Vyashtisamasthi — Part and Whole in Fractions and Ratios

In this tutorial, you'll learn about Vyashtisamasthi. We cover key concepts, practical examples, and best practices.

Vyashtisamasthi ("Part and whole") decomposes complex fractional and ratio problems into simpler parts, then recombines the results — the Vedic equivalent of divide-and-conquer for arithmetic.

The Sutra: Part and Whole

Vyashtisamashti states: every whole can be split into its constituent parts, and operations on the whole can be performed by operating on the parts and recombining.

In arithmetic, this applies to:

• Splitting fractions into partial sums
• Distributing ratios across multiple terms
• Decomposing mixed numbers for easier computation

Part-Whole Flow

flowchart TD
    A["Complex problem
e.g., 3²/5 × 2³/4"] --> B["Split into parts
integer + fraction"]
    B --> C["Operate on parts
separately"]
    C --> D["3 × 2 = 6
3 × 3/4 = 9/4
2/5 × 2 = 4/5
2/5 × 3/4 = 6/20"]
    D --> E["Combine partial
results"]
    E --> F["Final answer"]

    style A fill:#1a73e8,color:#fff,stroke:none
    style B fill:#34a853,color:#fff,stroke:none
    style E fill:#fbbc04,color:#333,stroke:none
    style F fill:#46bdc6,color:#fff,stroke:none

Worked Examples

Example 1: Mixed number multiplication

Compute: 3²/5 x 2³/4

Step 1: Convert to improper fractions. 3²/5 = 17/5. 2³/4 = 11/4.

Step 2: Multiply: 17/5 x 11/4 = 187/20.

Step 3: But using Vyashtisamasthi, split each into whole + fraction: (3 + 2/5)(2 + 3/4) = 3x2 + 3x3/4 + 2/5x2 + 2/5x3/4 = 6 + 9/4 + 4/5 + 6/20.

Step 4: Combine: 6 + 2.25 + 0.8 + 0.3 = 6 + 3.35 = 9.35.

Answer: 187/20 = 9.35

Check: 187/20 = 9.35

Example 2: Ratio distribution

Divide 240 in the ratio 3:5.

Step 1: Vyashtisamasthi: the whole is 240, the parts are in ratio 3:5. Sum of parts = 3 + 5 = 8.

Step 2: Each part of the ratio = (ratio / sum) x whole.

Step 3: Part 1 = 3/8 x 240 = 3 x 30 = 90.

Step 4: Part 2 = 5/8 x 240 = 5 x 30 = 150.

Answer: 90 and 150

Check: 90 + 150 = 240, 90:150 = 3:5.

Example 3: Fraction addition with part decomposition

Compute: 7/12 + 5/18

Step 1: Vyashtisamasthi: split each fraction into parts that share a common denominator. 7/12 = 21/36, 5/18 = 10/36.

Step 2: Sum: 21/36 + 10/36 = 31/36.

Alternatively, split by decomposition: 7/12 = 1/3 + 1/4 (since 1/3 + 1/4 = 4/12 + 3/12 = 7/12). 5/18 = 1/6 + 1/9 (since 1/6 + 1/9 = 3/18 + 2/18 = 5/18).

Step 3: Sum the Egyptian fraction parts: (1/3 + 1/6) + (1/4 + 1/9).

Step 4: 1/3 + 1/6 = 1/2. 1/4 + 1/9 = 13/36.

Step 5: 1/2 + 13/36 = 18/36 + 13/36 = 31/36.

Answer: 31/36

Example 4: Profit-sharing ratio

A, B, and C invest 5000, 7000, and 8000 respectively. Total profit is 4000. Share proportionally.

Step 1: Whole investment = 5000 + 7000 + 8000 = 20000.

Step 2: A's share = (5000 / 20000) x 4000 = 1/4 x 4000 = 1000.

Step 3: B's share = (7000 / 20000) x 4000 = 7/20 x 4000 = 1400.

Step 4: C's share = (8000 / 20000) x 4000 = 2/5 x 4000 = 1600.

Answer: A = 1000, B = 1400, C = 1600

Check: 1000 + 1400 + 1600 = 4000.

Example 5: Converting a recurring decimal to fraction

Convert 0.142857142857... to fraction using part-whole.

Step 1: Recognize the repeating block: 142857.

Step 2: The decimal is 142857 / 999999 (since 1/7 = 0.142857...).

Step 3: Vyashtisamasthi: 142857 / 999999 = divide numerator and denominator by 142857.

Step 4: 142857/999999 = 1/7.

Answer: 1/7

Code Snippet: Python Implementation

def ratio_split(whole, ratios):
    """Split a whole value according to given ratios using Vyashtisamasthi."""
    total_ratio = sum(ratios)
    parts = []
    for r in ratios:
        part = whole * r / total_ratio
        parts.append(part)
    return parts


def mixed_operation(a_whole, a_frac_num, a_frac_den,
                    b_whole, b_frac_num, b_frac_den, op):
    """Operate on mixed numbers using part-whole decomposition."""
    a = a_whole + a_frac_num / a_frac_den
    b = b_whole + b_frac_num / b_frac_den

    if op == '+':
        result = a + b
    elif op == '-':
        result = a - b
    elif op == '*':
        result = a * b
    elif op == '/':
        result = a / b

    return result


# Test ratio split
print("Ratio split 240 in 3:5:")
parts = ratio_split(240, [3, 5])
print(f"  {parts}")

# Test mixed number operations
print("\nMixed number: 3²/5 × 2³/4:")
result = mixed_operation(3, 2, 5, 2, 3, 4, '*')
print(f"  {result} (expected 9.35)")

# Test profit sharing
print("\nProfit share (5000, 7000, 8000) on 4000:")
shares = ratio_split(4000, [5000, 7000, 8000])
print(f"  {shares}")
print(f"  Sum: {sum(shares)}")

Expected output:

Ratio split 240 in 3:5:
  [90.0, 150.0]

Mixed number: 3²/5 × 2³/4:
  9.35 (expected 9.35)

Profit share (5000, 7000, 8000) on 4000:
  [1000.0, 1400.0, 1600.0]
  Sum: 4000.0

Code Snippet: JavaScript Implementation

function ratioSplit(whole, ratios) {
    const total = ratios.reduce((a, b) => a + b, 0);
    return ratios.map(r => whole * r / total);
}

function mixedOperation(a, af, ad, b, bf, bd, op) {
    const va = a + af / ad;
    const vb = b + bf / bd;
    switch (op) {
        case '+': return va + vb;
        case '-': return va - vb;
        case '*': return va * vb;
        case '/': return va / vb;
    }
}

console.log("240 in ratio 3:5:", ratioSplit(240, [3, 5]));
console.log("3²/5 × 2³/4:", mixedOperation(3, 2, 5, 2, 3, 4, '*'));
console.log("Profit shares:", ratioSplit(4000, [5000, 7000, 8000]));

Common Errors

1. Forgetting to sum the ratios before dividing. To split a whole in ratio a:b:c, first compute the total a+b+c, then compute each share as (a/total) x whole.

2. Decomposing fractions incorrectly. 7/12 = 1/3 + 1/4 is valid because 4/12 + 3/12 = 7/12. But 7/12 does NOT equal 1/7 + 1/5. Always verify your decomposition sums correctly.

3. Applying part-whole to unrelated quantities. Part-whole decomposition only works when the parts genuinely compose the whole. A 3:5 ratio of boys to girls means 3/8 are boys, not 3/5.

4. Not simplifying intermediate fractions in mixed number operations. When computing (3 + 2/5)(2 + 3/4), keep intermediate results as simplified fractions, not decimals, to avoid rounding errors.

5. Confusing part-whole ratios with part-part ratios. A ratio of 3:5 has 3+5=8 total parts. Each part is a fraction of the whole, not a fraction of the other part.

Practice Questions

1. Divide 360 in the ratio 4:5.
2. Compute 2¹/3 x 1³/5 using part-whole decomposition.
3. Three partners invest in ratio 2:3:4. Profit is 900. How much does each get?

Answers:

1. 4+5=9, 360x4/9=160, 360x5/9=200.
2. (2+1/3)(1+3/5) = 2x1 + 2x3/5 + 1/3x1 + 1/3x3/5 = 2 + 6/5 + 1/3 + 3/15 = 2 + 1.2 + 0.333 + 0.2 = 3.733 = 56/15.
3. 2+3+4=9, 900x2/9=200, 900x3/9=300, 900x4/9=400.

Mini Project: General Ratio Splitter

def vyashtisamashti_solver(problem_type, *args):
    """General part-whole solver for multiple problem types."""
    if problem_type == 'ratio':
        whole, *ratios = args
        return ratio_split(whole, list(ratios))

    elif problem_type == 'mixed_mul':
        a_w, a_f, a_d, b_w, b_f, b_d = args
        return mixed_operation(a_w, a_f, a_d, b_w, b_f, b_d, '*')

    elif problem_type == 'fraction_add':
        n1, d1, n2, d2 = args
        # Decompose and add
        from math import gcd
        lcm = d1 * d2 // gcd(d1, d2)
        result_n = n1 * (lcm // d1) + n2 * (lcm // d2)
        result_d = lcm
        g = gcd(result_n, result_d)
        return (result_n // g, result_d // g)

    return None


# Examples
print("Ratio 360 in 4:5:", vyashtisamashti_solver('ratio', 360, 4, 5))
print("Mixed 2¹/3 × 1³/5:", vyashtisamashti_solver('mixed_mul', 2, 1, 3, 1, 3, 5))
print("Fraction 7/12 + 5/18:", vyashtisamashti_solver('fraction_add', 7, 12, 5, 18))

FAQ

Next Steps

Continue with Gunakasamuchya — verification sutras for checking your work using part-whole logic.

Related tutorials:

• Vedic Maths Fractions — rapid fraction operations with Vedic sutras
• Vedic Maths Overview — introduction to all Vedic sutras

Built by the developers of Doda Browser, DodaZIP, and Durga Antivirus Pro.