hours? 15. A person has money to buy 25 cycles worth Rs 500 each. How many cycles will he be able to buy if each cycle costs Rs 125 more? 90

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Identify the type of variation in the following: Fare per person for a certain distance. (A) Inverse proportion (B) Direct proportion (C) Both (A) and (B) (D) Neither (A) nor (B) 2. _____________ is constant in direct proportion between two quantities ‘x’ and ‘y’. [] x (A) y (B) x + y (C) xy (D) x – y 3. Choose a counter example for quantities in direct proportion. (A) Number of bags of rice and it’s cost. (B) Monthly expenditure and number of family members. (C)Number of men to complete a work and number of days to finish the work (D) Number of fruits and number of boxes to pack it in 4. Kishan bought a pen for Rs. 55. Which of these proportions gives the amount he has to spend if he buys 8 pens? (A)1: ? : : 55: 8 (B) 1: 55 : : 8 : ? (C) 8 : 55 : : 1: ? (D) 55 : 1 : : 8 : ? 5. Two quantities ‘p’ and ‘q’ are in inverse proportion. If p = 6, q = 12. What is the value of ‘q’ when p = 4? (A) 288 (B) 16 (C) 8 (D) 18 6. Two quantities ‘m’ and ‘n’ are in direct proportion. If m = 8, n = 40. What is the value of ‘m’ when n = 15? (A) 3 (B) 8 (C) 12 (D) 320 7. The missing number in the given table is ______. x 100 200 300 400 y 60 ? 20 15 (A) 20 (B) 30 (C) 22 (D) 25 8. In the given grid, if x and y are in direct proportion, find the missing number. x 20 17 14 11 8 y 40 34 28 22 ? (A) 11 (B) 15 (C) 16 (D) 12 9. Identify the statement that is an example of direct variation. (A) Decrease in the amount paid due to increase in the cost of an item. (B) Increase in the amount paid due to decrease in the number of items bought. (C) Increase in the amount paid due to increase in the cost of an item. (D) All of these 10. A train moving at 50 kmph covers_________ km in 2 hours. (A) 52 (B) 100 (C) 25 (D) 48 II. Short Answer Questions. 11. A bus ticket costs Rs. 15 for the first 5 km, Rs. 30 for the first 10 km and so on. How much does a ticket cost for 25 km? 12. If 15 workers can make a cupboard in 48 hours, how many workers will be required to make it in 24 hours? 13. A person covers 40 km cycling at a speed of 25 kmph. What distance would he cover in the same time, if he increases his cycling speed by 15 kmph? III. Long Answer Questions. 14. A machine in a factory packs 650 bottles in 5 hours. (i) How much time does the machine take to pack 1300 bottles? (ii) How many bottles can it pack in 13 hours? 15. A restaurant has enough groceries for 50 persons in 8 days. Find (i) The number of days the groceries would last for 40 persons. (ii) The number of persons that can be fed in 20 days. 91

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. If x varies directly to y, and x = 30 and y = 90, 5. A man has enough money to buy 90 articles at then: Rs 1500 each. How many articles can he buy (i) find x when y = 45 with the same amount of money if cost of each (ii) find y when x = 90 (2 Marks) article decreases by 20% ?  (3 Marks) 2. 55 cows can graze a field in 16 days. How many cows will graze the same field in 10� days? (2 Marks) 6. Arun has a road map which represents 1 cm as 25 km. If he drives 100 km, what distance has he covered on the map? If a place is 10 cm on the map, how far is the place actually? (3 Marks) 3. A man has enough money to buy 75 machines of Rs 200 each. How many machines can he buy if he gets a discount of Rs 50 on each machine. (2 Marks) 4. Anupama takes 125 minutes in walking a distance of 100 m. How far will she walk in 315 minutes? How much time will she take to cover 625 m?(3 Marks) 92

14. Factorisation Learning Outcome By the end of this chapter, you will be: • Find the common factors of algebraic terms • Factorize algebraic expressions by grouping the terms • Factorize algebraic expressions by using suitable identities. Concept Map Key Points expression. Example: • Factors of algebraic expression a) 2x+4 An algebraic expression can also be expressed as 2 is the HCF OF 2 and 4 Products of factors. 2x + 4 = 2(x + 2) Example: b) 11x2y + 5x First reduce it to irreducible factors form 6xy 6uxy 6uxuy 2u3uxy 2u3uxuy 11x2y + 5x =11 × x × x × y + 5 × x Now we can see that x is the common factor We can see that 2, 3, x, y cannot be further expressed 11x2y + 5x = x(11xy + 5) as products of factor.  So they are called the prime ii) Factorisation by regrouping terms factors. We call them irreducible factors in terms of 1) First we see common factor across all the terms algebraic expressions. in the algebraic expression. • Factorization of algebraic expression 2) We look at grouping the terms and check if • When we factorize an algebraic expression, we we find binomial factor from both the groups. write it as a product of factors. These factors 3) Take the common Binomial factor out. may be numbers, algebraic variables or algebraic Example: expressions. a) 2xy + 3x + 2y + 3 Example The expression 6x(x−2) First reduce it to irreducible factors form. There is The above expression can be written as a product no common factor across all the terms. of factors 2, 3, x and (x−2). =2×x×y+3×x+2×y+3 6x (x−2) = 2 × 3 × x × (x−2) Now we think about grouping the terms. We can The factors 2, 3, x and (x +2) are irreducible factors of see that we have common factor between first two 6x (x + 2). terms. • Method of Factorisation = x × (2y + 3) + 1 × (2y + 3) i) Common factor method = (2y + 3) (x + 1) 1) We can look at each of the term in the algebraic b) 6xy−4y + 6−9x expression and factorize each term into irreducible There are no common factors across all the terms, factors. 2) Then find common factors to factorize the 93

14. Factorisation so we think of grouping and rearranging the terms. This resembles the second identity = 6xy−4y−9x + 6 Here a = 2x and b = 3 = 2y(3x−2)−3(3x−2) So, 4x2−12x + 9 = (2x−3)2 = (2y−3) (3x−2) 2) 16x2−81 iii) Factorisation using identities We can write this as We can use the below identities to factorize (4x)2−92 algebraic expression This resembles the third identity (a + b)2 = a2 + 2ab + b2 Here a = 4x and b = 9 (a−b)2 = a2−2ab + b2 16x2−81 = (4x−9) (4x + 9) (a + b) (a−b) = a2−b2 Example: 1) 4x2−12x + 9 We can write this as (2x)2−2 × 2x × 3 + 32 Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Algebraic expressions • Algebraic expressions, factorization PS – 1 • HCF and LCM • Algebraic identities PS – 2 PS – 3 Factors of algebraic expressions • Factors of natural numbers PS – 4 • Factors of algebraic expressions Self Evaluation Sheet • Method of common factors • Regrouping of terms Factorization using identities • Factorization using standard identities: (a + b)2 = a2 + 2ab +b2 (a−b)2 = a2−2ab + b2 (a2−b2) = (a + b) (a−b) • Factorize using (x + a)(x + b) = x2 + (a + b)x + ab Division of algebraic expression • Division of monomial by monomial • Division of polynomial by monomial • Division of polynomial by polynomial Worksheet for “Factorisation” Evaluation with Self Check or ---- Peer Check* 94

PRACTICE SHEET - 1 (PS-1) 1. Write the greatest common factor in each of the following (i) –18a2, 108a (ii) 3x2y, 18xy2, –6xy (iii) qrxy, pryz, rxyz (iv) 13x2y, 169xy (v) 11x2, 12y2 2. Factorise: (i) 6ab + 12bc (ii) l2m2n−lm2n2−l2mn2 (iii) 4xy2−10x2y + 16x2y2 + 2xy (iv) a3 + a2 + a + 1 (v) 2x2−2y + 4yx−x (vi) y2 + 8zx−2xy−4yz (vii) a2b + a2c + ab + ac + b2c + c2b PRACTICE SHEET - 2 (PS-2) 1. Factorise using suitable identity: x2 + 14x + 49 2. Factorise using suitable identity: a2x2 + 2ax + 1 3. Factorise using suitable identity: 4x2 + 12x + 9 4. Factorise using suitable identity: a2x2 + 2abxy + b2y2 5. Factorise using suitable identity: 2x3 + 24x2 + 72x 6. Factorise using suitable identity: x2−10x + 25 7. Factorise using suitable identity: 4a2−4ab + b2 8. Factorise using suitable identity: a2y3−2aby2 + b2y 9. Factorise using suitable identity: 4x2−25y2 10. Factorise using suitable identity: 63ay2−175ax2 11. Factorise using suitable identity: 1331x3y−11y3x 12. Factorise using suitable identity: 9x2−(3y + z)2 13. Factorise using suitable identity: x4−y4 + x2−y2 14. Find ‘a’ if 8a= 352–272. 15. Find the value of 6.25u 6.25  1.75u1.75 4.5 95

PRACTICE SHEET - 3 (PS-3) 1. Factorise: x2 + 15x + 26 2. Factorise: x2 + 9x + 20 3. Factorise: y2−4y−21 4. Factorise: x2 + 4x−77 5. Factorise: x2 + 17x−60 6. Factorise: x2−13x + 30 7. Divide : 76x3yz3 19x2y2 8. Divide : x2  10x  21 x2  5x  6 9. Divide : x2  9x  20 x2  x  12 10. Area of a square is 4x2 + 12xy + 9y2. Find the side of the square. PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. The prime factor form of 48 is _______. (A) 4 × 2 × 6 (B) 2 × 3 × 2 × 4 (C) 2 × 3 × 12 (D) 2 × 2 × 2 × 2 × 3 2. 14a³b + 21 a²b² = _______. (A) 7a²b(2a² + 3b²) (B) 7a²b(2a + 3b) (C) 7ab(2a² + 3ab) (D) 7ab(2a + 3a²b) 3. 36p² + 36pq + 9q² = _____________ (A) (6p + 3q)(6p - 3q) (B) (6p – 3q)² (C) (6p + 3q)² (D) 6pq(6p + 3q) 4. m² + n² – p² – 2mn expressed as factors is _______________. (A) (m – n – p) (m – n + p) (B) (m + n – p) (m – n + p) (C) (m – n + p) (m + n + p) (D) (m + n – p) (m + n + p) 5. The formula used to simplify (-22)⁴ - (-15)⁴ is _______. (A) (a – b)² = a² + b² – 2ab (B) (a + b)³ = a³ + b³ + 3ab(a + b) (C) (a + b)² = a² + b² + 2ab (D) a² – b² = (a + b) (a – b) 6. k² + 36k + 180 = ______________. (A) (k + 30) (k – 6) (B) (k – 30) (k – 6) (C) (k + 30) (k + 6) (D) (k – 30) (k + 6) 7. (x⁴y² – x²y⁴) ÷ x²y² = ____________. (A) x²y² (B) (x – y) (x + y) (C) (y – x) (y + x) (D) 1 8. The expanded form of (a – 4) (a – 3) is _____. (A) a² – 7a + 12 (B) a² – 7a – 12 (C) a²+ 7a – 12 (D) a² + 7a + 12 9. Identify the correct one. (A) (t – 5)² = t² + 25 (B) (g – 6)² = g² + 12g +36 (C) (r + 6)² = r² + 12r +36 (D) (k + 6)² = k² + 36 10. Identify the incorrect one. (A) (s + 11)2 = s2 + 121 (B) (d + 8)² = d² + 16d + 64 (C) (f – 2)2 = f2 – 4f + 4 (D) (a + 9) (a – 9)= a² – 81 96

PRACTICE SHEET - 4 (PS-4) II. Short Answer Questions. 11. Simplify: (w² + 24w + 144) ÷ (w + 12) 12. Find the quotient in 40(p³q²r² + p²q³r²+ p²q²r³) ÷ 5p²q²r² 13. Factorise: a) (c + d)² – 4cd b) (m² + 2mn + n²) – (p² – 2pq+ q²) III. Long Answer Questions. 14. Factorise: a) g⁴ – (g + h)⁴ b) e⁴ – (f – g)⁴ 15. Divide 77(x⁴ – 5x³ – 24x²) by 11x(x – 8). 97

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Find the common factors of 3x−12 and x−4. (1 Mark) 6. Evaluate (2x + 3y)2−16z2(2 Marks) 2. Factorise: 7a2b + 21ab2  (1 Mark) 7. Area of a parallelogram is x2 + 18x + 65. Find the 3. Find the value of 4x2−25y2. (1 Mark) possible base and height if x=2.  (3 Marks) 4. Factorise x2–22x+121.  (2 Marks)  8. 45x3 32x2  98  (3 Marks) Divide : 15x4x  7 5. Factorise: x2−9x−52.  (2 Marks) 98

15. Introduction to Graphs • Plot line graphs from given data. • Plot linear graphs from given data. Learning Outcomes • Plot coordinates on the graph. At the end of this chapter, you will be able to: • Identify types of graphs. • Interpret data from line graphs. Concept Map Key Points intervals. • A line graph displays data that changes • Graphical presentation of data is easier to understand. continuously over periods of time. • A line graph which is a whole unbroken line is • A bar graph is used to show comparison among categories. called a linear graph. • For fixing a point on the graph sheet we need, • A pie graph is used to compare parts of a whole. x-coordinate and y-coordinate. • The relation between dependent variable • A histogram is a bar graph that shows data in and independent variable is shown through a Work Plan graph. CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Line Graphs • Read line graphs PS – 2 • Draw line graphs PS – 3 PS – 4 Cartesian coordinates • Plot points on Cartesian plane • Identify coordinates of given points Self-evaluation Sheet Application questions • Plot linear graphs for day to day applications Worksheet for “Introduction to Graphs” Evaluation with self- check or Peer check* 99

PRACTICE SHEET - 1 (PS-1) 1. Following graph represents the speed of bike every hour from city A to city B. a. What is the average speed? b. What is the stoppage time of the rider and when? c. What speed did the rider travel in the first 3 hours? 250 200 Distance 150 100 50 0 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12:00 PM 1:00 PM 2:00 PM Time Sales2. Observe the given graph and answer the questions that follow. Sales Report 600 500 400 300 200 100 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Months a. What is the total sales in the first quarter? b. In which month the sales in minimum and maximum. What was it? c. Calculate the mean of sales in the first 6 months? d. What is the average of sales in the full year? 3. Observe the given graph and answer the questions that follow. Class VII Class VIII Medals 10 9 8 Long jump 100mts race 400mts race 1000mts race 7 Games 6 5 4 3 2 1 0 High jump a. Which class has won the maximum medals? b. What is the mean medals won by class VII in running race? c. What is the mean medals won by class VIII in running race? 100

PRACTICE SHEET - 1 (PS-1) d. Find the mean medals won by class VII and class VIII. e. Find the ratio of medals won by class VII and class VIII in long jump. 4. Observe the given graph and answer the questions that follow. Scale is 1cm = 10 students No. of Students 10 Boys Girls 8 6 VB VC VD VE 4 Section of Class V 2 0 VA a. Find the total number of boys and girls in class V. b. In which sections girls are more? c. Find the ratio of girls and boys in class V. 5. Construct a line graph with the following data. Days of the Mon Tue Wed Thurs Fri Sat Sun week 85 92 55 43 73 91 65 Items sold a. Calculate the average sale during the week b. Find the ratio of the minimum and maximum sale. c. On how many days of the week was the scale above the average sales? 6. Construct a double line graph for the following data of matches won by India and Australia. India 2010 2012 2014 2016 2018 Australia 85 78 80 60 50 78 80 76 70 50 a. Find the ratio of matches won by India and Australia. b. Which team has won the maximum matches? 101

PRACTICE SHEET - 2 (PS-2) 1. Plot the following points on a graph sheet. Verify if they lie on a line. a. A (5, 0), B (5, 3), C (5, 7), D (5, 5.5) b. P (2, 2), Q (3, 3), R (4, 4), S (5, 5) c. K (3, 4), L (6, 4), M (6, 6), N (3, 6) 2. Draw the line passing through (1, 2) and (2, 1). Find the coordinates of the points at which this line meets the x-axis and y-axis. 3. Write the coordinates of the vertices of each of below figures. a. 2.5 Y-Values 2F 1.5 1G 0.5 E H 0 0 0.5 1 1.5 2 2.5 b. 2.5 Y-Values 2B 1.5 1AC 0.5 D 0 0 0.5 1 1.5 2 2.5 c. 2.5 Y-Values 2 P 1.5 1 R 1.5 Q 0.5 1 2 2.5 0.5 0 0 102

PRACTICE SHEET - 2 (PS-2) 4. Find the co-ordinates from the following graph. 6 D 5 4A C E 3 B F 2 1 0G 012345678 PRACTICE SHEET - 3 (PS-3) 1. A bank gives 10% simple interest (S.I.) on deposits by senior citizens. Draw a graph to illustrate the relation between the sum deposited and simple interest earned. Find from your graph the following. a. The annual interest obtainable for an investment of Rs. 250 b. The investment one has to make to get an annual Simple interest of Rs. 70 2. Mayank deposited Rs. 1400 in a bank at the rate of 10% per annum. Draw a linear graph which shows the relationship between time and the interest earned by Mayank. 3. Parul is driving a car constantly at a speed of 30 km/h. Draw a distance-time graph in this case. Also, find the time taken by Parul to cover a distance of 120 km. 103

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Which of these has horizontal or vertical rectangular bars to show comparison among different catego- ries? (A) A pie chart (B) A histogram (C) A pictograph (D) A bar graph 2. Which of these graphs shows data in intervals? (A) A histogram (B) A bar graph (C) A double bar graph (D) A pie chart 3. A ______________ displays data that changes continuously over periods of time. (A) histogram (B) line graph (C) bar graph (D) pie chart 4. In an experiment, the change in the temperature of a solution with the passage of time is as shown. Choose the correct statement from the graph. (A) The temperature of the solution is increased with increase in time. (B) The temperature of the solution is decreased with decrease in time. (C) The temperature of the solution is decreased with decrease in time. (D) The temperature of the solution is decreased with increase in time. 5. From the given linear graph, identify the correct one. Point (A) W X Y Z Ordered pairs (B) (2, 6) (3, 5) (5, 3) (6, 2) (C) (6, 2) (2, 6) (3, 5) (5, 3) (D) (6, 2) (3, 5) (5, 3) (2, 6) (6, 2) (5, 3) (2, 6) (3, 5) 6. The coordinates of the origin are ________. (C) (0, 1) (D) (0, 0) (A) (0, 3) (B) (3, 0) 7. Identify the point that lies on the x – axis. (C) (-2, 6) (D) (8, 0) (A) (-2, 0) (B) (-2, -4) 104

PRACTICE SHEET - 4 (PS-4) 8. From the graph given, what is the amount earned through sales in the year 2004? (A) Rs. 8.5 crores (B) Rs. 9 crores (C) Rs. 6 crores (D) Rs. 8 crores 9. In a chemistry experiment, Tina observed that the temperature of the solution remained constant for a peri- od of 3 hours. If she drew a graph for this situation, how would the graph look like? (A) (B) (C) (D) 10. From the given line graph, find the time duration in which the fall of temperature is maximum. (A) Between 12 noon and 2 p.m. (B) Between 10 a.m. and 11 a.m. (C) Between 12 noon and 1 p.m. (D) Between 1 p.m. and 2 p.m. II. Short Answer Questions. 11. From the given graph, find write the ordered pairs of the points with (i) the same x-coordinates (ii) the same y-coordinates 105

PRACTICE SHEET - 4 (PS-4) 12. The distance covered by a train from Warangal to Secunderabad is shown in the table. Time ( in hours) 6 a.m. 7 a.m. 8 a.m. 9 a.m. Distance (in km) 40 80 120 160 Study the values in the given grid and estimate the time when the train crossed 100 km. 13. The production of oranges in six consecutive years on Rahul’s farm is given in the table. Year 2012 2013 2014 2015 2016 2017 75 60 85 100 Tons of oranges produced 40 55 Draw a graph with suitable scales on the axes. III. Long Answer Questions. 14. Draw the graph of the following data. Quantity of milk (in litres) 20 25 30 35 40 45 Cost (in Rs.) produced 480 600 720 840 960 1080 15. The simple interests earned by Mrs. Komala on the amount deposited in a bank are as given in the table. Amount (in Rs.) 10000 20000 30000 40000 50000 60000 Simple interest (in Rs.) 900 1800 2700 3600 4500 5400 Draw the graph and from it, find the amount to be deposited to earn an interest of Rs. 4000. Also find the interest when Rs. 35000 is deposited. 106

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Define line graph. (1 Mark) 6. Observe the given graph and answer the questions that follow. Scale is 1cm = 10 students 2. Define linear graph. (1 Mark)  (3 Marks) 3. Give one real life explanation of a linear graph Boys GirlsNo. of Students and explain it. (2 Marks) 10 4. Side of square is plotted on one axis and area of 8 6 4 2 0 Cricket Hockey Football Badminton Swimming Favourite Sport a. Which sports is most preferred by boys and girls? b. Find the mean of sports played by boys. c. Find the ratio of girls playing cricket and badminton. square on the other axis. Draw the graph. Is the graph linear? (2 Marks) 5. Table shows maximum temperature of a city on 7. Find the co-ordinates from the following graph. [3 M] each day of a week. Represent the given data by a line graph. (3 Marks) 5 Days Mon Tue Wed Thu Fri Sat Sun D 4 3A C E F Max Temp 34 35 37 38 41 39 36 2B (deg Celsius) 1 0 0 1 2 3 4 5 6 7 8 G9 -1 -2 107

16. Playing with Numbers Divisibility Test Learning Outcome By the end of this chapter, you will be: • Learn and understand the tricks • Recollect and use divisibility of tests Concept Map Proofs for Irrationality of Numbers Key Points Missing Numbers • Types of Numbers o Natural numbers: The group of the positive Solution: numbers which are countable are known as The given number is 34. natural numbers. On reversing it we have 43. The examples of natural numbers will be 1, 2, 3, Now, let us add both the numbers, so we have 34 + 43 4, 5,……….. etc. o Whole numbers: The group of natural numbers = 77. with inclusion of zero in it, are known as whole Dividing 77 by 11 will result into a zero remainder. numbers. • For a given two digit number, reverse the number The examples of whole numbers will be 0, 1, 2, 3, 4, 5,…….. etc. and subtract it from original number. Then o Integers: The group of positive and negative on dividing it by 9, will always result in a zero numbers along with zero are known as integer remainder. numbers. Example: Reversing two digit trick and dividing by 9 The examples of integer numbers will be for 43. ........−3, −2, −1, 0, 1, 2, 3, 4, 5,...... etc. Solution: o Rational numbers: The numbers which can The given number is 43. On reversing it we have 34. be expressed as ratio of integers are known as Now, let us subtract both the numbers, so we have rational numbers. 43−34 = 9. The examples of rational numbers will be Dividing 9 by 9 will result into a zero remainder. 1 2 3 34 • For a given 3 digit number, reverse the number and , , − , , etc. subtract it from original number. Then on dividing 4 7 10 7 it by 99 will always result in a zero remainder. Example: Reversing three digit trick and dividing by • Any 2 - digit number ab can be written as 99 for 987. ab = 10 × a + b = 10a + b Solution: • Any 3 - digit number abc can be written as The given number is 987. On reversing it we have abc = 100 × a +10 × b + c = 100a + 10b + c 789. • For any given two digit number, reverse the number Now, let us subtract both the numbers, so we have and add it to the original number. Then on dividing 987−789 = 198. it by 11 will always result in a zero remainder. Dividing 198 by 99 will result into a zero remainder. Example: Reversing two digit trick and dividing by • Letters for Digits 11 for 34. (i)Each digit must stand for just one letter i.e. a single letter represents a single digit not multiple 108

16. Playing with Numbers digits. For any number to be the multiple of 3, the (ii) The digit represented by a letter cannot have its sum of its digit must also be multiple of 3. first digit as zero i.e. the first digit must be a non- Thus, 3 + 1 + z + 5 = 9 + z must be multiple zero number. of 3. Follow simple arithmetic and multiplication rules to find the answers. Here, z is a single digit number, now since z Example: Find the value of A for the following is added with 9, so any multiple of 3 can take addition. place i.e. z can be 3 or 6 or 9. Moreover, z can   A  2 also be 0 as 9 itself is multiple of 3. + A  9   3  A Hence, value of z can be 0, 3, 6 or 9. Answer: o Divisibility by 9: For any given number, if the We know that, here there is only one possible value of A. sum of its digits is divisible by 9, then that Hence, for the given addition to be true, A = 1 is the number will always be divisible by 9. only possible value. Example: Check divisibility by 9 test for 740.   1  2 Sum of digits of 740 is 7+4+0 = 11. Now, we + 1  9 know that 11 is not divisible by 9. Thus, 740 is   3  1 not divisible by 9 • Divisibility tests o Divisibility by 11: When the sum of odd digits o Divisibility by 10: Any number whose unit’s and even digits are same or they differ by a multiple of 11, then the given number is digit is ‘0’ will always be divisible by 10 divisible by 11. o Divisibility by 5: Any number whose unit’s digit Example: Check if 31267 is divisible by 11 3+2+7=12, 1+6=7 is either ‘0’ or ‘5’ will always be divisible by 5. 12−7 = 5, since it is not a multiple of 11, it is o Divisibility by 2: Any number whose unit’s digit not divisible by 11. is even (0, 2, 4, 6 and 8) will always be divisible by 2. o Divisibility by 3: For any given number, if the sum of its digits is divisible by 3, then that number will always be divisible by 3. Example 1: Divisibility by 3 test for 630 Sum of digits of 630 is 6+3+0 = 9. Now, we know that 9 is divisible by 3. Thus, 630 is divisible by 3 Example 2: If 31z5 is a multiple of 3, where z is a digit, what might be the values of z? 109

16. Playing with Numbers Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites Number System • Natural numbers, whole numbers, basic operations on natural numbers Tricks Divisibility test • Concept of two digit and three digit number. PS−1 • Use the properties to find the missing term • Use divisibility test to find the missing term in the numbers given • Basic multiplication and addition to find missing terms Worksheet for “Playing with Numbers” PS−2 Evaluation with Self Check or ---- Self Evaluation Peer Check* Sheet 110

PRACTICE SHEET - 1 (PS-1) 1. Generalised form of a 2-digit number is? ___________ 2. A 4-digit number aabb is divisible by 55. Then, possible value(s) of b are________. 3. If abc is a 3-digit number, then number abc -a-b-c is divisible by________. 4. If the sum of digits of a number is divisible by three, then the number is always divisible by ? 5. The sum of a 2-digit number and the number obtained by reversing the digits is always divisible by_______________ 6. The difference of 2-digit number and the number obtained by reversing its digits is always divisible by_______________ 7. If AB × 4 = 192, find A + B. 8. Check if 3n+2 is divisible by 3? PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. 1. Which of these is the general form of a 2-digit number? (A) ab = 10a + b (B) ab = 10b + a (C) ba = 10a + b (D) ab =10a + 10b 2. The sum of a 2-digit number and the number obtained by reversing its digits is a ____________. (A) multiple of 9 (B) multiple of 10 (C) multiple of 5 (D) multiple of 11 3. Identify the number that divides 22812 exactly. (A) 2 (B) 3 (C) 5 (D) 9 4. X and Y in 2XY7 are distinct numbers. If it is exactly divisible by 3, which of the following is the least value of (X + Y)? (A) 6 (B) 0 (C) 3 (D) 9 5. The 3-digit number 24N is exactly divisible by 5. What is the value of the digit N? (A) 0 or 5 (B) 5 (C) 4 (D) 1 6. M is a number such that M ÷ 5 gives a remainder of 2. Which of the following is the one’s digit of M? (A) 1 (B) 2 or 7 (C) 7 (D) 3 7. The value of A in the product 7A × 6 = AAA is __________. (A) 2 (B) 3 (C) 4 (D) 6 8. The number 100a + 10b + c is a ____ number. (A) 4 – digit (B) 2 – digit (C) 1 – digit (D) 3 – digit 9. Which statement is correct? (A) 32745 is divisible by 9. (B) 5176 is divisible by 4. (C) 200241 is divisible by 3 and 9. (D) 3263 is divisible by 10. 10. The number 496 expressed in the form 10a + b. The values of ‘a’ and ‘b’ respectively are__________. (A) 49 and 6 (B) 4 and 96 (C) 6 and 40 (D) 40 and 6 II. Short Answer Questions. 11. Find the least value of ‘p’ if 16p6 is a multiple of 3. 12. The product of two non- zero 1-digit numbers is a 1- digit number, while their sum is a 2 – digit num- ber. Find the numbers. 13. A number is such that its cube is the same as the number itself, but its square is not. Find the number. III. Long Answer Questions. 14. If 531z is a multiple of 3, find the value of the digit ‘z’. 15. A 2-digit number is such that it exceeds the sum of its digits by 18. Also, the digit in its ones place is twice the digit in its tens place. Find the number. 111

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. What is the divisibility test for 11? (1 Mark) 5. If 85 +4A= BC3, find A, B, C. (2 Marks) 2. If 20x3 is a multiple of 3, find x. (2 Marks) 6. If 1AB + CCA = 697 and there is no carry-over in addition, find the value of A + B + C. (3 Marks) 3. If b × b = ab, find a and b. (1 Mark) 7. If 213a27 is divisible by 9, then find the (3 Marks) value of ‘a’ 4. If 1p × p = q6, where q−p = 3, find the value of p and q. (3 Marks) 112

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