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The question in the image is in Hindi and can be translated as follows:

**Question 23:** If 12, 28, 21, and 45 are given, and if *x* is added to each, then the obtained numbers are in proportion. What will be the mean proportional between \((x + 3)\) and \((4x + 1)\)?

The answer options provided are:
(a) 15  
(b) 12  
(c) 10  
(d) 18  

### Solution:

We are told that by adding *x* to the numbers 12, 28, 21, and 45, the obtained numbers are in proportion. This means the resulting sequence must follow a proportional (or ratio-based) relationship.

Let us denote the new numbers as:
\[
(12 + x), (28 + x), (21 + x), (45 + x)
\]

For them to be in proportion, the ratios must be equal:
\[
\frac{12 + x}{28 + x} = \frac{21 + x}{45 + x}
\]

Now, cross-multiply to get an equation:
\[
(12 + x)(45 + x) = (28 + x)(21 + x)
\]

Expand both sides:
\[
12(45) + 12x + 45x + x^2 = 28(21) + 28x + 21x + x^2
\]
Simplifying:
\[
540 + 57x + x^2 = 588 + 49x + x^2
\]
Cancel out the \(x^2\) terms on both sides:
\[
540 + 57x = 588 + 49x
\]
Now, simplify further:
\[
57x - 49x = 588 - 540
\]
\[
8x = 48
\]
\[
x = 6
\]

Now that we know \(x = 6\), we need to find the mean proportional between \((x + 3)\) and \((4x + 1)\).

Substitute \(x = 6\) into these expressions:
\[
x + 3 = 6 + 3 = 9
\]
\[
4x + 1 = 4(6) + 1 = 24 + 1 = 25
\]

The mean proportional between 9 and 25 is given by:
\[
\sqrt{9 \times 25} = \sqrt{225} = 15
\]

Thus, the mean proportional is **15**.

### Final Answer:
(a) 15

---

Would you like more details or have any other questions? Here are five related questions you might find helpful:

1. What is the concept of mean proportional?
2. How do we solve equations involving proportions?
3. Can we always apply cross-multiplication when dealing with proportions?
4. How is the mean proportional used in geometric problems?
5. What are the properties of proportional relationships?

**Tip:** When dealing with proportions, always ensure the ratios are set up correctly before applying cross-multiplication.

यदि 12, 28, 21 और 45 में से, प्रत्येक में x जोड़ा जाता है, तो इस क्रम में, प्राप्त संख्याएँ समानुपात में हैं। (x + 3) और (4x + 1) के बीच मध्यानुपाती (mean proportional) क्या होगा?

Learn how to find the mean proportional between two algebraic expressions derived from a sequence of numbers. This problem involves adding a variable to numbers and ensuring they are in proportion, followed by calculating the mean proportional. Suitable for students in Grades 8-10.

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