Calculating the Square Root of a Fraction with Decimals

The question asks us to find the value of the square root of the fraction \(\frac{36.1}{102.4}\).

To make the calculation easier, we can first remove the decimal points from the numerator and the denominator. We can do this by multiplying both the numerator and the denominator by 10.

The fraction becomes:

\[ \frac{36.1}{102.4} = \frac{36.1 \times 10}{102.4 \times 10} = \frac{361}{1024} \]

Now, we need to find the square root of this new fraction:

\[ \sqrt{\frac{361}{1024}} \]

To find the square root of a fraction, we can find the square root of the numerator and the square root of the denominator separately, and then divide the results:

\[ \sqrt{\frac{361}{1024}} = \frac{\sqrt{361}}{\sqrt{1024}} \]

Finding the Square Root of the Numerator

We need to find the square root of 361. We are looking for a number that, when multiplied by itself, gives 361.

Let's try some numbers:

• \(10^2 = 100\)
• \(20^2 = 400\)

The number is between 10 and 20. Let's try numbers ending in digits that square to a number ending in 1 (like 1 or 9):

• \(11^2 = 121\)
• \(19^2 = 361\)

So, the square root of 361 is 19.

\[ \sqrt{361} = 19 \]

Finding the Square Root of the Denominator

Next, we need to find the square root of 1024. We are looking for a number that, when multiplied by itself, gives 1024.

Let's try estimating:

• \(30^2 = 900\)
• \(35^2 = 1225\)

The number is between 30 and 35. Let's try numbers ending in digits that square to a number ending in 4 (like 2 or 8):

• \(31^2 = 961\)
• \(32^2 = 1024\)

So, the square root of 1024 is 32.

\[ \sqrt{1024} = 32 \]

Final Calculation of the Square Root

Now we can substitute the square roots back into the expression:

\[ \sqrt{\frac{361}{1024}} = \frac{\sqrt{361}}{\sqrt{1024}} = \frac{19}{32} \]

The value of the square root of \(\frac{36.1}{102.4}\) is \(\frac{19}{32}\).

Comparing with Options

Let's look at the given options:

Our calculated value, \(\frac{19}{32}\), matches Option 3.

Revision Table: Square Root Calculation Steps

Additional Information: Understanding Square Roots

The square root of a number \(x\) is a number \(y\) such that \(y \times y = x\). It is denoted by the symbol \(\sqrt{}\). For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).

For positive numbers \(a\) and \(b\), the square root of a fraction \(\frac{a}{b}\) can be calculated as the square root of the numerator \(a\) divided by the square root of the denominator \(b\):

\[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

This property is very useful when dealing with square roots of fractions.

To find the square root of a number like 361 or 1024, it is helpful to know the squares of common integers or use prime factorization methods for larger numbers. In this case, recognizing that 361 and 1024 are perfect squares makes the calculation straightforward.