Comparing Square Root Of 130 And 111/8 Using Inequalities

Hey guys! Today, we're diving into a fun little math problem where we need to compare two numbers: $\sqrt{130}$ and $\frac{111}{8}$. Our mission is to figure out which one is bigger, smaller, or if they're equal. We'll use the symbols $<$, $>$, or $=$ to show our findings. So, let's get started and break this down step by step!

Understanding the Numbers

First, let's get a good grasp of what these numbers actually represent. Square root of 130 ($\sqrt{130}$): This is the number that, when multiplied by itself, gives us 130. We know that $11^2 = 121$ and $12^2 = 144$, so $\sqrt{130}$ lies somewhere between 11 and 12. It's closer to 11 because 130 is closer to 121 than it is to 144.

Fraction 111/8: This is an improper fraction, which means the numerator (111) is larger than the denominator (8). To get a better sense of this number, we can convert it into a mixed number. If we divide 111 by 8, we get 13 with a remainder of 7. So, $\frac{111}{8}$ is equal to $13\frac{7}{8}$. This tells us that $\frac{111}{8}$ is a bit less than 14, which is useful information for our comparison. Now that we have a basic understanding of our numbers, let's dive deeper into comparing them. Estimating the square root involves finding perfect squares around the number under the root. In our case, 130 lies between 121 ($11^2$) and 144 ($12^2$). This tells us that $\sqrt{130}$ is between 11 and 12. To get a more accurate estimation, we can consider that 130 is closer to 121 than 144. We might guess that $\sqrt{130}$ is around 11.4. Using calculators or approximations, we can refine this estimation further, but for our comparison's sake, knowing it's between 11 and 12 is a great start. The fraction $\frac{111}{8}$ can be understood better by converting it to a mixed number. Dividing 111 by 8 gives us 13 with a remainder of 7. Thus, $\frac{111}{8}$ is equal to $13\frac{7}{8}$. This mixed number representation immediately tells us that $\frac{111}{8}$ is greater than 13 but less than 14. The fraction $\frac{7}{8}$ is quite close to 1, so $\frac{111}{8}$ is almost 14. This precise understanding will be crucial when we compare it with $\sqrt{130}$.

Method 1: Squaring Both Numbers

One way to compare these numbers is to square both of them. This works because the squaring function is increasing for positive numbers. That means if $a > b$, then $a^2 > b^2$ (when both a and b are positive). Let's square $\sqrt{130}$: $(\sqrt{130})^2 = 130$. Now, let's square $\frac{111}{8}$: $(\frac{111}{8})^2 = \frac{111^2}{8^2} = \frac{12321}{64}$. To compare these results, we need a common denominator or a decimal representation. Let's convert $\frac{12321}{64}$ to a mixed number or a decimal. Dividing 12321 by 64, we get approximately 192.515625. Alternatively, we can express 130 with a denominator of 64: $130 = \frac{130 \times 64}{64} = \frac{8320}{64}$. Now we're comparing $\frac{8320}{64}$ and $\frac{12321}{64}$. Clearly, 12321 is much bigger than 8320. So, $(\frac{111}{8})^2 > (\sqrt{130})^2$. Since squaring preserves the inequality for positive numbers, this means $\frac{111}{8} > \sqrt{130}$. This method gives us a direct comparison by eliminating the square root and working with simpler fractions or decimals. Squaring both values helps eliminate the square root, making the comparison more straightforward. Squaring $\sqrt{130}$ simply gives us 130. Squaring $\frac{111}{8}$ requires a bit more calculation. We square the numerator and the denominator separately to get $\frac{111^2}{8^2} = \frac{12321}{64}$. To effectively compare 130 and $\frac{12321}{64}$, we can either convert the fraction to a decimal or rewrite 130 as a fraction with the same denominator. Converting $\frac{12321}{64}$ to a decimal gives us approximately 192.515625. Alternatively, converting 130 to a fraction with a denominator of 64 requires multiplying 130 by 64, resulting in $\frac{8320}{64}$. Now we can clearly compare the two squared values: $\frac{8320}{64}$ and $\frac{12321}{64}$. Since 12321 is significantly larger than 8320, we can conclude that $(\frac{111}{8})^2 > (\sqrt{130})^2$. Because the squaring function preserves the inequality for positive numbers, this leads us to the conclusion that $\frac{111}{8} > \sqrt{130}$.

Method 2: Estimating and Comparing

Another approach is to estimate the values and compare them directly. We already know that $\sqrt{130}$ is between 11 and 12, and we estimated it to be around 11.4. We also know that $\frac{111}{8} = 13\frac{7}{8}$, which is very close to 14. Let's get a more precise decimal value for $\frac{111}{8}$: $\frac{111}{8} = 13.875$. Now we can directly compare 11.4 (our estimate for $\sqrt{130}$) and 13.875. It's clear that 13.875 is much larger than 11.4. This method relies on good estimations and a clear understanding of the numbers. If our estimations are accurate enough, this can be a quicker way to compare. Estimating the square root involves identifying perfect squares close to the number under the root. We know that $11^2 = 121$ and $12^2 = 144$. Since 130 is between 121 and 144, $\sqrt{130}$ lies between 11 and 12. To refine our estimation, we note that 130 is closer to 121, so $\sqrt{130}$ is likely closer to 11 than to 12. A reasonable estimate might be 11.4 or 11.5. For comparing to $\frac{111}{8}$, this level of precision is quite helpful. Converting the fraction to a decimal provides a straightforward value for comparison. Dividing 111 by 8 gives us 13.875. This decimal representation makes it immediately clear how $\frac{111}{8}$ stacks up against our estimate of $\sqrt{130}$. With $\sqrt{130}$ estimated at around 11.4 or 11.5, and $\frac{111}{8}$ precisely at 13.875, the difference is quite apparent. Comparing the estimated value of $\sqrt{130}$ (approximately 11.4) with the decimal value of $\frac{111}{8}$ (13.875), we see a significant gap. This direct comparison makes it evident that 13.875 is considerably larger than 11.4. Therefore, this estimation method quickly confirms that $\frac{111}{8} > \sqrt{130}$.

Method 3: Using a Calculator

The most straightforward method is to simply use a calculator to find the decimal approximations of both numbers. This gives us: $\sqrt{130} \approx 11.40175425...$ and $\frac{111}{8} = 13.875$. Comparing these decimal values, we can clearly see that 13.875 is greater than 11.40175425.... This method is quick and accurate, but it's always good to understand the underlying math and be able to estimate and compare without relying solely on a calculator. Using a calculator is a straightforward method for finding the decimal approximations of both numbers, which facilitates a direct comparison. When we use a calculator to find the decimal value of $\sqrt{130}$, we get approximately 11.40175425.... This precise value gives us a clear point of comparison. Similarly, using a calculator to convert the fraction $\frac{111}{8}$ to a decimal yields 13.875. This exact decimal value is easy to work with. Comparing the decimal approximations obtained from the calculator, we have $\sqrt{130} \approx 11.40175425...$ and $\frac{111}{8} = 13.875$. It is immediately evident that 13.875 is greater than 11.40175425.... Therefore, this method confirms that $\frac{111}{8} > \sqrt{130}$.

Conclusion

Using all three methods, we've consistently found that $\frac{111}{8}$ is greater than $\sqrt{130}$. So, we can confidently say: $\sqrt{130} < \frac{111}{8}$. Guys, it’s awesome how we can use different approaches to solve the same problem and arrive at the same answer! Whether we squared both numbers, estimated and compared, or used a calculator, the result is the same. Keep exploring and happy math-ing! In conclusion, through various methods such as squaring both numbers, estimating and comparing, and using a calculator, we have consistently determined that $\frac{111}{8}$ is greater than $\sqrt{130}$. This consistent outcome across different methods reinforces the accuracy of our findings. Therefore, we can confidently state that $\sqrt{130} < \frac{111}{8}$. This exercise highlights the importance of using multiple approaches to verify mathematical comparisons, ensuring a comprehensive understanding and accurate results. The ability to solve problems through different techniques showcases a deeper mathematical insight and builds confidence in the solutions obtained.