Question

Which one number is closest to 193 which is divisible by 18 is: (a) 180 (b) 195 (c) 198 (d) 108

Short answer

Answer: (c) 198

Step-by-step solution

Divide 193 by 18

First, we need to divide 193 by 18: \[ \frac{193}{18} \approx 10.72 \]

Round the result up and down

Now, we need to round 10.72 up and down to get the closest integers: \[ \lfloor 10.72 \rfloor = 10 \] \[ \lceil 10.72 \rceil = 11 \]

Multiply the rounded integers by 18

Now, multiply the rounded integers (10 and 11) by 18 to find the closest multiples of 18: \[ 10\times 18 = 180 \] \[ 11\times 18 = 198 \]

Compare the multiples to the given options

Compare the two multiples 180 and 198 to the given options and select the closest one to 193: (a) 180 (b) 195 (c) 198 (d) 108 The closest multiple of 18 to 193 is 198 (option c). So, the correct answer is (c) 198.

Key concepts

Quantitative Aptitude

Quantitative aptitude relates to an individual's ability to perform numerical and mathematical calculations. It's an essential skill for problem-solving and reasoning tasks, frequently tested in academic and professional settings. It encompasses various arithmetic concepts, including basic operations like addition, subtraction, multiplication, and division.

When approaching a textbook exercise involving quantitative aptitude, one may need to apply several mathematical principles. For example, when asked to find a number closest to 193 that is divisible by 18, the problem requires an understanding of division and multiplication, as well as knowledge of rounding numbers.

Improving quantitative aptitude involves practicing a wide range of numeric exercises, ensuring that the steps involved in solving such problems are understood, not just memorized. To build this aptitude, focus on fundamental concepts first, such as the division algorithm and rounding numbers, and progressively tackle more complex problems.

Division Algorithm

The division algorithm is a basic mathematical concept that lays the foundation for divisibility rules and operations. It essentially breaks down how division of integers is systematically carried out. When a number, say 'a', is divided by another non-zero number 'b', the division algorithm states that there exist unique integers, 'q' and 'r', such that

\( a = bq + r \) where '\r' is the remainder and 0 ≤ r < b. In the context of our sample exercise, when 193 is divided by 18, the goal is to find such integers 'q' and 'r' to identify the closest number divisible by 18.

Understanding the division algorithm is crucial as it describes the relationship between the divisor, the quotient, and the remainder, which is pivotal when solving problems on divisibility.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify complex or lengthy numbers into approximate values that are easier to work with. There are several methods of rounding, but the most common involve rounding up (ceiling function) or rounding down (floor function).

In the exercise mentioned, 10.72 is not an integer, and thus we round it to the nearest integers, up and down. To round down, or take the floor of 10.72, we simply discard the decimal part, ending up with 10. Conversely, to round up, or take the ceiling of 10.72, we move to the next integer, which is 11.

Rounding is not only practical but also essential in everyday life for making quick estimates and is an important aspect of quantitative aptitude.

Multiples of Integers

Multiples of integers are simply numbers that can be expressed as the product of that integer and some other integer. For instance, multiples of 18 include numbers like 36, 54, 72, and so on, because they can be obtained by multiplying 18 with 2, 3, 4, etc., respectively.

Back to our exercise, once we've determined the numbers to round up and down to (10 and 11), we multiplied them by 18 to find the closest multiples of 18 to 193. Through multiplication, we got 180 and 198. Since we're looking for the number closest to 193 that's divisible by 18, we select 198.

Recognizing and working with multiples is crucial for solving many quantitative problems, as it helps in understanding the number's divisibility, patterns, and in performing efficient calculations.