# Worksheet on Mean Proportional | Free Printable Mean Proportional Worksheet with Answers

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 = âˆšac. This free printable Worksheet on Mean Proportional with Answers PDF helps you understand the problem-solving techniques and feels fun to practice.

## Mean Proportional Worksheet PDF with Solutions

Example 1:Â
Find the mean proportional of the following sets of positive integers:
(i) x- y, xÂ³- xÂ²y
(ii) xÂ³y, xyÂ³

Solution:Â

(i) Given x- y, xÂ³- xÂ²y
Now, to find the mean proportion.
Let p be the mean proportional between x- y, xÂ³- xÂ²y.
So, x- y: p:: p: xÂ³- xÂ²y
Product of extremes = Product of means.
Now,
pÂ² = (x- y)(xÂ³- xÂ²y)
â‡’ pÂ² = xÂ²(x- y)(x- y)
â‡’ pÂ² = xÂ²(x- y)Â²
â‡’ p = âˆš(xÂ²(x- y)Â²)
â‡’ p = x(x- y)
Thus, the mean proportion of x- y, xÂ³- xÂ²y is x(x- y).

(ii) Given xÂ³y, xyÂ³
Let m be the mean proportion of xÂ³y, xyÂ³.
So, xÂ³y: m:: m: xyÂ³.
Product of extremes = Product of means.
Product of extremes = xÂ³y Ã— xyÂ³
Product of means = m Ã— m = mÂ²
mÂ² = xÂ³y Ã— xyÂ³
â‡’ mÂ² = (xÂ³ Ã— x) Ã— (yÂ³ Ã— y)
â‡’ mÂ² = x4 Ã— y4
â‡’ m = âˆš(x4 Ã— y4)
â‡’ m = âˆš(xÂ²yÂ²)Â²
â‡’ m = xÂ²yÂ²
Hence, the mean proportion of xÂ³y, xyÂ³ is xÂ²yÂ².

Example 2:Â Â
Find the mean proportional of the following:
(i) 8 and 32
(ii) 0.04 and 0.56
(iii) 4 and 25

Solution:Â

(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 Ã— 32 = a Ã— a
â‡’ aÂ² = 256
â‡’ a = âˆš256
â‡’ 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 Ã— 0.56 = p Ã— p
â‡’ pÂ² = 0.02
â‡’ p = âˆš0.02
â‡’ 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 Ã—25 = x Ã— x
â‡’ xÂ² = 100
â‡’ x = âˆš100
â‡’ x = 10
Hence, the mean proportion of 4 and 25 is 10.

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Example 3:Â
If b is the mean proportion between a and c, show that a4+ a2b2+ b4/b4+ b2c2+ c4Â = $$\frac{aÂ²}{cÂ²}$$.

Solution:Â

Given b is the mean proportion between a and c, then we have bÂ² = ac.
Now, we have to prove that a4+ a2b2+ b4 / b4+ b2c2+ c4 = $$\frac{aÂ²}{cÂ²}$$ i.e., L.H.S = R.H.S.
LHS = a4+ a2b2+ b4/ b4+ b2c2+ c4
Let us substitute bÂ² = ac in LHS.
LHS = a4+ aÂ²(ac)+ (ac)Â²/(ac)Â²+ (ac)cÂ²+ c4
â‡’ LHS = $$\frac{aÂ²( aÂ²+ ac+ cÂ² )}{cÂ²( aÂ²+ ac+ cÂ² )}$$
â‡’ LHS = $$\frac{aÂ²}{cÂ²}$$ = RHS
â‡’ LHS = RHS
Therefore, a4+ a2b2+ b4/ b4+ b2c2+ c4= $$\frac{aÂ²}{cÂ²}$$.

Example 4:Â
Find the mean proportion of the following
(i) 4$$\frac{4}{5}$$, 2$$\frac{1}{2}$$
(ii) aÂ²b, abÂ²

Solution:Â

(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.
$$\frac{24}{5}$$ Ã— $$\frac{5}{2}$$ = m Ã— m
â‡’ mÂ² = $$\frac{24}{2}$$
â‡’ mÂ² = 12
â‡’ m = âˆš12
â‡’ m = 3.46
Thus, the value of mean proportion is 3.46.

(ii) Given aÂ²b, abÂ²
Let k be the mean proportion of aÂ²b and abÂ².
So, aÂ²b: k:: k: abÂ².
In mean proportion, Product of extremes = Product of means.
Now, aÂ²b Ã— abÂ² = kÂ²
â‡’ kÂ² = aÂ³bÂ³
â‡’ kÂ² = (ab)Â¹ Ã— (ab)Â²
â‡’ k = âˆš((ab)Â¹)Â²
â‡’ k = ab
Therefore, ab is the mean proportion of aÂ²b, abÂ².

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