The Correct Answer and Explanation is:
Here are the correct answers for each comparison, followed by a detailed explanation.
- 31/4 > √35 is True
- √7 < √49 is True
- 1/3 > 1/2 is False
- √64 < √20 is False
Explanation:
To determine whether each comparison is true or false, we need to evaluate and compare the numbers on both sides of the inequality sign.
For the first comparison, 31/4 > √35, we can convert the fraction to a decimal. 31 divided by 4 equals 7.75. For the other side, we need to estimate the square root of 35. We know that the square root of 36 is 6, so the square root of 35 must be a number slightly less than 6, approximately 5.92. Comparing the two values, 7.75 is clearly greater than 5.92. Therefore, the statement is true.
In the second comparison, √7 < √49, we are comparing two square roots. The square root function is an increasing function, which means that for positive numbers, if one number is smaller than another, its square root will also be smaller. Since 7 is less than 49, it follows that the square root of 7 is less than the square root of 49. We can also calculate the values directly. The square root of 49 is exactly 7, and the square root of 7 is approximately 2.65. The inequality 2.65 < 7 is correct, so this statement is true.
For the third comparison, 1/3 > 1/2, we are comparing two fractions. To compare them, we can find a common denominator, which is 6. The fraction 1/3 is equivalent to 2/6, and 1/2 is equivalent to 3/6. The comparison becomes 2/6 > 3/6. Since 2 is not greater than 3, this statement is false. Alternatively, in decimal form, 1/3 is approximately 0.333 and 1/2 is 0.5. The statement 0.333 > 0.5 is false.
Finally, for the fourth comparison, √64 < √20, we can evaluate each square root. The square root of 64 is exactly 8. For the square root of 20, we know it must be between the square root of 16 (which is 4) and the square root of 25 (which is 5). The value is approximately 4.47. The comparison is 8 < 4.47, which is incorrect. Therefore, the statement is false.
