Question

Evaluate the expression. Use estimation to check your answer. $$ 5.3-(-2.5)-4.7 $$

Short answer

The solution to the expression is 3.1.

Step-by-step solution

Understanding the Expression

The given expression is \(5.3-(-2.5)-4.7\). In order to simplify, one needs to consider the fact that subtracting a negative value is the same as adding a positive value.

Removing Parentheses

Start with the part within the parentheses: \(-(-2.5)\). Using the rule: subtracting a negative value equals adding a positive value results in: \(5.3 + 2.5 - 4.7\).

Perform the Calculation

Adding 5.3 and 2.5 gives 7.8 which is subtracted by 4.7 leads to \(7.8 - 4.7 = 3.1\).

Estimating the Result

Even if performing a simple estimation, already the detailed calculation shows that the subtraction of 4.7 from a value of around 8 leads to an approximate result of 3. Therefore, the calculation seems correct.

Key concepts

Evaluating Expressions

Evaluating mathematical expressions involves computing their values using the fixed order of operations and understanding the function of parentheses and signs. When faced with an expression like (5.3-(-2.5)-4.7), it's important to carefully interpret each component.

The presence of a negative sign before the parentheses indicates a subtraction, yet the negative number within suggests a 'double negative'. This is crucial as the 'rule of signs' tells us that a negative sign before parentheses effectively turns the subtraction into addition, thus transforming the negative number into its positive counterpart. This process is an essential part of simplifying expressions, guiding us to rewrite (5.3-(-2.5)-4.7) as (5.3 + 2.5 - 4.7).

This rewriting is not merely about following rules, but understanding the mechanics behind them. When students grasp that subtracting a negative is equivalent to adding a positive, they're building their conceptual understanding of number operations, all which serve as foundation for more complex mathematics.

Arithmetic Operations

Arithmetic operations form the backbone of most math calculations and understanding them is pivotal for performing more complex tasks. Four basic operations are addition, subtraction, multiplication, and division. In the context of our example, we're dealing with subtraction and the nuance of subtracting negative numbers.

When we adjust the original expression from a subtraction of a negative to the addition of a positive, we apply the fundamentals of arithmetic operations. (5.3+2.5-4.7) highlights a sequential approach: start by combining terms that are being added - (5.3+2.5) which equals (7.8), and then subtract the remaining term (4.7) to achieve the final value of 3.1.

This sequence is not just a rote procedure but an application of the commutative property of addition; the order in which we add numbers does not affect the sum, making the calculation process flexible and efficient. Teaching students the meaningful application of these properties can solidify their arithmetic skills and enhance their problem-solving abilities.

Mathematical Estimation

Estimation is a powerful tool that allows us to quickly gauge the 'about' value of an expression without needing to compute it exactly - a skill that becomes particularly useful in checking our work or when precise values are unnecessary.

In estimating (5.3-(-2.5)-4.7), we could round each number to its nearest whole, converting the expression to (5-(-2)-5). This rough approximation guides us towards an expected outcome, which in this case is 3, since (5+2-5) equals 2. Despite this being slightly different from the exact answer due to rounding, it's a reasonable estimate, affirming that our original precise calculation yielding 3.1 is on the right track.

Effective estimation strikes a balance between speed and accuracy, honing a sense for magnitude that's both valuable for academic purposes and a practical life skill when dealing with numbers in everyday contexts. Encouraging students to use estimation validates their answers independently and enhances their numerical intuition.