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Continued Proportion

Introduction

A continued proportion links several numbers so that each consecutive pair has the same ratio. Unlike a simple proportion that only compares two ratios, continued proportion creates a chain of equal ratios.

These questions may look complex, but if you understand the idea that the terms follow a geometric progression (GP), you can solve them easily.

Pattern: Continued Proportion

Pattern

Key ideas:

• In a continued proportion, each term is obtained by multiplying the previous term with a common ratio r.

• If a : b = b : c = c : d = r → the terms form a GP: b = a·r, c = a·r², d = a·r³, and so on.

• If the first and last terms are known, you can find r by using roots: r = (last ÷ first)^(1/n), where n is the number of equal ratios.

Step-by-Step Example

Question

If a : b = b : c = c : d and a = 2, d = 54, find b and c.

Solution

  1. Step 1: Express the relation.

    From the pattern: b = a·r, c = a·r², d = a·r³.

  2. Step 2: Use a and d to find r.

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

  3. Step 3: Find b and c.

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

  4. Step 4: Final Answer.

    The missing terms are: b = 6, c = 18.

  5. Step 5: Quick Check.

    Ratios: 2 : 6 = 1 : 3, 6 : 18 = 1 : 3, 18 : 54 = 1 : 3 → all equal ✅ So, the solution is correct.

Quick Variations

If four or more terms are given, apply the same idea: b² = a × c, c² = b × d, and so on.

If only the first and last terms are known, use the formula r = (last ÷ first)^(1/n) to find the common ratio, then calculate the missing terms.

Trick to Always Use

  • Step 1: Recognise the GP structure.
  • Step 2: Write terms as b = a·r, c = a·r², etc.
  • Step 3: Find r using roots from first and last terms.
  • Step 4: Solve for missing terms and verify ratios.

Summary

Summary

In continued proportion, the terms follow a geometric progression. The property is:

b = a·r, c = a·r², d = a·r³

  • Step 1: Write each term using the common ratio r.
  • Step 2: Find r using roots from first and last terms.
  • Step 3: Calculate intermediate values.
  • Step 4: Verify that all consecutive ratios are equal.

Once you know this, even long chains of proportions can be solved easily.

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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