SSAT Middle Level Exam Prep

Explanation
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (C) is equal to the sum of the squares of the lengths of the other two sides (A and B). This means that C^2 = A^2 + B^2. Therefore, the given answer "The square of C is the sum of the squares of A and B" is true under the Pythagorean theorem.

Explanation
The cost of matches varies directly with the number of boxes. This means that as the number of boxes increases, the cost of matches also increases proportionally. In this case, the cost of five boxes of matches is £100, which means that the cost of one box of matches is £20. Therefore, the cost of matches is directly proportional to the number of boxes.

Explanation
The sum of P, X, Y, and 16 is 58, which means P + X + Y + 16 = 58. Since the numbers are consecutive, we can assume that X is the number immediately after P, and Y is the number immediately after X. Therefore, P + (P+1) + (P+2) + 16 = 58. Simplifying this equation, we get 3P + 19 = 58. Solving for P, we find that P = 13. Therefore, the consecutive numbers are 13, 14, and 15, which matches the given answer.

Explanation
The given question asks to perform a series of operations on the numbers 20 and 6. First, we find the product of 20 and 6, which is 120. Then, we subtract 55 from this result, giving us 65. Finally, we divide this outcome by its lowest multiple, which is 5. Therefore, the final answer is 13.

Explanation
The given expression is a polynomial with four terms. By grouping the terms, we can factor out common factors. In this case, we can factor out (a + 2b) from the first two terms and factor out (-1) from the last two terms. This gives us (a + 2b)(3 + x) - (a + 2b)(-1). Factoring out (a + 2b) from both terms, we get (3 + x) - (-1), which simplifies to 3 + x + 1. Therefore, the correct answer is (3 + x)(a + 2b).

Explanation
The probability of picking a ball that is neither black nor blue can be found by subtracting the probability of picking a black or blue ball from 1. Since there are 20 black balls and 5 blue balls out of a total of 50 balls, the probability of picking a black or blue ball is (20+5)/50 = 25/50 = 1/2. Therefore, the probability of picking a ball that is neither black nor blue is 1 - 1/2 = 1/2.

Explanation
To convert a decimal number to binary, we divide the decimal number by 2 repeatedly and record the remainders in reverse order. In this case, when we divide 105 by 2, the remainder is 1. Dividing the quotient (52) by 2 again, the remainder is 0. Repeating this process, we get the remainders 1, 0, 1, 0, 0, and 1. Writing these remainders in reverse order, we get the binary equivalent of 105 as 1101001.

Explanation
When 64 is divided by twice X, the result is greater than 5. This means that 64 divided by 2X is greater than 5. To find the highest values of X, we need to find the values that make 2X the smallest possible divisor of 64 that gives a result greater than 5. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64. The values of X that make 2X a divisor of 64 are 1, 2, 4, and 8. However, we are looking for values of X that make the result greater than 5. Therefore, the three highest values of X are 4, 5, and 6.

Explanation
To find the sum of 3.5 and the product of 2.8 and 5, we first calculate the product of 2.8 and 5, which is 14. Then we add 3.5 to 14, resulting in a sum of 17.5.

Explanation
To find one-quarter of the negative difference between 30 and 17, we first calculate the difference between the two numbers: 30 - 17 = 13. Then, we take the negative of this difference: -13. Finally, we find one-quarter of this negative difference by dividing it by 4: -13/4 = -3.25. Therefore, the correct answer is -3.25.