Ratio and Proportion (விகிதம் மற்றும் விகிதாச்சாரம்)

விகிதம் மற்றும் விகிதாச்சாரம் (Ratio and Proportion)

Overview

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This category involves problems related to comparing quantities (ratios) and equating two ratios (proportions). The general approach is to understand the relationship between the given quantities and use formulas to find unknown values or combined ratios.

Definitions and Formulas (வரையறைகள் மற்றும் சூத்திரங்கள்)

1. Ratio (விகிதம்):

   • A ratio is a comparison of two numbers (or quantities) by division. The ratio of 'a' to 'b' is written as a:b.
   • In the ratio a:b, 'a' is the antecedent (முன்னிகழ்வு) and 'b' is the consequent (பின் நிகழ்வு).

2. Proportion (விகிதாச்சாரம்):

   • A proportion is an expression stating that two ratios are equal.
   • If a:b = c:d, we can write it as a:b::c:d.
   • The terms a and d are called the extremes (முனையுறுப்புகள்), and b and c are called the means (இடைநிலை உறுப்புகள்).
   • Fundamental Rule: Product of means = Product of extremes. $$a:b = c:d \implies bc = ad$$

3. Dividing a Quantity in a Ratio (கொடுக்கப்பட்ட எண்ணை விகிதத்தில் வகுத்தல்):

   • To divide a number A in the ratio a:b:
     • First Part = $A \times \frac{a}{a+b}$
     • Second Part = $A \times \frac{b}{a+b}$

4. Fourth Proportional (நான்காவது விகிதம்):

   • If a:b::c:d, then d is the fourth proportional to a, b, and c. $$d = \frac{bc}{a}$$

5. Third Proportional (மூன்றாம் விகிதம்):

   • If a:b::b:c, then c is the third proportional to a and b. $$c = \frac{b^2}{a}$$

6. Mean Proportional (சராசரி விகிதம்):

   • The mean proportional between a and b is x such that a:x::x:b. $$x^2 = ab \implies x = \sqrt{ab}$$

Example Problem (உதாரண கணக்கு)

Question: If $a:b = 5:9$ and $b:c = 4:7$, find $a:b:c$.

Solution:

1. Write down the given ratios:

   • $a:b = 5:9$
   • $b:c = 4:7$

2. Identify the common term: The common term is b. The values for b are 9 and 4.

3. Find the LCM (Least Common Multiple) of the values of the common term:

   • LCM of 9 and 4 is 36.

4. Adjust each ratio to make the common term equal to the LCM:

   • To make b equal to 36 in the first ratio, multiply by 4: $a:b = (5 \times 4) : (9 \times 4) = 20:36$
   • To make b equal to 36 in the second ratio, multiply by 9: $b:c = (4 \times 9) : (7 \times 9) = 36:63$

5. Combine the adjusted ratios:

   • Now that b is common, we can combine them: $a:b:c = 20:36:63$

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Practice Questions

1. a:b = 2:3, b:c = 4:5 எனில் a:b:c =

   • a) 9:11:15
   • b) 9:17:18
   • c) 12:15:25
   • d) 8:12:15

   Answer: d) 8:12:15

   Solution

   $$a:b = 2:3 b:c = 4:5$$

   To make the 'b' term common, we find the LCM of 3 and 4, which is 12.

   $$a:b = (2:3) \times 4 = (2 \times 4) : (3 \times 4) = 8:12 b:c = (4:5) \times 3 = (4 \times 3) : (5 \times 3) = 12:15$$

   Now, since b is 12 in both ratios:

   $$a:b:c = 8:12:15$$
   Why this question belongs to Ratio and Proportion

   This question involves combining two separate ratios (a:b and b:c) into a single continuous ratio (a:b:c), which is a fundamental problem type in this category.

2. a:b = 3:5, b:c = 4:9 எனில் a:b:c =

   • a) 10:20:43
   • b) 12:22:45
   • c) 12:20:45
   • d) 9:17:35

   Answer: c) 12:20:45

   Solution

   $$a:b = 3:5 b:c = 4:9$$

   LCM of b values (5 and 4) is 20.

   $$a:b = (3:5) \times 4 = (3 \times 4) : (5 \times 4) = 12:20 b:c = (4:9) \times 5 = (4 \times 5) : (9 \times 5) = 20:45$$

   Therefore:

   $$a:b:c = 12:20:45$$
   Why this question belongs to Ratio and Proportion

   This question uses keywords like a:b and b:c and asks to find the combined ratio a:b:c, a core concept of Ratio and Proportion.

3. a:b = 1:3, b:c = 2:5 எனில் a:b:c =

   • a) 2:6:15
   • b) 9:12:14
   • c) 8:10:15
   • d) 4: 7:10

   Answer: a) 2:6:15

   Solution

   $$a:b = 1:3 b:c = 2:5$$

   LCM of b values (3 and 2) is 6.

   $$a:b = (1:3) \times 2 = (1 \times 2) : (3 \times 2) = 2:6 b:c = (2:5) \times 3 = (2 \times 3) : (5 \times 3) = 6:15$$

   Therefore:

   $$a:b:c = 2:6:15$$
   Why this question belongs to Ratio and Proportion

   This question requires finding a continuous proportion from two given ratios, which is a classic problem in this topic.

4. a:b = 2:5, b:c = 3:8 எனில் a:b:c =

   • a) 4:10:15
   • b) 5:12:15
   • c) 6:15:35
   • d) 6:15:40

   Answer: d) 6:15:40

   Solution

   $$a:b = 2:5 b:c = 3:8$$

   LCM of b values (5 and 3) is 15.

   $$a:b = (2:5) \times 3 = (2 \times 3) : (5 \times 3) = 6:15 b:c = (3:8) \times 5 = (3 \times 5) : (8 \times 5) = 15:40$$

   Therefore:

   $$a:b:c = 6:15:40$$
   Why this question belongs to Ratio and Proportion

   This question involves the keyword a:b:c and the process of equating the middle term b to combine two ratios, a standard procedure in Ratio and Proportion problems.

5. a:b = 3:5, b:c = 3:7 எனில் a:b:c =

   • a) 8:12:35
   • b) 9:15:35
   • c) 9:11:15
   • d) 12:20:45

   Answer: b) 9:15:35

   Solution

   $$a:b = 3:5 b:c = 3:7$$

   LCM of b values (5 and 3) is 15.

   $$a:b = (3:5) \times 3 = (3 \times 3) : (5 \times 3) = 9:15 b:c = (3:7) \times 5 = (3 \times 5) : (7 \times 5) = 15:35$$

   Therefore:

   $$a:b:c = 9:15:35$$
   Why this question belongs to Ratio and Proportion

   This question involves keywords like a:b, b:c and asks to find the combined ratio a:b:c, a core concept of Ratio and Proportion.

6. a:b = 2:3, b:c = 4:5 மற்றும் c:d = 6:7 எனில் a🅱️c:d =

   • a) 16:24:30:35
   • b) 8:15:30:32
   • c) 12:25:30:36
   • d) 12:24:28:33

   Answer: a) 16:24:30:35

   Solution

   First, find a:b:c.

   $$a:b = 2:3 b:c = 4:5 a:b:c = 8:12:15$$

   Now, combine a:b:c = 8:12:15 with c:d = 6:7. The common term is c with values 15 and 6. LCM of 15 and 6 is 30.

   $$a:b:c = (8:12:15) \times 2 = 16:24:30 c:d = (6:7) \times 5 = 30:35$$

   Therefore:

   $$a:b:c:d = 16:24:30:35$$
   Why this question belongs to Ratio and Proportion

   This question extends the concept of combining two ratios to combining three ratios (a:b, b:c, c:d) to form a single continuous proportion (a:b:c:d).

7. a:b = 2:3, b:c = 5:8 மற்றும் c:d = 6:7 எனில் a🅱️c:d =

   • a) 10:14:15:18
   • b) 8:12:15:30
   • c) 10:15:24:28
   • d) 8:12:18:27

   Answer: c) 10:15:24:28

   Solution

   First, find a:b:c.

   $$a:b = 2:3 b:c = 5:8$$

   LCM of b values (3, 5) is 15.

   $$a:b = (2:3) \times 5 = 10:15 b:c = (5:8) \times 3 = 15:24 a:b:c = 10:15:24$$

   Now, combine with c:d = 6:7. Common term c has values 24 and 6. LCM is 24.

   $$a:b:c = 10:15:24 c:d = (6:7) \times 4 = 24:28$$

   Therefore:

   $$a:b:c:d = 10:15:24:28$$
   Why this question belongs to Ratio and Proportion

   This question involves keywords like a:b, b:c, c:d and asks to find the combined ratio a:b:c:d, a standard multi-step problem in this topic.

8. a:b = 4:5, b:c = 5:6 மற்றும் c:d = 2:3 எனில் a🅱️c:d =

   • a) 3:6:10:14
   • b) 3:8:10:12
   • c) 4:5:6:9
   • d) 4:6:2:3

   Answer: c) 4:5:6:9

   Solution

   First, a:b = 4:5 and b:c = 5:6. The term b is already common (5). So, a:b:c = 4:5:6.

   Now, combine with c:d = 2:3. The common term is c with values 6 and 2. LCM is 6.

   $$a:b:c = 4:5:6 c:d = (2:3) \times 3 = 6:9$$

   Therefore:

   $$a:b:c:d = 4:5:6:9$$
   Why this question belongs to Ratio and Proportion

   This question requires creating a continuous ratio a:b:c:d from three individual ratios, a fundamental skill in this category.

9. a:b = 5:6, b:c = 9:10 மற்றும் c:d = 4:7 எனில் a🅱️c:d =

   • a) 5:9:10:7
   • b) 15:18:20:35
   • c) 5:10:12:15
   • d) 15:18:24:35

   Answer: b) 15:18:20:35

   Solution

   First, find a:b:c.

   $$a:b = 5:6 b:c = 9:10$$

   LCM of b values (6, 9) is 18.

   $$a:b = (5:6) \times 3 = 15:18 b:c = (9:10) \times 2 = 18:20 a:b:c = 15:18:20$$

   Now, combine with c:d = 4:7. Common term c has values 20 and 4. LCM is 20.

   $$a:b:c = 15:18:20 c:d = (4:7) \times 5 = 20:35$$

   Therefore:

   $$a:b:c:d = 15:18:20:35$$
   Why this question belongs to Ratio and Proportion

   This problem involves forming a four-term proportion (a:b:c:d) by sequentially combining simpler ratios, which is a key application of ratio concepts.

10. a:b = 1:2, b:c = 2:4 மற்றும் c:d = 3:8 எனில் a🅱️c:d =

    • a) 3:6:12:32
    • b) 4:7:10:12
    • c) 1:2:3:8
    • d) 4:6:2:3

    Answer: a) 3:6:12:32

    Solution

    First, find a:b:c. b is already common (2). So, a:b:c = 1:2:4.

    Now, combine with c:d = 3:8. The common term is c with values 4 and 3. LCM is 12.

    $$a:b:c = (1:2:4) \times 3 = 3:6:12 c:d = (3:8) \times 4 = 12:32$$

    Therefore:

    $$a:b:c:d = 3:6:12:32$$
    Why this question belongs to Ratio and Proportion

    This question tests the ability to combine multiple ratios (a:b, b:c, c:d) into a single continuous ratio a:b:c:d, a core problem-solving technique in this topic.