a) In a day, how many times is a straight angle formed between minute and hour hands? b) In a day, how many ti

Exercise 2.2 · Q7

Q. a) In a day, how many times is a straight angle formed between minute and hour hands?

b) In a day, how many times is a right angle formed between minute and hour hands?

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Concept understanding — Average

What Does "Average" Really Mean?

Imagine you and three friends share a pizza. The pizza has 8 slices. If you divide it equally, each person gets 2 slices. That "2" is the average number of slices per person.

The core idea is simple: the average is a single number that represents a whole group of numbers — it's the "fair share" if everything were distributed equally.

The Intuition: Balancing the Scale

Think of a seesaw. If you put weights at different positions, the seesaw tilts. The average is the balancing point — the single position where the seesaw would stay level if all the weights were the same.

For example, consider the numbers 2, 4, and 6. Their average is 4. Notice how 2 is 2 below 4, and 6 is 2 above 4 — they balance perfectly around 4.

Note

The average doesn't have to be one of the numbers in the set. The average of 1, 2, and 6 is 3, which isn't in the list at all.

The Precise Definition

The most common average — called the arithmetic mean — is defined as:

\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}

That's it. Add everything up, then divide by how many things you added.

Example

Find the average of: 5, 8, 12, 15

Sum: $5 + 8 + 12 + 15 = 40$
Count: There are 4 numbers.
Divide: $40 \div 4 = 10$

The average is 10.

Why Does This Work?

The formula is just a compact way of saying: "If every value were the same, what would that common value be?"

Suppose you have test scores: 70, 80, 90. The sum is 240. If all three scores were equal, each would be $240 \div 3 = 80$ . So 80 is the average — the number that, repeated three times, gives the same total.

Tip

To check your average: multiply it by the number of values. You should get back the original sum. For the example above: $10 \times 4 = 40$ ✓

A Common Misunderstanding

Many students think the average is "the most common value" or "the middle value." It's not.

Average (mean): the balancing point — sensitive to every number.
Median: the middle value when sorted.
Mode: the most frequent value. …

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Doubling the standard 12-hour counts (11 straight-angle events, 22 right-angle events) for a full 24-hour day gives 22 and 44 respectively.

180^\circ

Given: one full day $=24$ hours $=2\times12$ hours.

(a) Straight angle ( $180^\circ$ ): occurs $11$ times in every $12$ -hour cycle (the minute hand gains a full $360^\circ$ on the hour hand $11$ times in $12$ h, being exactly opposite once per such gain-cycle).

\text{In 24 h: } 11 \times 2 = 22\ \text{times}

(b) Right angle ( $90^\circ$ ): the hands are perpendicular twice in each of those $11$ relative-revolution cycles (once approaching, once receding), giving $22$ times in $12$ h. $\text{In 24 h: } 22 \times 2 = 44\ \text{times}$ …

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Q.a) In a day, how many times is a straight angle formed between minute and hour hands?

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