6th Grade Benchmark Cluster

Explanation
The probability of Jennifer selecting a king can be calculated by dividing the number of favorable outcomes (the number of kings in the deck) by the total number of possible outcomes (the total number of cards in the deck). In this case, there are 4 kings in a standard deck of 52 cards, so the probability is 4/52. This fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor, which is 4. Therefore, the probability of Jennifer selecting a king is 1/13.

Explanation
Given that Jackson is in a race with three other people, the probability of Jackson finishing first depends on the assumption that each racer has an equal chance of winning. In this context, with four racers in total, the probability of Jackson finishing first is calculated as the ratio of the number of favorable outcomes (Jackson finishing first) to the total number of possible outcomes (four racers).
Since there is only one favorable outcome (Jackson finishing first) and four racers in total, the probability is 1/4, which translates to approximately 25%. This assumes an equal chance for each racer to win the race.

Explanation
Ann has three choices for each digit in the combination: 2, 6, or 9. Since she can repeat digits and can choose not to use any of them, there are 3 options for each digit. Therefore, the total number of combinations she can make is 3 x 3 x 3 = 27.

Explanation
To make $.65 using at least one quarter and no pennies or half dollars, we can consider the different combinations of coins. We can use one quarter (25 cents) and then use nickels (5 cents) to make up the remaining 40 cents. So, we can have 1 quarter and 8 nickels, 1 quarter and 6 nickels, 1 quarter and 4 nickels, and so on until 1 quarter and 0 nickels. This gives us a total of 7 different ways to make $.65 using at least one quarter and no pennies or half dollars.

Explanation
The dimensions of a bathroom are typically measured in feet or inches, not in miles. Therefore, the option of 2 miles x 3 miles is not a reasonable dimension for a bathroom. However, the option of 9 feet x 12 feet is a reasonable dimension for a bathroom as it falls within the typical range of bathroom sizes. The dimensions of 40 in x 30 in could also be reasonable for a smaller bathroom, as it is within the range of standard bathroom sizes.

Explanation
Each player has 3 choices for picking a door and 4 choices for picking a curtain behind the door. Therefore, the total number of choices for each player is the product of the number of choices for picking a door and the number of choices for picking a curtain, which is 3 * 4 = 12.

Explanation
The probability of getting heads on a coin toss is 1/2. Since there are 4 aces in the deck and one of them is chosen, the probability of choosing the ace of hearts is 1/4. To find the probability of both events happening together, we multiply the probabilities: (1/2) * (1/4) = 1/8. Therefore, the probability of getting heads on the coin and the ace of hearts is 1/8.

Explanation
The rule given is to multiply the input by 3 and then subtract 5. Therefore, if the input is 6, we multiply 6 by 3 to get 18 and then subtract 5 to get the output of 13.

Explanation
The given rule is x3 - 5. When the input is 1, substituting it into the rule, we get 13 - 5 = 8. Therefore, the output is 8.

Explanation
The rule given is x3 - 5. To find the input if the output is 10, we need to solve the equation 10 = x3 - 5. By adding 5 to both sides of the equation, we get 15 = x3. Taking the cube root of both sides, we find that x is equal to 5. Therefore, the input for an output of 10 is 5.

Explanation
The statement "The digit is prime" is true because the number 1 is only divisible by 1 and itself, which are the criteria for a prime number.

Explanation
In a group of 4 judges, each judge shakes hands with the other 3 judges. So, in the first group of judges, there are 4*(4-1)/2 = 6 unique handshakes. Similarly, in the second group of judges, there are also 6 unique handshakes. However, there may also be handshakes between judges from different groups. Since there are 4 judges in each group, there can be 4*4 = 16 unique handshakes between judges from different groups. Therefore, the total number of unique handshakes is 6 + 6 + 16 = 28.

Explanation
The question asks for the number of ways to pick a President, Vice President, and Secretary from the five members. To find the number of ways, we multiply the number of choices for each position. Since there are 5 members to choose from for the President position, 4 members left to choose from for the Vice President position, and 3 members left to choose from for the Secretary position, we multiply 5x4x3 to get the total number of ways, which is 60.