Question

What is the angle between the hands of a clock when the time is 15 minutes past 6 ? (a) \(97.5^{\circ}\) (b) \(87.5^{\circ}\) (c) \(77^{\circ}\) (d) none of these

Short answer

Answer: The angle between the hands of a clock at 6:15 is 97.5 degrees.

Step-by-step solution

Calculate the angle made by the hour hand

Using the formula mentioned above, we can calculate the angle made by the hour hand at 6:15. Replace Hour with 6 and Minutes with 15 in the formula: Hour hand angle = \((Hour × 30) + (Minutes × 0.5)\) = \((6 × 30) + (15 × 0.5)\) = \(180 + 7.5\) = \(187.5\) degrees

Calculate the angle made by the minute hand

Next, we'll calculate the angle made by the minute hand using the formula mentioned above. Replace Minutes with 15 in the formula: Minute hand angle = \(Minutes × 6\) = \(15 × 6\) = \(90\) degrees

Calculate the angle between the hour and minute hands

To find the angle between the hour and minute hands, calculate the absolute difference between the angles calculated in steps 1 and 2: Angle between the hands = \(|Hour \thinspace hand \thinspace angle - Minute \thinspace hand \thinspace angle|\) = \(|187.5 - 90|\) = \(97.5\) degrees Hence, the angle between the hands of a clock when the time is 15 minutes past 6 is \(97.5^{\circ}\), which corresponds to option (a).

Key concepts

Quantitative Aptitude

Quantitative aptitude is an essential area of study for improving problem-solving and numerical ability. It entails the capacity to perform calculations accurately and efficiently, and to understand logical reasoning behind numerical patterns and relationships. The clock angle problem, like the one we explore here, is a notable example of a quantitative aptitude exercise. It requires a blend of basic arithmetic and the ability to apply a formula to ascertain the angle between the hands of a clock at a given time.

When attempting a problem that asks for the angle between the clock hands at 15 minutes past 6, students need first to translate this scenario into a mathematical framework. This requires knowledge of how the clock functions mathematically, which is part of clock arithmetic. Solving these problems improves one's aptitude and prepares students for more complex mathematical challenges, as well as standardized tests and competitive exams that often include similar questions.

Clock Arithmetic

Clock arithmetic, often called modular arithmetic, is a fascinating area of mathematics that deals with the cyclical nature of time as represented by a clock face. In terms of clocks, modular arithmetic is used to calculate angles based on the position of the hour and minute hands.

To tackle clock angle problems, we must recognize that the clock is a circle of 360 degrees, and that each hour mark represents 30 degrees of that circle (360 degrees/12 hours). Furthermore, every minute represents 6 degrees (360 degrees/60 minutes). This allows us to establish formulas such as the ones used in our original exercise. By multiplying the hour by 30 and adding half a degree for each minute, we can find the hour hand angle. The minute hand angle can be found simply by multiplying the minutes by 6. Understanding this arithmetic enables students not only to solve the problems correctly but also to make sense of the underlying principles of modular arithmetic applied in daily life.

Mathematical Reasoning

Mathematical reasoning refers to the logical thinking process used to solve mathematical problems. It involves formulating hypotheses, arriving at conclusions through logical steps, and validating the correctness of those conclusions. In the context of the clock angle problem, mathematical reasoning is crucial when deriving and applying the formulas necessary for calculating hand angles.

For instance, to obtain the angle between the hour and minute hands, one must reason through several steps as demonstrated in the provided solution. Identifying that the situation is indicative of an absolute difference, understanding how to apply the arithmetic formulas, and recognizing the relationship between time and angles on a clock face all involve mathematical reasoning. Such exercises are excellent for enhancing students' ability to reason through problems that may at first seem complex. As learners improve their reasoning skills, they become more adept at approaching and solving diverse mathematical challenges.