Mean Proportional: Formula, Methods and Examples

Mean Proportional is the number placed between two quantities so that the three quantities form a continued proportion. It is also called the geometric mean of the two quantities. This page covers the mean proportional formula, methods to find it, key properties, shortcuts and numerical examples used in quantitative aptitude.

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What is Mean Proportional?

If x is the mean proportional between two positive numbers a and b, then a : x = x : b. Therefore, x² = ab and x = √(ab).

The mean proportional is the geometric mean of the two given numbers. To find it, multiply the numbers and take the positive square root. For 4 and 25, the mean proportional is √(4 × 25) = √100 = 10, because 4 : 10 = 10 : 25.

Mean Proportional Formula & Tricks

Important Formulas

Basic mean proportional formula
x = √(ab)

The mean proportional x between positive numbers a and b is the positive square root of their product.

Continued proportion form
a : x = x : b ⇒ x² = ab

Cross-multiplication gives x² = ab, which leads to the mean proportional formula.

Mean proportional between fractions
x = √((p/q) × (r/s)) = √(pr/qs)

Multiply the fractions first and then find the square root, after simplifying the product.

Finding an unknown endpoint
b = x²/a or a = x²/b

If the mean proportional and one endpoint are known, divide the square of the mean proportional by the known endpoint.

Quick Tricks

Use nearby square numbers

When the product is a perfect square, identify its square root directly. This avoids long calculation.

Example: The mean proportional between 12 and 27 is √324 = 18.
Use prime-factor pairing

For a non-perfect-square product, pair equal prime factors inside the square root and leave unpaired factors outside.

Example: √(18 × 8) = √144 = 12. Alternatively, 18 × 8 = 2⁴ × 3², so the square root is 12.
Check using the product rule

After finding x, verify that x² equals the product of the two endpoints.

Example: For endpoints 9 and 16, x = 12 because 12² = 144 = 9 × 16.

Mean Proportional Concepts

Finding the Mean Proportional Between Two Numbers

Multiply the two positive numbers and take the positive square root to find their mean proportional.

For a and b, write a : x = x : b. Cross-multiplication gives x² = ab. Hence, x = √(ab). The result lies between a and b when a and b are unequal positive numbers.

Example: Between 8 and 18, x = √(8 × 18) = √144 = 12. Thus, 8 : 12 = 12 : 18.

Mean Proportional as Geometric Mean

The mean proportional is the geometric mean of two positive numbers.

The arithmetic mean of a and b is (a + b)/2, whereas their geometric mean is √(ab). These are generally different. For unequal positive numbers, the geometric mean is less than the arithmetic mean.

Example: For 4 and 16, the geometric mean is √64 = 8, while the arithmetic mean is (4 + 16)/2 = 10.

Finding a Missing Extreme Term

If x is the mean proportional between a and b, the unknown endpoint equals x² divided by the known endpoint.

From x² = ab, if a is known, then b = x²/a. If b is known, then a = x²/b. The units of the unknown endpoint must match the units of the known endpoint.

Example: If 15 is the mean proportional between 5 and b, then b = 15²/5 = 225/5 = 45.

Mean Proportional in Continued Proportion

In the continued proportion a : b = b : c, the middle term b is the mean proportional between a and c.

Cross-multiplication gives b² = ac. Therefore, b = √(ac) for positive terms. The first, middle and third terms are not generally in arithmetic progression.

Example: In 3 : 6 = 6 : 12, 6 is the mean proportional between 3 and 12 because 6² = 3 × 12.

Mean Proportional Video Lessons

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Practice Mean Proportional Questions

Practise published questions related to this topic.

1A positive number exceeds its square root by 132. Find the number given that it is also a multiple of 12.→ 2A person sells an article at the profit of 30 percent. Cost price and selling price both are decreased by 40 percent. What will be the new profit percentage?→ 3The simplified value of the expression 0.5² × 5⁵ × 2⁻⁴ × 4⁴ × 10⁻³ ÷ a is 1, then the value of a is ______.→ 4What is the Least Common Multiple of 0.75 and (25/30)?→ 5Three numbers are in the ratio 17 : 13 : 8, and their LCM is 7072. Their HCF is:→ 6Marked price of an article is Rs. 4000. After giving two successive discounts of 25 percent and A percent, the article is sold for Rs. 2400. What is the value of A?→ 7A shopkeeper offers the following four schemes. A) Two successive discounts of 19% and 25% B) Buy 7, get 6 free C) Single discount of 50% D) Two successive discounts of 3% and 39% Which scheme is the best for the shopkeeper?→ 8The difference between cost price and selling price of an article is Rs. 420. Loss percentage is 16 percent. What is the cost price of the article?→ 9If on a sale, there is 40% discount on the marked price of ₹1,200, but the sale is done at ₹520 only, then how much additional discount did the customer get?→ 101/(6 + √3) + 1/(6 - √3) = ?→

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Quick Revision Notes

Mean Proportional Revision Points

Recall these formulas and checks for quick calculations.

  • If x is the mean proportional between a and b, then a : x = x : b.
  • The main formula is x = √(ab), for positive a and b.
  • The mean proportional is also called the geometric mean.
  • To find an unknown endpoint, use b = x²/a or a = x²/b.
  • For unequal positive numbers, the geometric mean is less than the arithmetic mean.
  • Verify every answer by checking whether x² = ab.

Mean Proportional FAQs

What is the mean proportional formula?

If x is the mean proportional between a and b, then x = √(ab). This follows from a : x = x : b.

What is the mean proportional between 6 and 24?

√(6 × 24) = √144 = 12. Therefore, the mean proportional is 12.

Is the mean proportional the same as the arithmetic mean?

No. The mean proportional is √(ab), while the arithmetic mean is (a + b)/2. They are equal only when a = b for positive numbers.

How do you find the missing number if the mean proportional is known?

Use the relation x² = ab. For example, if x = 10 and a = 4, then b = 10²/4 = 25.

Can the mean proportional be found between fractions?

Yes. Multiply the fractions and take the positive square root. For 1/4 and 9/16, x = √(9/64) = 3/8.

What is the mean proportional between 9 and 25?

The mean proportional is √(9 × 25) = √225 = 15.

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