Problem 41 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{2}{5}(0.3)+\frac{3}{5}(0.3)$$

Step-by-Step Solution

Answer

The expression simplifies to 0.30.

Step 1: Distribute Terms

First, we need to distribute the terms in the expression to simplify it. We have two terms: \(\frac{2}{5}(0.3)\) and \(\frac{3}{5}(0.3)\). Distribute \(0.3\) in both fractions:\[\frac{2}{5} \times 0.3 + \frac{3}{5} \times 0.3\]

Step 2: Multiply Each Term

Now, multiply each fraction by \(0.3\). Calculate each product separately:\[\frac{2}{5} \times 0.3 = 0.12\] \[\frac{3}{5} \times 0.3 = 0.18\]

Step 3: Add the Products

Add the results from Step 2 to find the final simplified form. Add \(0.12\) and \(0.18\):\[0.12 + 0.18 = 0.30\]

Step 4: Write as a Decimal

The final result is already in a decimal form. Thus, the calculation simplifies to \(0.30\).

Key Concepts

Understanding Decimal Representation

Decimals are a way to express fractions or parts of a whole in a different form. Instead of using a numerator and denominator (like fractions), decimals use a decimal point to separate the whole number part from the fractional part.
For example, the decimal representation of the fraction \(\frac{2}{5}\) can be found by dividing 2 by 5, resulting in 0.4. This means that \(\frac{2}{5}\) is equal to 0.4 in decimal form. Similarly, the fraction \(\frac{1}{10}\) is expressed as 0.1 in decimal form.
Decimals are particularly useful for simplifying calculations and making numbers easier to understand and compare. Some key points about decimals include:

• They are used in everyday applications like money, measurements, and scientific notations.
• Converting between fractions and decimals can help in performing arithmetic operations more easily.
• Decimals allow for precise representation of non-whole numbers.

When simplifying expressions, understanding decimal representation helps us effortlessly switch between fractions and decimals, as seen in the original exercise.

Fraction Multiplication Simplified

Multiplication involving fractions is a fundamental math skill, but when decimals are mixed in, it might seem a bit more complex. Not to worry—it's all about following a simple process:
When multiplying a fraction by a decimal, you can either convert the fraction to a decimal before multiplying or directly multiply as shown in the step-by-step solution.

• First, understand that multiplying \(\frac{2}{5}\) by 0.3 is the same as multiplying \(\frac{2}{5}\) by 3 and then dividing the product by 10 (since 0.3 is \(\frac{3}{10}\)).
• Calculate: \(\frac{2}{5} \times \frac{3}{10} = \frac{6}{50}\). Simplify \(\frac{6}{50}\) to \(\frac{3}{25}\), which is 0.12 when converted to a decimal.
• Perform the same steps for \(\frac{3}{5} \times 0.3\) to get 0.18.

Fraction multiplication becomes more straightforward when you remember that you can multiply numerators together and denominators together, then simplify if possible, or convert the fraction to a decimal and multiply directly. This method is an essential concept for tackling mixed numerical forms like in the given exercise.

Adding Decimals for Simplification

Adding decimals is simple and similar to adding whole numbers, but you must align the decimal points correctly. When adding decimal numbers like 0.12 and 0.18, make sure each decimal point is directly above or below each other.
This ensures the numbers' place values are accurately added together:

• Start by stacking the numbers:
      0.12
  +0.18
  ------
  
• Add the numbers starting from the rightmost position (hundredths place in this example).
• The sum of 2 and 8 is 10 (write 0 and carry 1 to the tenths place).
• Then add 1 (carried over) to 1 and 1 in the tenths place, giving a sum of 3.
• Write the result: 0.30.

After performing addition, double-check your work to ensure accuracy. Proper alignment of decimal places and careful addition leads to the correct solution. This step is crucial in scenarios like the original exercise, where the final expression simplifies by adding two decimals.