Problem 103 - Elementary and Intermediate Algebra

Question

Look Alikes . . . a. \(2.7+(-0.9)\) b. \(2.7-(-0.9)\) c. \(2.7(-0.9)\) d. \(\frac{2.7}{-0.9}\)

Step-by-Step Solution

Answer

a. 1.8, b. 3.6, c. -2.43, d. -3.

Step 1: Addition of Signed Numbers

To solve part (a) \(2.7 + (-0.9)\), notice that adding a negative number is the same as subtracting its absolute value. Thus, \(2.7 + (-0.9)\) is the same as \(2.7 - 0.9\). Calculate \(2.7 - 0.9 = 1.8\).

Step 2: Subtraction of a Negative Number

In part (b) \(2.7 - (-0.9)\), subtracting a negative number is equivalent to adding its absolute value. Therefore, this expression becomes \(2.7 + 0.9\). Calculate this as \(2.7 + 0.9 = 3.6\).

Step 3: Multiplication of a Positive and Negative Number

For part (c) \(2.7 \times (-0.9)\), when you multiply numbers with different signs, the result is negative. Multiply the absolute values, \(2.7 \times 0.9 = 2.43\), and then apply the negative sign to get \(-2.43\).

Step 4: Division of a Positive and Negative Number

In part (d) \(\frac{2.7}{-0.9}\), dividing a positive number by a negative number gives a negative result. Divide the absolute values, \(\frac{2.7}{0.9} = 3\), and include the negative sign to yield \(-3\).

Key Concepts

Addition of Signed Numbers

Adding signed numbers might seem tricky at first, but it's quite simple when you understand the concept. When you add a positive and a negative number, it is similar to subtracting. In the example of part (a) from the original exercise, we see an addition of 2.7 and -0.9:
- First, focus on the signs of the numbers. We have a positive 2.7 and a negative 0.9. - To add these, you envision moving from positive territory towards zero by the magnitude of the negative number. - Calculate as: 2.7 - 0.9, which equals 1.8.
This process is simple: positive + negative = the difference (with the sign of the larger absolute value). Thus, 2.7 + (-0.9) becomes 1.8. Remember this rule whenever you face addition involving signed numbers.

Subtraction of Negative Numbers

Subtracting a negative number is often where students hit a bump, but the trick lies in understanding what really happens. Much like in part (b) of the original exercise:
- If you have an expression such as 2.7 - (-0.9), it's essential to understand that subtracting a negative is equivalent to adding the positive. - Why? Because two negatives make a positive. - Update the equation to 2.7 + 0.9. - Adding these gives you 3.6.
This method is a lifesaver in arithmetic. Always transform subtraction of a negative number into addition of the positive version, simplifying the problem and reducing error potential.

Multiplication with Negative Numbers

When multiplying numbers with different signs, such as in part (c) of the original exercise, the outcome is negative. Here's why:
- Consider 2.7 multiplied by -0.9. - You multiply the absolute values first: 2.7 times 0.9 equals 2.43. - Since we have a negative number involved, the result becomes negative. - Hence, 2.7 (-0.9) equals -2.43.
The rule here is simple:

• Positive 15 Negative = Negative
• Negative 15 Positive = Negative
• Negative 15 Negative = Positive

This rule is consistent; remember it, and multiplication involving negative numbers will never confuse you.

Division with Negative Numbers

Like multiplication, division with a negative number can seem troublesome, but it follows a straightforward pattern. Let's look at part (d) of the original exercise:
- You have 2.7 divided by -0.9. - Start as you would with division of two positive numbers, - Divide their absolute values: 19(2.7 / 0.9) = 3. - The rule for division is similar to multiplication concerning signs:

• Positive f Negative = Negative
• Negative f Positive = Negative
• Negative f Negative = Positive

Thus, 2.7 divided by -0.9 results in -3. Observing this circular logic in division can simplify many problems, helping you steer clear of common pitfalls.