Question

check all statements that are true.
□ since it is a ratio of two integers, $\frac{10}{15}$ is rational.
□ since it is a terminating decimal, 4.6 is irrational.
□ since 10 is not a perfect square, $sqrt{10}$ is rational.
□ since 9 is a perfect square, $sqrt{9}$ is rational.
□ since it is an integer, 12 is irrational.
□ none of the above statements are true.

Explanation:

Step1: Analyze $\frac{10}{15}$

A rational number is a ratio of two integers (denominator non - zero). $\frac{10}{15}=\frac{2}{3}$, and 2 and 3 are integers. So the statement "Since it is a ratio of two integers, $\frac{10}{15}$ is rational" is true.

Step2: Analyze 4.6

A terminating decimal can be written as a fraction of two integers. $4.6=\frac{46}{10}=\frac{23}{5}$, so it is rational. The statement "Since it is a terminating decimal, 4.6 is irrational" is false.

Step3: Analyze $\sqrt{10}$

If a number is not a perfect square, its square root is irrational. Since 10 is not a perfect square, $\sqrt{10}$ is irrational. The statement "Since 10 is not a perfect square, $\sqrt{10}$ is rational" is false.

Step4: Analyze $\sqrt{9}$

$\sqrt{9} = 3$, and 3 is an integer (a rational number as it can be written as $\frac{3}{1}$). Since 9 is a perfect square, $\sqrt{9}$ is rational. The statement "Since 9 is a perfect square, $\sqrt{9}$ is rational" is true.

Step5: Analyze 12

Integers are rational numbers (as they can be written as a ratio of themselves and 1, e.g., $12=\frac{12}{1}$). The statement "Since it is an integer, 12 is irrational" is false.

Answer:

• Since it is a ratio of two integers, $\frac{10}{15}$ is rational.
• Since 9 is a perfect square, $\sqrt{9}$ is rational.