Number Operations and Properties

If A solves 24 problems in 15 minutes, her rate is 24/15 = 1.6 problems per minute. In 40 minutes, she can solve 1.6 × 40 = 64 problems.

Dividing 30 by 1/2 gives 60 (because 30 ÷ 1/2 = 30 × 2 = 60), and adding 10 gives 70. The common mistake is to think dividing by 1/2 means halving, when it actually means multiplying by 2.

Let the number be x. Following steps: multiply by 2 gives 2x; add 5 gives 2x + 5; multiply by 3 gives 6x + 15; subtract x gives 5x + 15; divide by 5 gives x + 3; subtract x again gives 3. The operations are designed to cancel out the original number, leaving 3. This is an algebraic identity.

Let the number be $x$. The equation is $2x = x + 1/x$. Subtracting $x$ from both sides gives $x = 1/x$, which means $x^2 = 1$. Since it is a natural number, $x = 1$.

Let the number be $x$. The equation is $x/2 = x - 13$. Multiplying by 2 gives $x = 2x - 26$. Solving for $x$ yields $x = 26$. This is a natural number, satisfying the condition.

Let the number be $x$. The equation is $x = 4(1/x)$, which simplifies to $x^2 = 4$. Solving for a positive integer gives $x = 2$. The value -2 is excluded because the question specifies a positive integer.

Let the positive integer be x. Then x(x+1) = 12. Testing options: 3 × 4 = 12, which works. This is the product of consecutive integers that equals 12. Option A (3) is correct.

Let the number be x. Then x + 2x = 21, which simplifies to 3x = 21. Therefore x = 7. Option C (7) is correct.

Interpreting 'half of two' as 2/2=1, then adding 2 gives 3. The phrase 'half of X plus Y' is ambiguous - it could mean (half of 2) + 2 = 3, or half of (2+2) = 2, or (half of 2) + (half of 2) = 2. The answer assumes the first interpretation. This is a poorly worded question.

5×2=10 is a basic multiplication fact. This is foundational arithmetic that students memorize as part of times tables. The other options (9, 5, 20) are incorrect.

4+7=11 is a basic addition fact. This requires either mental math or knowledge of addition facts. The other options (4, 7, 9) are incorrect.

Dividing by 1/2 is the same as multiplying by 2. So 30 ÷ (1/2) = 30 × 2 = 60. Then adding 10 gives 70. Many people mistakenly calculate 30 ÷ 2 = 15 instead. This tests understanding of fraction division.

The pattern 6 9 + + 9 6 X X 6 9 + + 9 6 X X 6 9 + + ? 6 X ? shows a repeating structure: 6 9 + + 9 6 X X appears twice, then 6 9 + +, then ? 6 X ?. The missing elements 9 X would complete the pattern, making it 6 9 + + 9 6 X X to maintain the established structure.