Problem 46 - Prealgebra

Question

Evaluate each expression. $$(3 \cdot 4)+(9 \cdot 5)$$

Step-by-Step Solution

Answer

57.

Step 1: Evaluate the first multiplication

The expression given is \((3 \cdot 4) + (9 \cdot 5)\). Start by evaluating the first multiplication. Calculate \(3 \cdot 4\).

Step 2: Solve the first multiplication

Multiply 3 by 4 to get 12, so \(3 \cdot 4 = 12\).

Step 3: Evaluate the second multiplication

Now, evaluate the second multiplication in the expression, which is \(9 \cdot 5\).

Step 4: Solve the second multiplication

Multiply 9 by 5 to get 45, so \(9 \cdot 5 = 45\).

Step 5: Add results from both multiplications

Add the results of the two multiplications together. So, add 12 and 45.

Step 6: Solve the addition

Calculate \(12 + 45 = 57\). This is the result of the expression.

Key Concepts

Multiplication

Multiplication is one of the four basic arithmetic operations. It involves finding the product of two numbers, known as factors. When you see a multiplication expression like \(3 \cdot 4\), the dot represents the operation of multiplying the two numbers.
To solve \(3 \cdot 4\), you simply calculate 3 times 4. This means you add 3, four times (3 + 3 + 3 + 3) to obtain 12. This kind of calculation is essential when dealing with problems that involve repeated addition.
Understanding multiplication helps us simplify complex expressions, reduce repetition, and solve problems involving scaling or comparisons quickly. In our exercise, after calculating each multiplication part, the results become components of a simpler addition statement.

Addition

Addition is the process of bringing two or more numbers together to make a new total. It's often the first arithmetic operation that students learn and is fundamental in all mathematics. In solving \((3 \cdot 4) + (9 \cdot 5)\), addition comes into play after you've dealt with multiplication tasks.
Once you have individual results from the multiplication, like 12 from \(3 \cdot 4\) and 45 from \(9 \cdot 5\), you add these results together. So, this involves adding 12 and 45.
The addition here functions as collecting the outcomes of previous operations. You combine the results to conclude the problem as 57. This echoes the importance of ensuring each part of a complex expression is dealt with in sequence before culminating at addition for a final total.

Arithmetic Expressions

Arithmetic expressions are combinations of numbers and operations like addition and multiplication. They can be simple or complex. In our exercise, the expression \((3 \cdot 4) + (9 \cdot 5)\) presents a mini challenge that requires the use of both multiplication and addition.
You must follow 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)). This is vital for reaching the correct result.
By first tackling the multiplication operations within the expression, you rightly simplify it before moving on to addition. Understanding arithmetic expressions involves recognizing these individual steps and ensuring they are executed in order, allowing for complex number problems to be broken down and solved systematically. This approach makes mathematics manageable and more intuitive.