Problem 52 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \((10+5) \cdot(10+5)-4 \cdot(60-4)\)

Step-by-Step Solution

Answer

The value is 1.

Step 1: Solve Inside the Parentheses

Firstly, solve the expressions inside each set of parentheses. For \((10+5)\), the value is \(15\). For \((60-4)\), the value is \(56\).

Step 2: Apply Operations on Parentheses Results

With the solved values from Step 1, now substitute them back into the equation:\((15) \cdot (15) - 4 \cdot (56)\).

Step 3: Perform Multiplication

Perform the multiplication operations first.\(15 \cdot 15 = 225\).\(4 \cdot 56 = 224\).

Step 4: Final Calculation with Subtraction

Now, subtract the second result from the first result to complete the operation:\(225 - 224 = 1\).

Step 5: Verify Result with a Calculator

Enter the original expression \((10+5) \cdot (10+5) - 4 \cdot (60-4)\) into a calculator to verify the result is indeed \(1\).

Key Concepts

Parentheses in Mathematics

Parentheses are an essential tool in mathematics, providing a way to group numbers and operations to indicate which calculations should be performed first. In mathematics, operations enclosed within parentheses must be completed before handling any calculations outside them. This is part of the order of operations often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In the given exercise, parentheses are used to group the numbers for addition and subtraction:

• \( (10+5) \) means that 10 and 5 should be added together first.
• \( (60-4) \) directs us to perform the subtraction of 4 from 60 before using the result for further calculations.

By solving what’s inside the parentheses first, you ensure that the calculations follow a structured and correct approach. This step simplifies complex equations, making them easier and less error-prone to solve.

Multiplication in Mathematics

Multiplication is one of the basic arithmetic operations and involves calculating the total of one number added to itself a specified number of times. When handling equations, multiplication typically comes after solving operations within parentheses and before performing addition or subtraction.

In the given problem, after finding the values inside the parentheses:

• The expression becomes \( 15 \cdot 15 \) for the first part.
• For the second set, it becomes \( 4 \cdot 56 \).

These multiplications should be performed before any subtraction occurs in the equation. By doing this, you first get:

• \( 15 \cdot 15 = 225 \)
• \( 4 \cdot 56 = 224 \)

Completing multiplication before addition or subtraction ensures mathematical operations follow the correct sequence, preventing errors and establishing accurate results.

Subtraction in Mathematics

Subtraction is the process of finding the difference between numbers. When performing operations, subtraction is one of the last steps, following multiplication or division and any operations inside parentheses.

In this exercise, after solving the multiplications:

• We have \( 225 \) from multiplying \( 15 \cdot 15 \)
• And \( 224 \) from \( 4 \cdot 56 \)

Now, subtraction is applied to these results, \( 225 - 224 = 1 \). It’s crucial to following the order of operations to guide which calculations to execute first. This ensures the solution is correct and aligns with mathematical principles. Subtraction finalizes the calculation, providing the solution's overall result.