Problem 68 - Algebra 1

Question

Evaluate the expression. Use estimation to check your answer. $$ -7.85+5.96-(-2.49) $$

Step-by-Step Solution

Answer

The evaluated expression is \(0.60\).

Step 1: Add positive numbers

Start the problem by adding the two positive numbers in the expression to get a preliminary result: \(5.96 + 2.49 = 8.45\).

Step 2: Add negative number

For the next step, remember the rule that when a negative number is added to a positive number, it reduces the positive number. So, add the negative number to the result from step 1: \(8.45 - 7.85 = 0.60\).

Step 3: Estimation

To check the result through estimation, round the decimals to the nearest whole number: \( -8 + 6 - (-3) = 1\). The estimate is close to the actual result, indicating the solution is likely correct.

Key Concepts

Algebraic Estimation

Algebraic estimation is a technique used to approximate the value of expressions to quickly gauge their correctness without needing a calculator or exact calculation.
For instance, when evaluating the expression \( -7.85+5.96-(-2.49) \), a student can use estimation to check if their solution is plausible. To do this, you round the numbers to the nearest whole number or to a decimal place that makes them easy to compute mentally, such as \( -8 + 6 - (-3) \).

• Rounding each term: \( -7.85 \) rounds to \( -8 \), \( 5.96 \) rounds to \( 6 \), and \( -2.49 \) rounds to \( -3 \).
• Change of signs: Remember that subtracting a negative is the same as adding a positive.
• Quick calculation: \( -8 + 6 + 3 = 1 \).

This estimation gives you a reference point. If your exact calculation is close to this estimate, your answer is likely correct; if not, you should recheck your calculations.

Adding Positive Numbers

Adding positive numbers is a straightforward process where you simply combine the values to get a sum.
Using the example \( 5.96 + 2.49 \), you add the two positive numbers by aligning the decimal points and adding each column as you would with whole numbers. The sum of these is \( 8.45 \).

• Decimal alignment: Make sure that the decimal points of the numbers line up vertically.
• From right to left: Add starting from the rightmost digit and move left, carrying over as necessary.
• Place the decimal in the sum directly below the other decimals.

This straightforward method ensures that you're adding the correct place values and getting the right sum.

Adding Negative Numbers

Adding negative numbers might seem complex, but it can be made simple with the right approach.
In our expression, you're actually subtracting a negative number \( -7.85 \), which is equivalent to adding its opposite positive number. Remember that 'two negatives make a positive'. So, \( 8.45 - 7.85 \) becomes \( 8.45 + 7.85 \), and now you have to subtract this absolutely.

• Sign reversal: Understand that subtracting a negative is the same as adding a positive.
• Absolute value: Imagine the negative sign isn't there, and just subtract the absolute values as you normally would.

The result here will be the difference, which in this case can be perceived as the smaller number being subtracted from the larger number to arrive at the answer \( 0.60 \).

Arithmetic with Decimals

Arithmetic with decimals includes adding, subtracting, multiplying, and dividing numbers that are not whole.
For precision, it's important to place decimal points carefully. When adding or subtracting, align the decimal points and proceed as with whole numbers. When multiplying, the number of decimal places in the product is the sum of the decimal places in the numbers being multiplied. Division with decimals might require you to move the decimal to make a whole number.

• Alignment is crucial: Decimal points should be in a vertical line when adding or subtracting.
• Number of decimal places: After multiplication, count decimal places in both operands to place the decimal point correctly in the result.
• Moving decimals: When dividing, you may shift the decimal point in both the dividend and the divisor to simplify the operation.

For our expression, when performing the addition and subtraction, keep the decimal points aligned to reach the correct result \( 0.60 \).