Third Proportional: Formula, Methods and Examples

Third Proportional is the number that forms a continued proportion with two given numbers. If a, b and c are in continued proportion, then a:b = b:c, so c is the third proportional to a and b. This page explains the formula, calculation methods, properties, shortcuts and numerical examples.

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What Is Third Proportional?

The third proportional to two non-zero numbers a and b is a number c such that a:b = b:c. Therefore, a, b and c are in continued proportion and c = b²/a.

From a:b = b:c, cross-multiplication gives ac = b². Hence, the third proportional is found by squaring the second number and dividing by the first number. For 4 and 6, c = 6²/4 = 36/4 = 9, so 9 is the third proportional to 4 and 6.

Third Proportional Formula & Tricks

Important Formulas

Basic third proportional formula
c = b²/a

If c is the third proportional to a and b, calculate it by dividing the square of b by a.

Cross-multiplication form
a:b = b:c ⟺ ac = b²

The proportion can be converted into ac = b², which is useful for checking or solving the value of any unknown term.

Fraction form
a/b = b/c ⟺ c = b ÷ (a/b)

When the given ratio is expressed as fractions, retain the order a:b = b:c before solving for c.

Quick Tricks

Square the second term first

For the third proportional to a and b, use the order exactly as given: square b and divide by a.

Example: For 8 and 12, c = 12²/8 = 144/8 = 18.
Simplify before squaring

If b/a contains a common factor, simplify the division where possible to reduce calculation.

Example: For 15 and 20, c = 20²/15 = 400/15 = 80/3. Alternatively, c = 20 × (20/15) = 20 × 4/3 = 80/3.
Check using the proportion

Verify the answer by checking whether a:b and b:c are equal.

Example: For 4, 6 and 9, 4:6 = 2:3 and 6:9 = 2:3, so 9 is correct.

Third Proportional Concepts

Finding the Third Proportional Directly

Use c = b²/a when the first two terms a and b are known.

The first given number is the first term of the first ratio, and the second given number is repeated as the middle term. Thus, for the third proportional to 9 and 12, c = 12²/9 = 144/9 = 16.

Example: 9:12 = 12:16 because both ratios equal 3:4.

Using Cross-Multiplication

The proportion a:b = b:c gives ac = b², which can be rearranged to find the missing term.

To solve a question, write the ratios in their given order, cross-multiply, and isolate the unknown. If a = 5 and b = 10, then 5c = 10², so c = 20.

Example: 5:10 = 10:20, since 5 × 20 = 10 × 10 = 100.

Third Proportional with Fractions and Decimals

The same formula c = b²/a applies to fractions and decimals after preserving their numerical values.

Convert fractions to a common form only when it makes calculation easier. For a = 1/2 and b = 3/4, c = (3/4)² ÷ (1/2) = 9/16 × 2 = 9/8.

Example: (1/2):(3/4) = (3/4):(9/8), because each ratio equals 2/3.

Difference Between Third and Fourth Proportional

A third proportional repeats the second term, while a fourth proportional uses three given terms in a:b = c:d.

For a, b, c, the third proportional d satisfies a:b = b:d, so d = b²/a. The fourth proportional x to a, b and c satisfies a:b = c:x, so x = bc/a.

Example: The third proportional to 4 and 6 is 9. The fourth proportional to 4, 6 and 8 is 12 because 4:6 = 8:12.

Third Proportional Video Lessons

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Practice Third Proportional Questions

Practise published questions related to this topic.

1Find the smallest number which when increased by 20 is completely divisible by 12, 20, 15, 18 and 24.→ 2Given that 119^(0.49) = x , 119^(0.62) = y and x^z = y⁶, then the value of z is close to:→ 3What is the minimum value that must be assigned to A so that the 8-digit number 322A4828 is divisible by 4?→ 4The marked price of an article is 80 percent more than its cost price. If 40 percent discount is given on the marked price, then what will be the profit percentage?→ 5A table was purchased for ₹850 and sold at a 5% loss. What was the selling price?→ 6Given that √15 = 3.87 and √150 = 12.24, then find the approximate value of √15 + √1500 + √0.015.→ 7If 2^a = 32 , b⁴ = 81 , and b > 0, then what is the value of a^b ?→ 8A data analyst is studying a series of zoomed-in satellite resolutions. The base square root reference she uses is: √18225 = 135 She now wants to quickly compute the total of square roots for four scaled-down values (by factors of 10): (√182.25 + √1.8225 + √0.018225 + √0.00018225) What is the value of this sum?→ 9How many numbers up to 200 are divisible by 4 and 3 both?→ 10An item is marked at Rs. 2000 and sold after two successive discounts of 20% and 10%. What is the selling price?→

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Quick Revision Notes

Third Proportional Revision Points

Use these formulas and checks for quick revision.

  • If a:b = b:c, then c is the third proportional to a and b.
  • The direct formula is c = b²/a, where a is non-zero.
  • The equivalent relation is ac = b².
  • The middle term b is repeated in the two ratios.
  • Always preserve the order of the two given numbers.
  • Check the answer by verifying a:b = b:c.
  • For positive numbers, the third proportional is positive.

Third Proportional FAQs

What is the formula for the third proportional to a and b?

The formula is c = b²/a. It follows from a:b = b:c and ac = b².

What is the third proportional to 6 and 15?

c = 15²/6 = 225/6 = 75/2 = 37.5. Therefore, the third proportional is 37.5.

How do you find the third proportional to 3/5 and 6/5?

c = (6/5)² ÷ (3/5) = 36/25 × 5/3 = 12/5. Thus, the third proportional is 12/5.

What is the third proportional to 0.4 and 0.8?

c = 0.8²/0.4 = 0.64/0.4 = 1.6. Hence, the third proportional is 1.6.

How can the answer be checked?

Check whether a:b equals b:c, or verify that ac = b². For 7, 14 and 28, 7 × 28 = 196 = 14².

What is the difference between mean proportional and third proportional?

The mean proportional x between a and b satisfies a:x = x:b, so x² = ab. The third proportional c to a and b satisfies a:b = b:c, so c = b²/a.

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