Getting confused while solving the mean proportional problems then practice more with this Mean Proportional Worksheet pdf and learn the concept of mean proportional efficiently. Mean proportional is also known as Geometric Mean and it is not similar to the arithmetic mean. The Mean Proportion is calculated between two terms of a ratio by taking the square root of the product of those two quantities in terms of ratio.
Let us understand more about mean proportional by taking help from this free printable mean proportional of two numbers worksheet. For example, if a, b, and c are in continued proportion then b is called the mean proportional of a and c. The mean proportion is expressed as b = √ac. This free printable Worksheet on Mean Proportional with Answers PDF helps you understand the problem-solving techniques and feels fun to practice.
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Mean Proportional Worksheet PDF with Solutions
Example 1:
Find the mean proportional of the following sets of positive integers:
(i) x- y, x³- x²y
(ii) x³y, xy³
Solution:
(i) Given x- y, x³- x²y
Now, to find the mean proportion.
Let p be the mean proportional between x- y, x³- x²y.
So, x- y: p:: p: x³- x²y
Product of extremes = Product of means.
Now,
p² = (x- y)(x³- x²y)
⇒ p² = x²(x- y)(x- y)
⇒ p² = x²(x- y)²
⇒ p = √(x²(x- y)²)
⇒ p = x(x- y)
Thus, the mean proportion of x- y, x³- x²y is x(x- y).
(ii) Given x³y, xy³
Let m be the mean proportion of x³y, xy³.
So, x³y: m:: m: xy³.
Product of extremes = Product of means.
Product of extremes = x³y × xy³
Product of means = m × m = m²
m² = x³y × xy³
⇒ m² = (x³ × x) × (y³ × y)
⇒ m² = x4 × y4
⇒ m = √(x4 × y4)
⇒ m = √(x²y²)²
⇒ m = x²y²
Hence, the mean proportion of x³y, xy³ is x²y².
Example 2:
Find the mean proportional of the following:
(i) 8 and 32
(ii) 0.04 and 0.56
(iii) 4 and 25
Solution:
(i) Given 8 and 32
Let the mean proportion between 8 and 32 is a.
Now, 8: a:: a: 32
We know that, Product of extremes = Product of means.
Here, extremes are 8 and 32 and means are a and a.
So, 8 × 32 = a × a
⇒ a² = 256
⇒ a = √256
⇒ a = 16
Thus, the value of mean proportion ‘a’ is 16.
(ii) Given 0.04 and 0.56
Let the mean proportion be p.
Now, 0.04: p:: p: 0.56
We know that, Product of extremes = Product of means.
The extremes are 0.04 and 0.56 and the means are p and p.
So, 0.04 × 0.56 = p × p
⇒ p² = 0.02
⇒ p = √0.02
⇒ p = 0.14
Therefore, the value of the mean proportion p is 0.14.
(iii) Given 4 and 25
Let the mean proportion between 4 and 25 be x.
Thus, 4: x:: x: 25
We know that, Product of extremes = Product of means.
Here, the extremes are 4 and 25 and the means are x and x.
So, 4 ×25 = x × x
⇒ x² = 100
⇒ x = √100
⇒ x = 10
Hence, the mean proportion of 4 and 25 is 10.
Example 3:
If b is the mean proportion between a and c, show that a4+ a2b2+ b4/b4+ b2c2+ c4 = \(\frac{a²}{c²}\).
Solution:
Given b is the mean proportion between a and c, then we have b² = ac.
Now, we have to prove that a4+ a2b2+ b4 / b4+ b2c2+ c4 = \(\frac{a²}{c²}\) i.e., L.H.S = R.H.S.
LHS = a4+ a2b2+ b4/ b4+ b2c2+ c4
Let us substitute b² = ac in LHS.
LHS = a4+ a²(ac)+ (ac)²/(ac)²+ (ac)c²+ c4
⇒ LHS = \(\frac{a²( a²+ ac+ c² )}{c²( a²+ ac+ c² )}\)
⇒ LHS = \(\frac{a²}{c²}\) = RHS
⇒ LHS = RHS
Therefore, a4+ a2b2+ b4/ b4+ b2c2+ c4= \(\frac{a²}{c²}\).
Example 4:
Find the mean proportion of the following
(i) 4\(\frac{4}{5}\), 2\(\frac{1}{2}\)
(ii) a²b, ab²
Solution:
(i) Given 4\(\frac{4}{5}\), 2\(\frac{1}{2}\)
Now, change the mixed fraction into proper fraction
4\(\frac{4}{5}\) = \(\frac{24}{5}\)
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
Let m be the mean proportion of \(\frac{24}{5}\), \(\frac{5}{2}\).
Product of extremes = Prouct of means
Here, the extremes are \(\frac{24}{5}\) and \(\frac{5}{2}\), the means are m and m.
\(\frac{24}{5}\) × \(\frac{5}{2}\) = m × m
⇒ m² = \(\frac{24}{2}\)
⇒ m² = 12
⇒ m = √12
⇒ m = 3.46
Thus, the value of mean proportion is 3.46.
(ii) Given a²b, ab²
Let k be the mean proportion of a²b and ab².
So, a²b: k:: k: ab².
In mean proportion, Product of extremes = Product of means.
Now, a²b × ab² = k²
⇒ k² = a³b³
⇒ k² = (ab)¹ × (ab)²
⇒ k = √((ab)¹)²
⇒ k = ab
Therefore, ab is the mean proportion of a²b, ab².