Fourth Proportional: Formula, Methods and Examples

Fourth Proportional is the fourth term in a proportion when three terms are known. If a : b = c : x, then x is the fourth proportional and can be found by cross-multiplication. This page explains the formula, calculation methods, comparison with third proportional, and solved numerical examples.

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What Is the Fourth Proportional?

The fourth proportional to three numbers a, b and c is a number x such that a : b = c : x. Its value is x = (b × c) ÷ a, provided a is non-zero.

Writing the proportion as a/b = c/x and cross-multiplying gives a × x = b × c. Therefore, x = bc/a. For example, in 4 : 6 = 8 : x, x = (6 × 8) ÷ 4 = 12, so 12 is the fourth proportional.

Fourth Proportional Formula & Tricks

Important Formulas

Fourth proportional formula
If a : b = c : x, then x = (b × c) / a

Multiply the second and third terms and divide the product by the first term.

Cross-multiplication form
a × x = b × c

Cross-multiplication converts the proportion into an equation that can be solved for the unknown fourth term.

Fraction form
a/b = c/x ⇒ x = bc/a

The two ratios must be equal. Rearranging the equation gives the fourth proportional directly.

Quick Tricks

Use cross-products directly

For a : b = c : x, multiply b and c first, then divide by a. This avoids forming unnecessary decimal values.

Example: For 7 : 9 = 14 : x, x = (9 × 14) / 7 = 18.
Cancel common factors before multiplying

If the divisor a shares a factor with b or c, cancel it first to reduce calculation time.

Example: For 12 : 15 = 20 : x, x = (15 × 20) / 12. Cancelling 15 and 12 by 3 gives x = (5 × 20) / 4 = 25.
Check by substituting the answer

After finding x, compare b/a with x/c or verify that a × x equals b × c.

Example: For 5 : 8 = 15 : x, x = 24. The cross-products are 5 × 24 = 120 and 8 × 15 = 120.

Fourth Proportional Concepts

Finding the Fourth Proportional by Cross-Multiplication

Set the three given terms in the order a : b = c : x, then calculate x = bc/a.

The order of the terms must remain unchanged. For 6 : 10 = 9 : x, cross-multiplication gives 6x = 10 × 9 = 90. Hence, x = 90/6 = 15.

Example: The fourth proportional to 6, 10 and 9 is 15 because 6 : 10 = 9 : 15.

Using Fractions to Solve a Proportion

Convert both ratios into fractions and isolate the unknown term.

If 3/7 = 12/x, cross-multiplication gives 3x = 7 × 12 = 84. Dividing by 3 gives x = 28. The same method applies when the terms are fractions or decimals, after simplifying them if needed.

Example: For 2/5 = 8/x, 2x = 40, so x = 20.

Effect of the Order of Terms

The fourth proportional depends on the exact order of the first three terms.

The expression a : b = c : x is different from b : a = c : x. For example, the fourth proportional to 4, 6 and 8 is (6 × 8)/4 = 12, whereas reversing the first ratio gives (4 × 8)/6 = 16/3.

Example: 4 : 6 = 8 : 12, but 6 : 4 = 8 : 16/3.

Fourth Proportional versus Third Proportional

A fourth proportional satisfies a : b = c : x, while a third proportional satisfies a : b = b : x.

The fourth proportional uses three generally different known terms and has the formula x = bc/a. The third proportional uses the second term twice and has the formula x = b²/a. For 2 and 4, the third proportional is 8 because 2 : 4 = 4 : 8.

Example: Fourth proportional to 2, 4 and 6: x = (4 × 6)/2 = 12. Third proportional to 2 and 4: x = 4²/2 = 8.

Fourth Proportional Video Lessons

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Practice Fourth Proportional Questions

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1Find the single discount (approximately) equal to three consecutive discounts of 6%, 5% and 11%.→ 2If the square root of a number is 0.25 times the number itself, find the number.→ 3A wholesaler offers two options for a sofa set marked at ₹80,000: either a flat discount of 30% or two successive discounts of 20% and 12%. What is the difference in the selling price between the two options?→ 495P9 is divisible by 11 . What is the value of P ?→ 5Find the smallest digit x such that the 6-digit number 28x219 is divisible by both 9 and 11.→ 6Find a single discount (approximately) equal to three consecutive discounts of 20%, 4% and 14%.→ 7Successive discounts of 16% and 22% are equivalent to what single discount?→ 8Find the maximum number of students among whom 198 chips and 330 toffees can be distributed such that each student gets the same number of each.→ 9What is the Highest Common Factor of 5/12, 15/20 and 25/30?→ 10A shopkeeper finds that the profit from selling a watch for ₹1100 is equal to the loss incurred when selling it for ₹900. What will be the percentage loss if the watch is sold for ₹850?→

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Quick Revision Notes

Fourth Proportional: Quick Revision

Use these rules to solve fourth proportional questions accurately.

  • If a : b = c : x, then x = bc/a.
  • Cross-multiplication gives a × x = b × c.
  • Keep the first three terms in their given order.
  • Cancel common factors before multiplication to simplify calculations.
  • The first term must be non-zero for x = bc/a to be defined.
  • Do not confuse the fourth proportional, a : b = c : x, with the third proportional, a : b = b : x.
  • Always verify the result by checking that a × x equals b × c.

Fourth Proportional FAQs

What is the fourth proportional formula?

If a : b = c : x, the fourth proportional is x = (b × c) / a.

What is the fourth proportional to 5, 8 and 15?

Using x = bc/a, x = (8 × 15)/5 = 24. Therefore, the fourth proportional is 24.

How do you find the fourth proportional when the ratio is written as a fraction?

For a/b = c/x, cross-multiply to get ax = bc, then divide by a: x = bc/a. For 3/4 = 9/x, x = (4 × 9)/3 = 12.

What is the fourth proportional to 6, 9 and 14?

x = (9 × 14)/6 = 126/6 = 21. Thus, 6 : 9 = 14 : 21.

What is the difference between the third and fourth proportional?

For the fourth proportional, a : b = c : x, so x = bc/a. For the third proportional, a : b = b : x, so x = b²/a.

Can the first term be zero in a fourth proportional?

No. The formula x = bc/a requires a non-zero first term. A ratio with zero as the denominator in fraction form is undefined.

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