Problem 46 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{7}{8}+0.45\left(\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The simplified result is 1.2125.

Step 1: Simplify the Fraction Multiplication

First, we simplify the multiplication within the problem. We have: \( 0.45 \times \frac{3}{4} \). To calculate the product, multiply 0.45 by 3 to get 1.35, then divide by 4. \[ 0.45 \times \frac{3}{4} = \frac{1.35}{4} \approx 0.3375 \].

Step 2: Simplify Addition with the Fraction

Next, we need to add \( \frac{7}{8} \) to the decimal obtained in Step 1, which is \( 0.3375 \). To do this, we convert \( \frac{7}{8} \) to a decimal. \[ \frac{7}{8} = 0.875 \].

Step 3: Perform the Addition

Now, add the two decimal numbers obtained from the previous steps: \( 0.875 + 0.3375 \). The sum is \( 1.2125 \).

Key Concepts

Understanding Fraction Multiplication

When it comes to multiplying a fraction by a decimal, the process may seem tricky at first. It's all about converting the numbers into a format that is easily manageable. For our exercise, we need to multiply 0.45 by \( \frac{3}{4} \). You can start by imagining 0.45 as \( \frac{45}{100} \), but in many cases, it's simpler to multiply directly.
So, multiply 0.45 by 3 to get 1.35. This is the same as having 1.35 whole items. Now, divide this by 4, as the denominator indicates.

The idea here is basic: you're re-distributing 1.35 into four parts, calculating \( \frac{1.35}{4} \), which yields approximately 0.3375.

In essence, fraction multiplication such as \( 0.45 \times \frac{3}{4} \) involves scaling a particular quantity (0.45) by a portion of another (\( \frac{3}{4} \)). This gives you the scaled result (0.3375).

Mastering Fraction Addition

Adding fractions can appear daunting, especially when mixing decimals and fractions. However, it simplifies beautifully with conversion.

In this problem, we have \( \frac{7}{8} \) to add to 0.3375. The key move is to convert the fraction into a decimal first for smoother addition.
To do this, divide the numerator (7) by the denominator (8):

• \( \frac{7}{8} = 0.875 \)

With both numbers as decimals—0.875 and 0.3375—you simply carry out basic addition.
The calculations are straightforward because both are now unified as decimals. Simply add them:

• \( 0.875 + 0.3375 = 1.2125 \)

In this way, fraction addition transforms into decimal addition by a little conversion magic, making it a clear-cut operation.

Effective Simplification of Fractions

Simplifying fractions is a technique meant to make numbers more digestible, reducing fractions to their simplest form. However, this exercise dealt more with conversion than simplification, since we wanted a decimal output.

Often, when simplifying fractions, you look for the greatest common divisor of the numerator and the denominator to divide them by. For instance, if you had \( \frac{8}{12} \), you'd simplify by dividing them by 4, yielding \( \frac{2}{3} \). In our scenario, instead of traditional simplification, we focused on converting to a decimal.
This three-step process is vital:

• Convert any fractions to decimals if needed.
• Perform the operations—like addition and multiplication.
• Ensure the final result is in decimal form, which is easily understandable and concise.

This approach helps bridge the gap between fractions and decimals efficiently, enabling clear calculations and results.