Problem 67 - Algebra 1

Question

Evaluate the expression. Use estimation to check your answer. $$ -4.89+2.69-(-3.74) $$

Step-by-Step Solution

Answer

The result of the given expression is 1.54.

Step 1: Understand the Order of Operations

In this expression, we have two operations, addition and subtraction. There are no parentheses, exponents or multiplication/division operations, so the expression should be simply calculated from left to right.

Step 2: Simplifying the Negative Before a Bracket

The negative sign before a bracket means that we have to multiply each term in the bracket by -1. So, -(-3.74) will become +3.74. The expression then looks like this: -4.89+2.69+3.74.

Step 3: Perform the Sum and Subtraction

Add and subtract the numbers from left to right: First, add -4.89 and 2.69 to get -2.20. Then, add -2.20 and 3.74 to get 1.54.

Step 4: Estimating the Result

To check the answer, estimate the result: Round -4.89 to -5, round 2.69 to 3 and round 3.74 to 4. Then calculate -5+3+4 which gives 2. The actual answer, 1.54, is close to the estimate, so it is reasonable.

Key Concepts

Order of Operations

When evaluating mathematical expressions, it's crucial to follow the 'order of operations' to ensure you arrive at the correct answer. This rule is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given expression \( -4.89+2.69-(-3.74) \), although there are no parentheses affecting order, there is the presence of a negative number before parentheses, which affects the order in which you operate. In the absence of other operations, you should first deal with this negative and then proceed from left to right, handling addition and subtraction as they appear. This systematic approach prevents common errors and ensures consistency across calculations.

Negative Numbers

Understanding how to work with negative numbers is fundamental in mathematics. A negative number is a number that is less than zero, represented with a minus (-) sign. When dealing with negative numbers, certain rules apply. For instance, the subtraction of a negative number is the same as addition, which is why the expression \( -(-3.74) \) becomes \( +3.74 \).

This nuance is essential when simplifying expressions. If you misunderstand this concept, it could lead to significant errors. It's also worth noting the difference between negative numbers in operations versus negative signs indicating subtraction, as seen with \( -4.89 + 2.69 \) and \( -(-3.74) \) respectively. Grasping the behavior of negative numbers ensures accurate and confident calculations.

Estimation in Mathematics

Estimation is a valuable skill that allows you to quickly determine an approximate value or answer without needing to achieve a precise result. In the context of our expression, estimation serves as a way to verify the plausibility of our calculated answer by rounding numbers to the nearest whole number and performing the operations.

For \( -4.89+2.69-(-3.74) \), we round -4.89 to -5, 2.69 to 3, and -(-3.74) to 4 to get an estimated sum of \( -5 + 3 + 4 = 2 \). While the estimate isn't exactly accurate, it's close enough to the computed answer of 1.54, illustrating that our calculation is reasonably accurate. Estimation helps you to double-check your work and develop a better intuition for numbers. Importantly, being proficient at estimation can also be a practical tool in everyday situations when exact values aren't necessary.