Question

A so-called 7 -way lamp has three 60 -watt bulbs which may be turned on one or two or all three at a time, and a large bulb which may be turned to 100 watts, 200 watts or 300 watts. How many different light intensities can the lamp be set to give if the completely off position is not included? (The answer is not 7 .)

Short answer

There are 15 different light intensities.

Step-by-step solution

Calculate total power from the 60-watt bulbs

Determine the different intensities that can be achieved by the three 60-watt bulbs. The possible combinations are:- 1 bulb on: 60 watts- 2 bulbs on: 120 watts- 3 bulbs on: 180 wattsThis gives us three possible intensities: 60, 120, and 180 watts.

List the intensities of the large bulb

The large bulb has three set intensities:- 100 watts- 200 watts- 300 wattsThis gives us three more possible intensities: 100, 200, and 300 watts.

Combine intensities from 60-watt bulbs and large bulb

Next, find all possible combinations by adding the intensities from the 60-watt bulbs to the intensities from the large bulb. For each of the 3 intensities from the 60-watt bulbs (60, 120, and 180 watts), add each of the 3 intensities from the large bulb (100, 200, and 300 watts):- 60 + 100 = 160 watts- 60 + 200 = 260 watts- 60 + 300 = 360 watts- 120 + 100 = 220 watts- 120 + 200 = 320 watts- 120 + 300 = 420 watts- 180 + 100 = 280 watts- 180 + 200 = 380 watts- 180 + 300 = 480 wattsThis step gives us nine possible light intensities.

List all unique intensities

Combining the unique intensities from all previous steps, we get:60, 120, 180, 100, 200, 300, 160, 220, 280, 260, 320, 380, 360, 420, 480This step provides a complete list of potential light intensities in increasing order.

Count the total number of unique intensities

Since the completely off position is not included, count all the unique intensities listed in the previous step. We have a total of 9 + 3 (from step 3) = 15 potential light intensities.

Key concepts

combinations and permutations

Understanding the difference between combinations and permutations is crucial in combinatorial mathematics. When dealing with permutations, the order of elements matters. For example, choosing different light intensities in a specific sequence can represent permutations. However, in this exercise, we're dealing with combinations. Here, we are interested in the different totals we can get from the bulbs regardless of the order.

For instance, turning on one 60-watt bulb and two 60-watt bulbs gives us 60 and 120 watts, respectively. The combinations problem focuses on which and how many bulbs are on at any moment, not the order they are turned on.

This might seem trivial, but understanding whether you're working with a combination or permutation can change how you calculate possibilities and approach a problem.

problem-solving in mathematics

Problem-solving in mathematics is about breaking down a complex problem into manageable steps. In this exercise, to find the total number of light intensities, we split the problem into smaller parts:

* First, identify the different combinations of the 60-watt bulbs.
* Next, list out the possible settings for the large bulb.
* Then, combine these settings to find all potential light intensities.

This method allows us to methodically solve the problem, ensuring no combinations get overlooked. Throughout this process, we apply basic arithmetic to sum the intensities, showing that problem-solving often relies on breaking down a problem and tackling it step by step.

mathematical reasoning

Mathematical reasoning skills are essential to deduce solutions accurately. In this particular problem, reasoning comes into play when we combine the intensities. Observing and understanding the nature of the bulbs help us see how combining different bulb states can give us the total possible light intensities.

For example, when we consider the combinations for the 60-watt bulbs (one, two, or three bulbs on), it's a reasoning exercise to see how these can add up with intensities from the large bulb (100, 200, 300 watts). This involves seeing beyond just the numerical values and understanding their interplay in achieving different light intensities.

This logical approach is an essential part of teaching students how to approach and solve mathematical problems effectively.

light intensity calculations

Calculating light intensity in this problem involves combining the power ratings of different bulbs. The exercise gives us a series of simple operations to follow:

* Add the power of one 60-watt bulb, then two, then three, resulting in 60, 120, and 180 watts.
* Then, list the intensities of the large bulb (100, 200, and 300 watts).

By adding these values together (e.g., 60+100, 120+200), we get the intensities. This approach shows how to handle multiple factors simultaneously when calculating outcomes.

Here, we focus on ensuring no combination is missed, so the step-by-step addition and listing all possible outcomes provide a thorough way to check our answers.

step-by-step problem solving

Breaking the problem into steps is an essential technique in mathematics. Here's how it works for our light intensity problem:

* **Step 1:** Identify all possible intensities from the three 60-watt bulbs.
* **Step 2:** List the fixed intensities for the large bulb.
* **Step 3:** Combine these intensities to find all possible outcomes.

This method ensures we don't miss any possibilities and makes the problem more manageable. By approaching each part systematically, we build up to the full solution, ultimately giving us the complete list of potential light intensities and understanding the structure behind our final answer.