Problem 91 - Elementary and Intermediate Algebra

Question

Military Science. During a battle, an army retreated \(1,500\) meters, regrouped, and advanced \(2,400\) meters. The next day, it advanced another \(1,250\) meters. Find the army's net gain.

Step-by-Step Solution

Answer

The army's net gain is 2,150 meters.

Step 1: Calculate the Initial Retreat

The army initially retreated 1,500 meters. This retreat is considered a negative progress towards their advance.

Step 2: Calculate the First Advance

After regrouping, the army advanced 2,400 meters. This is a positive contribution towards their progress.

Step 3: Calculate the Second Advance

The next day, the army further advanced 1,250 meters. This is also a positive contribution towards their progress.

Step 4: Calculate the Net Gain

To find the net gain, add the positive advances together and subtract the retreat. This can be expressed as: \( \text{Net Gain} = 2,400\,\text{meters} + 1,250\,\text{meters} - 1,500\,\text{meters} \).

Step 5: Simplify the Net Gain Calculation

Perform the arithmetic operations: first, calculate the total advance: \(2,400 + 1,250 = 3,650\) meters. Then subtract the retreat to find the net gain: \(3,650 - 1,500 = 2,150\) meters.

Key Concepts

Addition and Subtraction in Algebra

Algebra frequently involves addition and subtraction to solve problems involving various quantities and their changes. An essential step in working with any algebraic problem, such as the one involving calculating net gain, is to correctly perform these operations. To calculate the net gain, you first identify the values you need to add or subtract. In this exercise, the positive values are the distances advanced, and the negative value is the one retreated.

• Identify what needs adding: In this case, advances of 2,400 meters and 1,250 meters.
• Identify what needs subtracting: The retreat of 1,500 meters.

Once these are identified, you perform the addition to get a sum, then subtract the retreat from this sum, giving a net result. This step-by-step approach clarifies the process and also helps prevent common errors. Remember, practicing these addition and subtraction steps is crucial for mastering algebraic expressions.

Positive and Negative Integers

In algebra, understanding positive and negative integers is key to successfully solving many problems. An integer represents a whole number, and when equipped with positive or negative signs, they signify direction or magnitude in a calculation. Using our problem as a practical example: - Positive integers are the distances the army advanced, namely 2,400 meters and 1,250 meters. - The negative integer ( -1,500 g) represents the retreat. When combining positive and negative integers:

• Add when integers have different signs by essentially calculating the difference without regard to sign, then give the sign according to the larger absolute value.
• Subtract when both numbers are positive, simply follow the subtraction rules.
• For adding two negative integers, sum the values as if they are positive, afterwards reapply the negative sign.

In this case, correctly handling these integer signs ensures accurate representation of the army's movements.

Distance Problems in Algebra

Distance problems in algebra often require a clear understanding of movement and direction to calculate net distances correctly. In algebra, not only is the distance important, but also whether it is positive or negative. This leads to determining the net effect or the total effective distance. For our exercise, the task was to determine an army's net gain in positioning: - First, account for and add positive advances, which are distance gains. - Then, subtract the backward movement or retreat (represented as a negative distance). This illustrates a broader application: whether dealing with navigating a battle or any scenario involving changes in position, calculating net distance requires:

• Summation of all positive movements.
• Subtraction of any retreats or backward steps.

Solving distance problems like this encourages one to precisely track all movements, demonstrating an essential skill in both personal and academic pursuits.