Problem 90 - Elementary and Intermediate Algebra

Question

Look Alikes... a. \(0.87+0.29\) b. \(-0.87+0.29\) c. \(-0.87+(-0.29)\) d. \(0.87+(-0.29)\)

Step-by-Step Solution

Answer

a. 1.16, b. -0.58, c. -1.16, d. 0.58

Step 1: Add Two Positive Numbers

Let's solve part (a). We need to add 0.87 and 0.29. Start by aligning the decimal points and adding the numbers as you would with whole numbers:\[\begin{array}{c} 0.87 \+0.29 \\hline 1.16\end{array}\]So, \(0.87 + 0.29 = 1.16\).

Step 2: Add a Negative and a Positive Number

Let's solve part (b). We need to add -0.87 and 0.29. Treat this as subtracting 0.29 from 0.87 because adding a negative is the same as subtraction:\[-0.87 + 0.29 = -(0.87 - 0.29)\]Calculate the difference:\[0.87 - 0.29 = 0.58\]Since the negative number has a greater absolute value:\[-0.87 + 0.29 = -0.58\].

Step 3: Add Two Negative Numbers

Let's solve part (c). We need to add -0.87 and -0.29. When adding two negative numbers, add their absolute values and place a negative sign in front:\[-0.87 + (-0.29) = -(0.87 + 0.29)\]Calculate the sum:\[0.87 + 0.29 = 1.16\]Thus:\[-0.87 + (-0.29) = -1.16\].

Step 4: Add Positive and Negative Numbers

Let's solve part (d). We need to add 0.87 and -0.29. This is equivalent to subtracting 0.29 from 0.87:\[0.87 + (-0.29) = 0.87 - 0.29\]Perform the subtraction:\[0.87 - 0.29 = 0.58\].Therefore:\[0.87 + (-0.29) = 0.58\].

Key Concepts

Adding Positive and Negative Numbers

Understanding how to add positive and negative numbers is a fundamental skill. When dealing with a positive and a negative number, think of it as a battle between the two numbers. The number with the higher absolute value "wins," and the result takes its sign. If we have

• a positive number and a negative number,
• or two numbers with different signs,

we treat this as a subtraction problem.
To solve an expression like \(-0.87 + 0.29\), compare the absolute values:

• The absolute value of \(-0.87\) is 0.87.
• The absolute value of 0.29 is 0.29.

Since 0.87 is greater, the result will be negative. Calculate the difference:
0.87 - 0.29 = 0.58, so the answer is -0.58. In terms of arithmetic rules:

• Positive + Negative = Subtract and take sign of larger absolute value,
• Negative + Positive = Same as above.

This concept helps simplify complex number operations. Keep the sign of the larger number's absolute value and subtract when the signs differ.

Decimal Subtraction

Working with decimals might seem tricky at first, but once you know the ropes, it’s quite simple. Decimal subtraction is much like subtracting whole numbers. Consider the problem breakdown for subtraction between decimals such as \(0.87 - 0.29\):

• Start by aligning the decimal points of both numbers.
• Imagine 0.87 minus 0.29 as if working on the whole number parts, while keeping decimal alignment in place.

This gives you:\[\begin{array}{c}0.87 \-0.29 \\hline0.58\end{array}\]It’s important to maintain the decimal points’ positions to ensure accuracy. Remember that the subtraction of one decimal from another is fundamentally about finding the difference between their values. Decimal numbers require careful attention but become easier with practice. Always double-check your placement and result. Aligning every digit correctly is key.

Absolute Value in Arithmetic

Understanding absolute value is crucial in arithmetic as it represents the "magnitude" of a number, regardless of its sign. It's like an anchor that shows how far a number is from zero without needing to consider direction (positive or negative).
In problems involving numbers with different signs, focus on the absolute values to determine which number "dominates." For example, in \(-0.87\), the absolute value is 0.87. Absolute values play a vital role in operations like:

• Adding negative and positive numbers: decide the dominant number.
• Subtracting numbers: help us understand the distance between them.

The absolute value helps simplify problems because:

• It turns any number into a non-negative number,
• Giving you a clearer picture for comparison.

Overall, absolute values convert complex problems into straightforward number comparisons, ensuring operations such as addition and subtraction are more manageable and accurately assessed.