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Continued Proportion

Introduction

Continued proportion कई numbers को इस तरह जोड़ता है कि हर consecutive pair का ratio समान होता है। Simple proportion केवल दो ratios compare करता है, जबकि continued proportion एक chain of equal ratios बनाता है।

ये questions देखने में complex लग सकते हैं, लेकिन अगर समझ लो कि ऐसे terms एक geometric progression (GP) बनाते हैं, तो इन्हें बहुत आसान से solve किया जा सकता है।

Pattern: Continued Proportion

Pattern

Key ideas:

• Continued proportion में हर term को पिछले term में एक common ratio r multiply करके पाया जाता है।

• अगर a : b = b : c = c : d = r हो, तो terms GP बनाते हैं: b = a·r, c = a·r², d = a·r³, आदि।

• अगर first और last terms पता हों, तो r root से निकाला जाता है: r = (last ÷ first)^(1/n), जहाँ n equal ratios की संख्या है।

Step-by-Step Example

Question

अगर a : b = b : c = c : d और a = 2, d = 54, तो b और c निकालो।

Solution

  1. Step 1: Relation लिखो।

    Pattern से: b = a·r, c = a·r², d = a·r³

  2. Step 2: a और d से r निकालो।

    d = a·r³ → r³ = d ÷ a = 54 ÷ 2 = 27 → r = ∛27 = 3

  3. Step 3: b और c निकालो।

    b = a·r = 2 × 3 = 6
    c = a·r² = 2 × 9 = 18

  4. Step 4: Final Answer.

    Missing terms: b = 6, c = 18

  5. Step 5: Quick Check.

    Ratios: 2 : 6 = 1 : 3, 6 : 18 = 1 : 3, 18 : 54 = 1 : 3 → सभी equal हैं ✅

Quick Variations

अगर चार या ज़्यादा terms हों, तो वही GP logic लागू होता है: b² = a × c, c² = b × d आदि relations काम आते हैं।

अगर केवल first और last terms हों, तो formula r = (last ÷ first)^(1/n) से common ratio निकालो और missing terms compute करो।

Trick to Always Use

  • Step 1: GP structure पहचानो।
  • Step 2: Terms को b = a·r, c = a·r² आदि लिखो।
  • Step 3: First और last terms से r निकालो।
  • Step 4: Missing terms solve करो और ratios verify करो।

Summary

Summary

Continued proportion में terms geometric progression follow करते हैं। मुख्य property:

b = a·r, c = a·r², d = a·r³

  • Step 1: हर term को common ratio r से लिखो।
  • Step 2: First और last terms से root लेकर r निकालो।
  • Step 3: Intermediate values निकालो।
  • Step 4: Verify करो कि सारे consecutive ratios equal हों।

यह concept आने के बाद लंबी ratio-chains भी आसानी से solve हो जाती हैं।

Practice

(1/5)
1. If a : b = b : c and a = 2, b = 6, find c.
easy
A. 18
B. 12
C. 24
D. 20

Solution

  1. Step 1: Use the three-term continued-proportion identity

    For three terms in continued proportion a, b, c we have b² = a × c.

  2. Step 2: Substitute the given values

    Substitute: 6² = 2 × c → 36 = 2c.

  3. Step 3: Solve for c

    c = 36 ÷ 2 = 18.

  4. Final Answer:

    18 → Option A

  5. Quick Check:

    a : b = 2 : 6 = 1 : 3 and b : c = 6 : 18 = 1 : 3 → continued proportion holds ✅
Hint: Use b² = a × c for three-term continued proportion.
Common Mistakes: Forgetting to square b or mixing up the positions of a, b, c.
2. If 3, x, 48 are in continued proportion, find x.
easy
A. 8
B. 12
C. 16
D. 20

Solution

  1. Step 1: Apply the mean-proportional rule

    For three terms in continued proportion a, x, c we have x² = a × c.

  2. Step 2: Substitute and compute

    x² = 3 × 48 = 144.

  3. Step 3: Take the square root

    x = √144 = 12.

  4. Final Answer:

    12 → Option B

  5. Quick Check:

    3 : 12 = 1 : 4 and 12 : 48 = 1 : 4 → continued proportion holds ✅
Hint: Mean proportional = √(first × third).
Common Mistakes: Taking arithmetic mean or neglecting to square/root correctly.
3. If 2, x, 50 are in continued proportion, what is x?
easy
A. 8
B. 9
C. 12
D. 10

Solution

  1. Step 1: Use the three-term continued-proportion identity

    For three terms a, x, c in continued proportion we have x² = a × c.

  2. Step 2: Substitute numbers

    x² = 2 × 50 = 100.

  3. Step 3: Take square root

    x = √100 = 10.

  4. Final Answer:

    10 → Option D

  5. Quick Check:

    2 : 10 = 1 : 5 and 10 : 50 = 1 : 5 → continued proportion holds ✅
Hint: Square root links first and third terms for three-term continued proportion.
Common Mistakes: Using wrong root or miscomputing the product a×c.
4. In a continued proportion of four terms a, b, c, d with a = 5 and d = 135, find c.
medium
A. 15
B. 30
C. 45
D. 60

Solution

  1. Step 1: Express terms using common ratio r

    For four terms in continued proportion let common ratio = r. Then b = a·r, c = a·r², d = a·r³.

  2. Step 2: Find r from first and last terms

    From d = a·r³ → r³ = d ÷ a = 135 ÷ 5 = 27 → r = ∛27 = 3.

  3. Step 3: Compute c using r

    c = a·r² = 5 × 3² = 5 × 9 = 45.

  4. Final Answer:

    45 → Option C

  5. Quick Check:

    Sequence: 5, 15, 45, 135 → each ratio = 3 → continued proportion holds ✅
Hint: Find r from r³ = d/a, then c = a·r² for 4-term sequences.
Common Mistakes: Applying square-root instead of cube-root for 4-term sequences.
5. If the first term is 4 and the last term is 64 with 4 equal ratios between them, find the common ratio r.
medium
A. 2
B. 3
C. √2
D. 4

Solution

  1. Step 1: Use n-th root formula for equal ratios

    If there are 4 equal ratios between first and last, r = (last ÷ first)^(1/4).

  2. Step 2: Substitute and compute

    r = (64 ÷ 4)^(1/4) = 16^(1/4).

  3. Step 3: Simplify

    16^(1/4) = 2 → r = 2.

  4. Final Answer:

    2 → Option A

  5. Quick Check:

    Sequence: 4, 8, 16, 32, 64 → each step multiply by 2 → r = 2 ✅
Hint: Use the n-th root: r = (last/first)^(1/n) where n is number of equal ratios.
Common Mistakes: Using wrong n (count of steps) when taking the root.

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