which of the following are rational numbers? 40.975, -95, -67, √92

Explanation:

Step1: Recall rational number definition

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p,q$ are integers and $q\neq0$. Decimals that terminate or repeat, integers, fractions are rational. Irrational numbers are non - repeating, non - terminating decimals (like square roots of non - perfect squares).

Step2: Analyze each number

• For $40.975$: It is a terminating decimal. A terminating decimal can be written as a fraction (e.g., $40.975=\frac{40975}{1000}$), so it is rational.
• For $- 95$: Integers are rational because $-95=\frac{-95}{1}$, so it is rational.
• For $-67$: Integers are rational as $-67=\frac{-67}{1}$, so it is rational.
• For $\sqrt{92}$: Simplify $\sqrt{92}=\sqrt{4\times23} = 2\sqrt{23}$. Since $\sqrt{23}$ is an irrational number (23 is not a perfect square), $2\sqrt{23}$ is also irrational. So $\sqrt{92}$ is not rational. So the incorrect selection here is $\sqrt{92}$. The correct rational numbers are $40.975$, $-95$, $-67$.

Answer:

The rational numbers are $40.975$, $-95$, $-67$ (the selection of $\sqrt{92}$ is incorrect as $\sqrt{92}$ is irrational). So the correct options to check are the boxes for $40.975$, $-95$, $-67$ and uncheck the box for $\sqrt{92}$.