Problem 57 - Prealgebra

Question

Perform the indicated operations. \(\frac{9}{5}(120)+32\)

Step-by-Step Solution

Answer

The result of the operation is 248.

Step 1: Analyze the Expression

The expression we need to solve is \(\frac{9}{5}(120) + 32\). We will start by handling the multiplication inside the parentheses first.

Step 2: Perform the Multiplication

Multiply \(\frac{9}{5}\) by 120. Calculate: \[ \frac{9}{5} \times 120 = \frac{9 \times 120}{5} = \frac{1080}{5} \]Divide 1080 by 5: \[ \frac{1080}{5} = 216 \]

Step 3: Add 32 to the Product

Now, add 32 to the result of the multiplication from Step 2:\[ 216 + 32 = 248 \]

Step 4: Arrive at the Final Result

The solution to the expression \(\frac{9}{5}(120) + 32\) is 248.

Key Concepts

Understanding Multiplication

Multiplication is an essential operation in mathematics. It involves finding the total of one number added repeatedly. To put it simply, if you want to multiply 9 by 5, you are essentially adding 9 to itself 5 times - 9 + 9 + 9 + 9 + 9.

Here are some key points about multiplication:

• Multiplication is often symbolized by an asterisk (*) or simply written as two numbers beside each other, like in 9 x 5 or 9(5).
• It helps in simplifying complex addition problems.
• Multiplication is commutative. That means the order of numbers doesn't change the result. For instance, 9 * 5 is the same as 5 * 9.

In our given exercise, we perform multiplication first because of the order of operations (often remembered by the acronym PEMDAS/BODMAS). This ensures we handle all arithmetic actions correctly before proceeding to other operations like addition.

The Significance of Addition

Once you've completed any multiplications, the next step is to use addition to combine values. Addition is simply the process of bringing two or more numbers together to make a new total.

Consider the highlights of addition:

• It is represented by the plus symbol (+).
• Addition is also commutative, so changing the order won't affect the outcome. For example, 3 + 4 is equal to 4 + 3.
• Unlike multiplication, addition acts directly on the numbers given without needing to think about repeated groupings.

In the original step-by-step solution, once the multiplication gives us a result, we simply add 32 to it to get the final total of 248. This straightforward approach keeps computations simple and accurate when following the correct order of operations.

Fraction Multiplication Explained

Fraction multiplication might initially seem tricky, but with practice, it can become as intuitive as whole number multiplication. It involves multiplying the numerators of the fractions (top numbers) and the denominators (bottom numbers) separately.

Here's how fraction multiplication works:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify the fraction if possible.

For instance, multiplying \( \frac{9}{5} \) by 120, we treat 120 as a fraction (\( \frac{120}{1} \)) and proceed by multiplying: \( \frac{9}{5} \times \frac{120}{1} = \frac{9 \times 120}{5 \times 1} \). This yields \( \frac{1080}{5} \), which simplifies to 216 after dividing.

By understanding the straightforward nature of multiplying fractions, you can tackle more complex numerical problems with ease, maintaining accuracy in calculations.