Problem 72 - Thinking Mathematically

Question

A firefighter spraying water on a fire stood on the middle rung of a ladder. When the smoke became less thick, the firefighter moved up 4 rungs. However it got too hot, so the firefighter backed down 6 rungs. Later, the firefighter went up 7 rungs and stayed until the fire was out. Then, the firefighter climbed the remaining 4 rungs and entered the building. How many rungs does the ladder have?

Step-by-Step Solution

Answer

The total number of rungs on the ladder is 19.

Step 1: Starting Point

The starting point of the firefighter is the middle rung of the ladder.

Step 2: Move Up

When the smoke became less thick, the firefighter moved up 4 rungs, which makes the position above the middle rung as 4.

Step 3: Backup

The firefighter then backed down 6 rungs due to the heat, which makes the current position as 4 (from step 2) - 6 = -2, relative to the middle rung.

Step 4: Climb Up Again

The firefighter then went up 7 rungs, which changes the current position to -2 (from step 3) + 7 = 5, which is above the middle rung.

Step 5: Final Climb

Finally, the firefighter climbed 4 more rungs to enter the building, which makes the current position as 5 (from step 4) + 4 = 9 rungs above the middle rung.

Step 6: Calculating Total Rungs

Since the firefighter finally ended up 9 rungs above the middle, and considering that their starting position was the middle rung, the total number of rungs is twice the number of rungs above the middle (since the ladder is symmetrical) plus one for the middle rung, which is calculated as (2 x 9) + 1 = 19.

Key Concepts

Problem-Solving Strategies

Understanding how to approach a problem is essential in solving mathematical exercises effectively. A key strategy involves identifying the starting point, tracking changes, and calculating based on the information provided. For the ladder rung scenario, let's break down how applying a structured approach aids in reaching the solution.

First, recognize the position of the firefighter on the ladder as the central pivot point. This is your starting reference. Knowing where you begin helps keep all subsequent movements relative and organized. Next, list each movement step and whether it's an upward or downward motion. Using a simple addition and subtraction method, calculate each change from the starting point. This sequential breakdown creates a clear map of the firefighter's movement to avoid confusion.

Another vital strategy is symmetry. With ladders, rungs are equidistant from the middle point, making the structure symmetrical. Identifying symmetry clarifies the calculation process by considering equal segments on either side of the central point. In this way, determining how far above the midpoint helps infer the total number of rungs, ensuring accuracy in mathematical reasoning.

Step-by-Step Solution

Breaking down a problem into manageable steps is a tried-and-true method in mathematics. Let's see how this works through the solution provided for the ladder rungs exercise.

To begin, identify the middle of the ladder as the firefighter's initial position. This point is crucial as it serves as the reference for all movements. Next, the firefighter ascends, so note the increase: 4 rungs up from the midpoint. Then, compute the retreat of 6 rungs, effectively placing the firefighter 2 rungs below the middle. Recording each move helps visualize the firefighter's position.

Following this, an upward movement by 7 rungs positions the firefighter 5 rungs above the midpoint. Conclude by adding the final ascent of 4 rungs, resulting in 9 rungs above the middle. The step-by-step approach ensures each movement's effect is distinct and valid, making it easier to backtrack or double-check calculations. This way, students can adapt similar scenarios confidently by applying the same logical sequence.

Ladder Rungs Calculation

Calculating the number of rungs on a ladder can seem tricky, but it becomes straightforward with a clear understanding of symmetry. Initially, envision the firefighter's moves on the ladder. Each upward or downward step is relative to a midpoint. Determining this point simplifies calculating total rungs.

At the highest point, the firefighter is 9 rungs above the midpoint. Knowing the ladder's symmetry, this high point is mirrored below the middle. Thus, you multiply 9 rungs by 2—representing both sides of the ladder—and add 1 for the middle point itself.

Therefore, with mathematical reasoning and understanding of ladder structure, compute the total rungs as 18 (from the double count) plus 1, equating to 19. This systematic approach reinforces the symmetry concept and ensures accurate rung counting. Such exercises build valuable problem-solving skills, enabling students to tackle diverse mathematical challenges confidently.