Ratio Proportion & Variation

Core Formula

Variation describes how one quantity changes in relation to another. For SSC CGL, understanding direct and inverse variation is crucial for quick problem-solving.

1. Direct Variation: When two quantities, say x and y, vary directly, it means that as x increases, y increases proportionally, and vice-versa. Their ratio remains constant. Formula: y ∝ x or y = kx Where k is the constant of proportionality.

2. Inverse Variation: When two quantities, say x and y, vary inversely, it means that as x increases, y decreases proportionally, and vice-versa. Their product remains constant. Formula: y ∝ 1/x or y = k/x Where k is the constant of proportionality.

Worked Example 1 (Direct Variation)

Question: If A varies directly as B, and A = 15 when B = 5, what is the value of A when B = 8?

Solution:

1. Identify Variation Type: Direct Variation (A ∝ B).
2. Write Formula: A = kB
3. Find 'k': Use the given values (A=15, B=5): 15 = k * 5 k = 15 / 5 = 3
4. Solve for New Value: Use k=3 and B=8: A = 3 * 8 A = 24

Worked Example 2 (Inverse Variation)

Question: If X varies inversely as Y, and X = 12 when Y = 4, what is the value of X when Y = 6?

Solution:

1. Identify Variation Type: Inverse Variation (X ∝ 1/Y).
2. Write Formula: X = k/Y
3. Find 'k': Use the given values (X=12, Y=4): 12 = k / 4 k = 12 * 4 = 48
4. Solve for New Value: Use k=48 and Y=6: X = 48 / 6 X = 8

Shortcuts & Tricks

For competitive exams, avoid finding 'k' explicitly every time. Use the ratio method:

• Direct Variation: If y1 = kx1 and y2 = kx2, then y1/x1 = y2/x2. This means y1/y2 = x1/x2.
  • Example 1 Shortcut: A1/B1 = A2/B2 => 15/5 = A2/8 => 3 = A2/8 => A2 = 24.
• Inverse Variation: If y1 = k/x1 and y2 = k/x2, then y1x1 = y2x2.
  • Example 2 Shortcut: X1Y1 = X2Y2 => 12 * 4 = X2 * 6 => 48 = 6X2 => X2 = 8.

Common Mistakes

1. Misidentifying Variation Type: Confusing direct with inverse variation, leading to incorrect formula application.
2. Calculation Errors: Simple arithmetic mistakes when finding 'k' or solving for the unknown value.
3. Ignoring Units: While less common in basic variation problems, always be mindful of units in more complex applications.
4. Not Using Shortcuts: Wasting time by always calculating 'k' when a direct ratio/product method is faster.

Ratio & Proportion is fundamental for SSC CGL, forming the base for many other topics like mixtures, ages, and time & work. Speed and accuracy here are critical.

Core Formulas

1. Ratio: A comparison of two quantities of the same unit. Represented as a:b or a/b. Always simplify to lowest terms.

   • Duplicate Ratio: a^2:b^2
   • Sub-duplicate Ratio: √a:√b
   • Triplicate Ratio: a^3:b^3
   • Sub-triplicate Ratio: ∛a:∛b
   • Compounded Ratio: The ratio of the product of antecedents to the product of consequents. For a:b, c:d, e:f, the compounded ratio is (a*c*e) : (b*d*f) or (a/b) * (c/d) * (e/f).

2. Proportion: An equality of two ratios. If a:b = c:d, then a, b, c, d are in proportion, written as a:b::c:d. This implies ad = bc (product of extremes = product of means).

   • Mean Proportional between a and b: √(ab)
   • Third Proportional to a and b: b^2/a (assuming a:b::b:x, where x is the third proportional)
   • Fourth Proportional to a, b, c: bc/a (assuming a:b::c:x, where x is the fourth proportional)

3. Componendo & Dividendo (C&D): If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). This is a powerful shortcut for equations involving fractions.

Worked Example 1

Q: If A:B = 3:4 and B:C = 8:9, find A:B:C and the compounded ratio of A:B and B:C. Solution:

1. Find A:B:C: To combine ratios, make the common term (B) equal. A:B = 3:4 (multiply by 2) => A:B = 6:8 B:C = 8:9 So, A:B:C = 6:8:9
2. Compounded Ratio: (A/B) * (B/C) = (3/4) * (8/9) = (3*8)/(4*9) = 24/36 = 2/3. So, 2:3.

Worked Example 2

Q: Find the mean proportional between 16 and 25, and the fourth proportional to 4, 12, and 15. Solution:

1. Mean Proportional: For 16 and 25, √(16 * 25) = √(400) = 20.
2. Fourth Proportional: For 4, 12, 15, let the fourth proportional be x. 4:12::15:x => 4x = 12 * 15 => x = (12 * 15) / 4 => x = 3 * 15 = 45.

Shortcuts & Tricks

• Combining Ratios (A:B, B:C): Write A:B:C. Multiply A by B's second value, B by both B values, C by B's first value. Example: A:B = 2:3, B:C = 4:5. A:B:C = (2*4) : (3*4) : (3*5) = 8:12:15.
• Direct Proportional Formulas: Memorize √(ab), b^2/a, bc/a for mean, third, and fourth proportional respectively. This saves significant time.
• Componendo & Dividendo: When you see (x+y)/(x-y) or similar structures, immediately think C&D. It simplifies complex fractional equations rapidly.

Common Mistakes

1. Confusing Proportional Types: Students often mix up mean, third, and fourth proportional formulas. Remember: Mean (two terms, middle term), Third (two terms, third term in continuous proportion), Fourth (three terms, fourth term).
2. Incorrectly Combining Ratios: Not making the common term equal before combining can lead to errors. Always ensure the value for the common element is consistent across all ratios.
3. Calculation Errors: Simple arithmetic mistakes in multiplication or division are common. Practice mental math and cross-check calculations, especially with large numbers or fractions.