## Forum Posts

Maths Platter

Mar 10, 2022

In Examples Permutation & Comb...

Question: Simplify [4 (45!) + 46!] / [150 (44!)] Question: Prove [15! / (5! x 10!)] + [15! / (4! x 11!)] = 16! / (5! x 11!) Question: Find 'n' if (16 - n)! / (14 - n)! = 12 Question: Find the LCM of 6!, 7! and 8! Question: Prove 1/9! + 1/10! + 1/11! = 122/11! Question: Simplify (10! - 9!) / 8! Question: If 1/4! + 1/5! = x/6!, find x? Question: Find x if x/10! = 1/8! + 1/9! Question: If (n + 2)! = 2550 (n!), find n? Question: Prove that n!(n + 2) = n! + (n + 1)! Question: Find the value of n if [2n! / 3! (2n - 3)!] : [n! / 2!(n - 2)!] = 44 : 3

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Maths Platter

Mar 09, 2022

In Concepts Permutation & Comb...

Learning Objective: What is Factorial? What is n x (n - 1)! Concept: The product of 'n' natural numbers is called factorial. For example if we multiply first 5 natural numbers, we obtain 5 factorial and if we multiply first 10 natural numbers, we obtain 10 factorial. Factorial is denoted as "!". So the product of n natural numbers is n! (read as n factorial). For example 1 x 2 x 3 .....n = n! For example (Reverse Order) n x (n-1) x (n-2) ..... x 3 x 2 x 1 = n! For example 1 x 2 x 3 x 4 x 5 = 5! For example (Reverse Order) 5 x 4 x 3 x 2 x 1 = 5! For example 1 x 2 x 3 x 4 x 5 x 6 = 6! For example (Reverse Order) 6 x 5 x 4 x 3 x 2 x 1 = 6! Important factorials to remember 6! = 720 5! = 120 4! = 24 3! = 6 2! = 2 1! = 1 0! = 1 Property of Factorials n x (n - 1)! = n! For example if we multiply 5 with 4! [5 x 4!] or [5 x 4 x 3 x 2 x 1], we obtain 5! For example if we multiply 6 with 5! [6 x 5!] or [6 x 5 x 4 x 3 x 2 x 1], we obtain 6! Video: What is Factorial?

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Maths Platter

Mar 02, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to finding values in ratio? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 12, CA Foundaton Study Material Question: If a : b = 3 : 4, the value of (2a+3b) : (3a+4b) is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 54 : 25 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 8 : 25 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 17 : 24 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : 18 : 25 Solution Video: Exercise 1A, Question 19, CA Foundaton Study Material Question: If x : y = 3 : 4, the value of x²y + xy² : x³ + y³ is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (13 : 12) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (12 : 13) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (21 : 31) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 22, CA Foundaton Study Material Question: If p : q = 2 : 3 and x : y = 4 : 5, then the value of 5px + 3qy : 10px + 4qy is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 71 : 82 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 27 : 28 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 17 : 28 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 02, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to ratios and math operations? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 13, CA Foundaton Study Material Question: Two numbers are in the ratio 2 : 3. If 4 be subtracted from each, they are in the ratio 3 : 5. The numbers are … ? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (16 , 24) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (4 , 6) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (2 , 3) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 17, CA Foundaton Study Material Question: The ratio of two numbers is 7 : 10 and their difference is 105. The numbers are … ? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (200, 305) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (185, 290) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (245, 350) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 23, CA Foundaton Study Material Question: The number which when subtracted from each of the terms of the ratio 19 : 31 reducing it to 1 : 4 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 15 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 02, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to ratios and angles? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 14, CA Foundaton Study Material Question: The angles of a triangle are in ratio 2 : 7 : 11. The angles are …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (20°, 70°, 90°) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (30°, 70°, 80°) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (18°, 63°, 99°) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 02, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to ratios and earnings? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 18, CA Foundaton Study Material Question: P, Q and R are three cities. The ratio of average temperature between P and Q is 11 : 12 and that between P and R is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (22 : 27) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (27 : 22) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (32 : 33) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 02, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to ratios and speed? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 25, CA Foundaton Study Material Question: The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 kms. in 5 hours, the speed of …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 10 Km/hr 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 50 Km/hr 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 70 Km/hr 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 01, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to solve word problem relating to ratios and earnings? Concept: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Both the terms of a ratio can be multiplied or divided by the same non-zero number. Usually a ratio is expressed in lowest terms. The order of the terms in a ratio is important. Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units. To compare two ratios, convert them into equivalent like fractions. Exercise 1A, Question 15, CA Foundaton Study Material Question: Division of ₹ 324 between X and Y is in the ratio 11 : 7. X & Y would get Rupees …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (204, 120) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (200, 124) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (180, 144) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 16, CA Foundaton Study Material Question: Anand earns ₹ 80 in 7 hours and Pramod ₹ 90 in 12 hours. The ratio of their earnings is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (32 : 21) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (23 : 12) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (8 : 9) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 24, CA Foundaton Study Material Question: Daily earnings of two persons are in the ratio 4:5 and their daily expenses are in the ratio 7 : 9. If each saves ₹ 50 per day, their daily incomes in ₹ are...? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : (40, 50) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : (50, 40) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : (400, 500) 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 01, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to find sub triplicate ratio? Concept: The sub-triplicate ratio of a : b is ³√a : ³√b Exercise 1A, Question 8, CA Foundaton Study Material Question: The sub-triplicate ratio of 8 : 27 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 27 : 8 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 24 : 81 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 2 : 3 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 11, CA Foundaton Study Material Question: The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and …. 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 4 : 512 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 3 : 32 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 : 12 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 01, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to find triplicate ratio? Concept: The triplicate ratio of a : b is a³ : b³ Exercise 1A, Question 7, CA Foundaton Study Material Question: The triplicate ratio of 2 : 3 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 8 : 27 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 6 : 9 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 3 : 2 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 10, CA Foundaton Study Material Question: The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 2 : 7 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 7 : 2 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 2 : 21 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 11, CA Foundaton Study Material Question: The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and …. 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 4 : 512 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 3 : 32 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 : 12 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Mar 01, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to find sub-duplicate ratio? Concept: The sub-duplicate ratio of a : b is √a : √b Exercise 1A, Question 6, CA Foundaton Study Material Question: The sub-duplicate ratio of 25 : 36 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 6 : 5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 36 : 25 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 50 : 72 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : 5 : 6 Solution Video: Exercise 1A, Question 11, CA Foundaton Study Material Question: The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and …. 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 4 : 512 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 3 : 32 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 : 12 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 20, CA Foundaton Study Material Question: If p : q is the sub-duplicate ratio of p - x² : q - x² then x² is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : p / p + q 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : q / p + q 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : pq / p + q 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Feb 28, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to find compound ratio? Concept: The duplicate ratio of a : b is a² : b² Exercise 1A, Question 5, CA Foundaton Study Material Question: The duplicate ratio of 3 : 4 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : √3 : 2 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 4 : 3 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 9 : 16 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 9, CA Foundaton Study Material Question: The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 1 : 4 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 1 : 3 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 3 : 1 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 10, CA Foundaton Study Material Question: The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 2 : 7 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 7 : 2 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 2 : 21 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 11, CA Foundaton Study Material Question: The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and …. 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 4 : 512 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 3 : 32 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 : 12 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 21, CA Foundaton Study Material Question: If 2s : 3t is the duplicate ratio of 2s – p : 3t – p then …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : p² = 6st 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : p = 6st 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 2p = 3st 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Feb 28, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: How to find compound ratio? Concept: The ratio compounded of the two ratios a : b and c : d is ac : bd. Exercise 1A, Question 1, CA Foundaton Study Material Question: The ratio compounded of 2 : 3, 9 : 4, 5 : 6 and 8 : 10 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 1 : 1 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 1 : 5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 3 : 8 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 9, CA Foundaton Study Material Question: The ratio compounded of 4 : 9 and the duplicate ratio of 3 : 4 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 1 : 4 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 1 : 3 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 3 : 1 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 10, CA Foundaton Study Material Question: The ratio compounded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3 and 9 : 7 is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 2 : 7 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 7 : 2 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 2 : 21 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 11, CA Foundaton Study Material Question: The ratio compounded of duplicate ratio of 4 : 5, triplicate ratio of 1 : 3, sub duplicate ratio of 81 : 256 and …. 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 4 : 512 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 3 : 32 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 1 : 12 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Feb 28, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: What is antecedent and consequent in ratios? Concept: If p and q are two quantities of the same kind, then the fraction p/q is called the ratio of p to q. The quantities p and q are called terms of the ratio. The first term p is called antecedent. The second term q is called consequent. Exercise 1A, Question 2, CA Foundaton Study Material Question: The ratio of two quantities is 3 : 4. If the antecedent is 15, the consequent is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 16 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : 60 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 22 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : 20 Solution Video: Exercise 1A, Question 3, CA Foundaton Study Material Question: The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : √5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 7 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Feb 28, 2022

In Ratio

Course: CA Foundation; Topic: Ratio Learning Objective: What is an inverse ratio? Concept: The inverse ratio of a : b is b : a Exercise 1A, Question 1, CA Foundaton Study Material Question: The inverse ratio of 11 : 15 is 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 15 : 11 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : √11 : √15 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 121 : 125 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video: Exercise 1A, Question 3, CA Foundaton Study Material Question: The ratio of the quantities is 5 : 7. If the consequent of its inverse ratio is 5, the antecedent is …? 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : 5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : √5 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : 7 𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these Solution Video:

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Maths Platter

Feb 25, 2022

In Examples Skewness & Kurtosis

Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution? Given: μ₂ = 16 μ₄ = 1024 Question: Find μ₂ for a distribution? Given: The distribution is mesokurtic μ₄ = 108 Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution? Given: The first four moments about 4 are 1, 4, 10 and 46 respectively. Solution Video:

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Maths Platter

Feb 21, 2022

In 💡 SKEWNESS AND KURTOSIS

Learning Objective: What is Skewness? What are measures of Skewness? What does Kurtosis relate to? How to measure Kurtosis? Concept: Skewness relates to asymmetry of a frequency distribution. Skewness can be of two types. If there is a positive tail in a frequency distribution, it is termed as postive skewness. If there is a negative tail in a frequency distribution, is it termed as negative skeness. Measures of Skewness can be Absolute or Relative. Absolute measure of skewness cannot compare series with different units. It is not widely used. Relative measure of skewness are of 4 types. Karl Pearson's Coefficeint of Skewness based on Mean, Median, Mode and Standard Deviation. Bowley's Coefficient of Skewness based on Quartiles. Kelly's Coefficient of Skewenss based on upper and lower Deciles and Median. Pearsonian's Coefficient of Skewness based on central Moments. If the value of above 4 relative measures of skewness is 0, the distribution is symmetric. If the value of above 4 relative measures of skewness is greater then 0, the distribution is positively skewed. If the value of above 4 relative measures of skewness is less then 0, the distribution is negatively skewed. In symmetric distribution, x̄ (Mean) = Median = Mode In positive skewness, x̄ (Mean) > Median > Mode In negative skewness, x̄ (Mean) < Median < Mode Kurtosis relates to bulginess, peakedness, tailedness of a frequency distribution. Leptokurtic frequency distribution is narrow with sharp peak and extended tails. Mesokurtic frequency distribution is normal with less sharp peak and not much extended tails. Platykurtic frequency distribution is like platypus without sharp peak and very less extended tails. Kurtosis is measured with Pearsonian's Coefficient of Kurtosis which is based on central moments. If γ₂ > 0, the frequency distribution is Leptokutic. If γ₂ = 0, the frequency distribution is Mesokurtic. If γ₂ < 0, the frequency distribution is Platukurtic. Formulas: Absolute Measure of Skewness = Sₖ = Mean - Mode Karl Pearson's Coefficient of Skewness = [Mean - Mode] / Standard Deviation Karl Pearson's Coefficient of Skewness = 3 x [Mean - Median] / Standard Deviation Bowley's Coefficient of Skewness = [Q₃ + Q₁ - 2Q₂] / [Q₃ - Q₁] Kelly's Coefficient of Skewness = [D₉ - D₁ - 2Q₂] / [D₉ - D₁] Pearsonian's Coefficient of Skewness = β₁ = μ₃² / μ₂³; γ₁ = ± √β₁; (sign of μ₃) Pearsonian's Coefficient of Kurtosis β₂ = μ₄ / μ₂² γ₂ = (β₂ - 3) Part - 1: Skewness and Measures of Skewness Concept Recap: Skewness relates to asymmetry of a frequency distribution. Skewness can be of two types. If there is a positive tail in a frequency distribution, it is termed as postive skewness. If there is a negative tail in a frequency distribution, is it termed as negative skeness. Measures of Skewness can be Absolute or Relative. Absolute measure of skewness cannot compare series with different units. It is not widely used. Relative measure of skewness are of 4 types. Karl Pearson's Coefficeint of Skewness based on Mean, Median, Mode and Standard Deviation. Bowley's Coefficient of Skewness based on Quartiles. Kelly's Coefficient of Skewenss based on upper and lower Deciles and Median. Pearsonian's Coefficient of Skewness based on central Moments. If the value of above 4 relative measures of skewness is 0, the distribution is symmetric. If the value of above 4 relative measures of skewness is greater then 0, the distribution is positively skewed. If the value of above 4 relative measures of skewness is less then 0, the distribution is negatively skewed. In symmetric distribution, x̄ (Mean) = Median = Mode In positive skewness, x̄ (Mean) > Median > Mode In negative skewness, x̄ (Mean) < Median < Mode Formulas: Absolute Measure of Skewness = Sₖ = Mean - Mode Karl Pearson's Coefficient of Skewness = [Mean - Mode] / Standard Deviation Karl Pearson's Coefficient of Skewness = 3 x [Mean - Median] / Standard Deviation Bowley's Coefficient of Skewness = [Q₃ + Q₁ - 2Q₂] / [Q₃ - Q₁] Kelly's Coefficient of Skewness = [D₉ - D₁ - 2Q₂] / [D₉ - D₁] Pearsonian's Coefficient of Skewness = β₁ = μ₃² / μ₂³; γ₁ = ± √β₁; (sign of μ₃) Part - 2: Kurtosis and Measures of Kurtosis Concept Recap: Kurtosis relates to bulginess, peakedness, tailedness of a frequency distribution. Leptokurtic frequency distribution is narrow with sharp peak and extended tails. Mesokurtic frequency distribution is normal with less sharp peak and not much extended tails. Platykurtic frequency distribution is like platypus without sharp peak and very less extended tails. Kurtosis is measured with Pearsonian's Coefficient of Kurtosis which is based on central moments. If γ₂ > 0, the frequency distribution is Leptokutic. If γ₂ = 0, the frequency distribution is Mesokurtic. If γ₂ < 0, the frequency distribution is Platukurtic. Formulas Pearsonian's Coefficient of Kurtosis β₂ = μ₄ / μ₂² γ₂ = (β₂ - 3) Part - 3: Solved Examples Question: Find Karl Pearson's Coefficient of Skweness? Given: Mean = 100 Mode = 80 Standard Deviation = 20 Question: Find Karl Pearson's Coefficient of Skweness? Given: Mean = 60 Median = 75 Variance = 900 Question: Find Mode and Median? Given: Mean = 50 Variance = 400 Karl Pearson's Coefficient of Skweness = -0.4 Part - 4: Solved Examples Question: Find Bowley's Coefficient of Skewness? Given: Lower Quartile for a distribution is 15 Upper Quartile for a distribution is 21 Median is 17 Question: Find the median? Given: Bowley's Coefficient of Skewness = -0.8 Q₁ = 44.1 Q₃ = 56.6 Question: Find Bowley's Coefficient of Skewness? Given: Q₃ - Q₂ = 100 Q₂ - Q₁ = 120 Part - 5: Solved Examples Question: Find Pearsonian's Coefficient of Skewness (γ₁) for a distribution? Given: μ₂ = 25 μ₃ = 100 Question: Find μ₃ for a distribution? Given: Standard Deviation = 4 Pearsonian's Coefficient of Skewness (γ₁) = 1 Question: Find Pearsonian's Coefficient of Skewness (γ₁) for a distribution? Given: The first three moments about 2 are 1, 16 and -40 respectively. Part - 6: Solved Examples Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution? Given: μ₂ = 16 μ₄ = 1024 Question: Find μ₂ for a distribution? Given: The distribution is mesokurtic μ₄ = 108 Question: Find Pearsonian's Coefficient of Kurtosis (γ₂) for a distribution? Given: The first four moments about 4 are 1, 4, 10 and 46 respectively. Part - 7: Solved Example Question: Find Karl Pearson's Coefficient of Skweness for a distribution? Given: Mean = 160 Mode = 157 Standard Deviation = 50 Part - 8: Solved Example Question: Find the mode and median of a frequency distribution? Given: Mean = 40 Variance = 625 Pearsonian's Coefficient of Skewness (Sₖₚ) = -0.2 Part - 9: Solved Example Question: Find the Coefficient of Skewness? Given: Sum of upper and lower quartiles is 200 Difference of upper and lower quartiles is 20 Median is 100

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Maths Platter

Feb 21, 2022

In 💡 THEORY OF ATTRIBUTES

Learning Objective: What are characteristics in Theory of Attributes? What are the types of Dichotomous Classification in Theory of Attributes? What are Attribute Classes? What are Attribute Frequencies? What are Contingency Table? Using Contingency Table. Consistency of Data. What is Independence of Attributes? What is Association of Attributes? What are methods to study Independence and Association of Attributes? How to construct Contingency Table for 3 Attributes? What is Coefficient of Colligation? Concepts Measurable characteristics can be termed as Quantitative Characteristics. Quantitative Characteristics are called as Variables. Non-Measurable characteristics can be termed as Qualitative Characteristics. Qualitative Characteristics can be called as Attributes. Attributes types can be Positive or Negative. Positive Attributes are denoted as "A", "B", "C".... Negative Attributes are denoted as "α", "β", "γ".... Attributes themselves represent a Group. Such groups can be termed as "Classes" Positive Attributes can have Positive Frequencies and Negative Attributes can have Negative Frequencies. Invalid classes can be "A" and "α"; "B" and "β"; "C" and "γ" Relationship of Frequency can be studied in the Contingency Table. Contingency Table can be for 2 OR 3 Attributes. Contingency Table for 2 Attributes is also termed as 9 Square Table or 2 x 2 Table. Any class frequencies cannot be greater than total population. Hence each class frequency must be less than or equal to "N" Any class frequencies cannot be negative. Hence each class frequency must be greater than or equal to 0. 9 square table can be used to check consistency of data for 2 attributes. Two attributes are said to be independent if there does not exist any kind of relation between them. Two attributes are said to be associated if they are not independent i.e. they are related in some way or another. Association of Attributes can be Positive or Negative. When two attributes are present or absent together in the data then the attributes are positively associated. When the presence of one attribute is associated with the absence of the other attribute then the attributes are negatively associated. Yule computed another coefficient called coefficient of colligation (γ) Yule's coefficient of colligation is generally restricted to 2 x 2 tables. Formulas Proportion Method Attributes are independent when (AB)/(B) = (Aβ)/(β) Attributes are positively associated when (AB)/(B) > (Aβ)/(β) Attributes are negatively associated when (AB)/(B) < (Aβ)/(β) If A and B are independent then (A and β), (α and B) and (α and β) are also independent. Comparison Method Attributes are independent when (AB) = (A) x (B) / N Attributes are positively associated when (AB) > (A) x (B) / N Attributes are negatively associated when (AB) < (A) x (B) / N Yule's Coefficient of Association Q🇦🇧 = [(AB) (αβ) - (Aβ) (αB)] / [(AB) (αβ) + (Aβ) (αB)] The value of Yule's Coefficient of Association lies between -1 and 1. If the value of Yule's Coefficient is 0, the attributes are independent. If the value of Yule's Coefficient lies between 0 and 1, the attributes are positively associated. If the value of Yule's Coefficient lies between 0 and 0.5, the attributes are weakly positively associated. If the value of Yule's Coefficient lies between 0.5 and 1, the attributes are strongly positively associated. If the value of Yule's Coefficient lies between -1 and 0, the attributes are negatively associated. If the value of Yule's Coefficient lies between 0 and -0.5, the attributes are weakly negatively associated. If the value of Yule's Coefficient lies between -0.5 and -1, the attributes are strongly negatively associated. If the value of Yule's Coefficient is 1, the attributes are completely associated. If the value of Yule's Coefficient is -1, the attributes are completely disassociated. Part - 1: Introduction Theory of Attributes Concept Recap: Measurable characteristics can be termed as Quantitative Characteristics. Quantitative Characteristics are called as Variables. Non-Measurable characteristics can be termed as Qualitative Characteristics. Qualitative Characteristics can be called as Attributes. Part - 2: Dichotomous Classification and Types of Attributes Concept Recap: Attributes types can be Positive or Negative. Positive Attributes are denoted as "A", "B", "C".... Negative Attributes are denoted as "α", "β", "γ".... Part - 3: Attributes Classes & Frequency and Contingency Table for 2 Concept Recap: Attributes can be Positive or Negative. Positive Attributes can be represented by "A", "B", "C".... Negative Attributes can be represented by "α", "β", "γ".... Attributes themselves represent a Group. Such groups can be termed as "Classes" Positive Attributes can have Positive Frequencies and Negative Attributes can have Negative Frequencies. Invalid classes can be "A" and "α"; "B" and "β"; "C" and "γ" Relationship of Frequency can be studied in the Contingency Table. Contingency Table can be for 2 OR 3 Attributes. Contingency Table for 2 Attributes is also termed as 9 Square Table or 2 x 2 Table. Part - 4: Contingency Table and Consistency of Data Concept Recap: Any class frequencies cannot be greater than total population. Hence each class frequency must be less than or equal to "N" Any class frequencies cannot be negative. Hence each class frequency must be greater than or equal to 0. 9 square table can be used to check consistency of data for 2 attributes. Part - 5: Independence and Association of Attributes Concept Recap: Two attributes are said to be independent if there does not exist any kind of relation between them. Two attributes are said to be associated if they are not independent i.e. they are related in some way or another. Association of Attributes can be Positive or Negative. When two attributes are present or absent together in the data then the attributes are positively associated. When the presence of one attribute is associated with the absence of the other attribute then the attributes are negatively associated. Part - 6: Methods to study Independence and Association of Attributes Formulas to Study Independence and Association of Attributes: Proportion Method Attributes are independent when (AB)/(B) = (Aβ)/(β) Attributes are positively associated when (AB)/(B) > (Aβ)/(β) Attributes are negatively associated when (AB)/(B) < (Aβ)/(β) If A and B are independent then (A and β), (α and B) and (α and β) are also independent. Comparison Method Attributes are independent when (AB) = (A) x (B) / N Attributes are positively associated when (AB) > (A) x (B) / N Attributes are negatively associated when (AB) < (A) x (B) / N Yule's Coefficient of Association Q🇦🇧 = [(AB) (αβ) - (Aβ) (αB)] / [(AB) (αβ) + (Aβ) (αB)] The value of Yule's Coefficient of Association lies between -1 and 1. If the value of Yule's Coefficient is 0, the attributes are independent. If the value of Yule's Coefficient lies between 0 and 1, the attributes are positively associated. If the value of Yule's Coefficient lies between 0 and 0.5, the attributes are weakly positively associated. If the value of Yule's Coefficient lies between 0.5 and 1, the attributes are strongly positively associated. If the value of Yule's Coefficient lies between -1 and 0, the attributes are negatively associated. If the value of Yule's Coefficient lies between 0 and -0.5, the attributes are weakly negatively associated. If the value of Yule's Coefficient lies between -0.5 and -1, the attributes are strongly negatively associated. If the value of Yule's Coefficient is 1, the attributes are completely associated. If the value of Yule's Coefficient is -1, the attributes are completely dissassociated. Part - 7: Contingency Table for 3 Attributes Part - 8: Understanding the frequencies of Contingency Table for 3 Attributes Part - 9: Obtaining missing frequencies using Contingency Table for 3 Attributes Question: Find the missing frequencies using contingency table? Given: (ABC) = 26 (AβC) = 18 (ABγ) = 38 (Aβγ) = 12 (αBC) = 12 (αBC) = 12 (αβC) = 6 (αBγ) = 16 (αβγ) = 9 Part - 10: Coefficient of colligation and Coefficient of contingency Concept Recap Yule computed another coefficient called coefficient of colligation (γ) Yule's coefficient of colligation is generally restricted to 2 x 2 tables. Part - 11: Solved Examples Question: Find the missing frequencies? Given: (αB) = 500 (β) = 600 (α) = 800 (B) = 1000 Part - 12: Solved Examples Question: Test for consistency of data when N = 1000, (A) = 150, (B) = 300 and (AB) = 200. Question: In a survey of 500 persons, 300 were married and 250 were successful executives. 198 successful executives were married. Is the data consistent? Part - 13: Solved Examples Question: State the type of association between the attributes when N = 126, (A) = 90, (B) = 56 and (AB) = 40. Question: State the type of association between the attributes when N = 180, (A) = 160, (β) = 130 and (αβ) = 6. Part - 14: Solved Examples Question: Find the Yule's coefficient of association when N = 35, (A) = 18, (AB) = 8 and (αβ) = 5. Question: Find the extent of association between intelligence of husband and the intelligence of wife when Intelligent husbands with intelligent wives = 50 Intelligent husbands with dull wives = 200 Dull husbands with intelligent wives = 100 Dull husbands with dull wives = 300 Part - 15: Solved Examples Question: Find the missing frequencies using contingency table? Given: (ABC) = 210 (αBC) = 280 (ABγ) = 180 (αBγ) = 240 (AβC) = 250 (αβC) = 160 (Aβγ) = 360 (αβγ) = 32

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