Problem 34 - Prealgebra

Question

Which is less, 11 or \(\sqrt{120} ?\) Explain your reasoning.

Step-by-Step Solution

Answer

\( \sqrt{120} \approx 10.95 \) is less than 11.

Step 1: Identify the Comparison

The task is to compare the numbers 11 and \( \sqrt{120} \) to determine which is smaller. To do this, we need to evaluate \( \sqrt{120} \) and see if it is greater or less than 11.

Step 2: Estimate or Calculate \( \sqrt{120} \)

We know that \( \sqrt{120} \) is between two perfect squares: \( \sqrt{121} = 11 \) and \( \sqrt{100} = 10 \). Thus, \( \sqrt{120} \) must be slightly less than 11. For a precise calculation, \( \sqrt{120} \approx 10.95 \).

Step 3: Compare the Values

Now, compare 11 to the calculated \( \sqrt{120} \) which is approximately 10.95. Since 10.95 is less than 11, we conclude that \( \sqrt{120} \) is indeed less than 11.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, indicating a value that, when multiplied by itself, yields the original number. For instance, the square root of 121 is 11, as 11 multiplied by 11 gives 121.
Understanding square roots is vital because they allow us to solve equations involving powers and roots. Here, the exercise requires comparing the number 11 with the square root of 120.
Our task is to understand whether this square root is larger or smaller than 11. This involves evaluating \( \sqrt{120} \)
by finding surrounding perfect squares that we are comfortable with—100 and 121 in this context. This bridge between known values assists in estimating the square root accurately.

Estimating Values

Estimating values embraces the art of making close-to-accurate assumptions about numbers that are not immediately obvious. For instance, decimals and square roots are common candidates for estimation.
In this example, since \( \sqrt{120} \) resides between the square roots of 100 and 121, we know \( 10 < \sqrt{120} < 11 \).
We use this boundary information to infer that \( \sqrt{120} \) is slightly less than 11.
Estimation is essential because it simplifies comparisons without requiring precise calculations. Recognizing that \( \sqrt{120} \approx 10.95 \) provides us with near certainty for comparisons.

Mathematical Reasoning

Mathematical reasoning involves drawing logical conclusions based on given information. It requires analytical thinking to assess, compare, and deduce outcomes. In this exercise, reasoning comes into play when evaluating \( \sqrt{120} \) compared to 11.
By identifying precise calculations are not always necessary, we deduced approximate values instead.

• The surrounding perfect squares guide our reasoning: \( \sqrt{100} = 10 \) and \( \sqrt{121} = 11 \).
• Knowing \( \sqrt{120} \) must fall between these provides a logical foothold for comparison.

This method enables quick and accurate decisions, demonstrating that mathematical reasoning relies heavily on recognizing patterns and relationships.