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Chapter 3: Problem 38

The function \(\&(\mathrm{x})\) is a clock function; it gives values between 0 and 12. \&(13) returns the value 1 , for example. What, then, is the result of \&(15) \(-\&(16)\) ? (A) 12 (B) \(-1\) (C) 11 (D) 7 (E) 1

Short Answer

Expert verified
Option (B): -1

Step by step solution

01

Understand the Function

The function \(\&{(x)}\) is a clock function that returns values between 0 and 12. It behaves like a 12-hour clock, where after 12 it resets to 1.
02

Calculate \(\&{(15)}\)

To find \(\&{(15)}\), consider that it behaves like a 12-hour clock. So, we take 15 mod 12, which gives us \(15 - 12 = 3\). Therefore, \(\&(15) = 3\).
03

Calculate \(\&{(16)}\)

Similarly, to find \(\&{(16)}\), we take 16 mod 12, which gives us \(16 - 12 = 4\). Therefore, \(\&(16) = 4\).
04

Compute the Difference

Now, we need to compute \(\&(15) - \&(16)\). Since \(\&(15) = 3\) and \(\&(16) = 4\), the expression is:\(3 - 4 = -1\).
05

Select the Correct Answer

The correct answer is \(-1\), which corresponds to option (B).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around when they reach a certain value—the modulus. It's like counting on a clock: after you pass 12, you start back at 1.

In mathematical terms, this is expressed as the remainder when one number is divided by another. For instance, in the given problem, to find \(\&{(15)}\), we calculate 15 mod 12. This means we divide 15 by 12, which leaves a remainder of 3. Hence, \(\&{(15)}\) is 3.

Similarly, \(\&{(16)}\) is calculated by finding 16 mod 12, which leaves a remainder of 4. Thus, \(\&{(16)}\) is 4.
Clock Arithmetic
Clock arithmetic is a practical application of modular arithmetic using a 12-hour clock as an example. Imagine a clock face: when you go past 12, the cycle starts again at 1.

The problem of finding \(\&{(15)}\) and \(\&{(16)}\) is akin to determining what hour it would be if you counted 15 or 16 hours from 12 o'clock. After 12 hours, the count resets, hence, \(\&{(15)}\) gives us 3, and \(\&{(16)}\) gives us 4.

In this context, subtracting one time from another is straightforward. For instance, \(\&{(15)}\) - \(\&{(16)}\) is analogous to figuring out the time difference between 15 o'clock and 16 o'clock on a 12-hour clock.
GMAT Math
The GMAT often includes questions that test your understanding of basic mathematical principles like modular arithmetic. It's crucial to grasp these concepts to solve problems efficiently.

In the given exercise, the function \(\&{(x)}\) represents a clock function, frequently used in GMAT problems to test modular arithmetic skills.

Here are the steps used in the solution:
  • Identify the function's behavior: \(\&{(x)}\) resets after 12.
  • Calculate individual function values using modulo operation.
  • Subtract the results to find the difference.

Knowing how to apply these steps can be beneficial in tackling similar GMAT math problems efficiently. Consistent practice with clock arithmetic and modular arithmetic will make these types of problems more intuitive.

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