Problem 50 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{3}{4}+3.7\)

Step-by-Step Solution

Answer

4.45

Step 1: Convert Fraction to Decimal

First, convert the fraction \( \frac{3}{4} \) to a decimal. To do this, divide the numerator by the denominator: \( 3 \div 4 = 0.75 \). So, \( \frac{3}{4} \) is equivalent to 0.75.

Step 2: Add the Decimals

Now, add the decimal equivalent of \( \frac{3}{4} \), which is 0.75, to 3.7. Perform the addition: \( 0.75 + 3.7 = 4.45 \).

Key Concepts

Converting Fractions to Decimals

To convert a fraction into a decimal, you need to divide its numerator by its denominator. This process helps express the value contained in the fraction as a decimal number, making it easier to combine with other decimal numbers. Let’s simplify this step. Without a calculator, you can do this by performing long division manually. For the fraction \( \frac{3}{4} \), this means you need to divide 3 by 4. - **Step 1**: Since 3 is less than 4, you start by adding a decimal point and a zero, turning it into 30.- **Step 2**: Determine how many times 4 can go into 30, which is 7 times. This gives you 28 when multiplied, leaving a remainder of 2.- **Step 3**: Bring down another zero making it 20, then divide by 4 again, which yields 5.- **Conclusion**: 4 goes 5 times into 20 perfectly without a remainder, giving you a decimal of 0.75.Thus, the fractional part \( \frac{3}{4} \) is precisely 0.75 when converted into a decimal. This method is crucial, as it forms the foundation for further operations with mixed numbers and decimals.

Adding Decimals

Adding decimals is similar to adding whole numbers, but it’s important to line up the decimal points. This ensures that digits representing tens, ones, tenths, hundredths, etc., align correctly. If you don’t align the decimals, your final sum might be incorrect. Here’s how you can approach adding decimals:- **Step 1**: Write the numbers vertically, ensuring that all decimal points are aligned.- **Example**: For adding 0.75 and 3.7, write them as: \[\begin{array}{c} 3.70 \ +0.75 \ \hline \end{array} \] - **Step 2**: Add the numbers from right to left, just like with whole numbers.- **Tip**: Add zeros to make the decimal places equal if necessary. This doesn't change the number's value but makes adding easier.- **Result**: Add starting from the rightmost column (hundredths) to the left, carrying over if needed.The sum of 0.75 and 3.7 is 4.45. It's straightforward once you ensure those decimals are lined up nice and neat!

Terminating Decimals

Terminating decimals are those that have a finite number of digits after the decimal point. Unlike repeating decimals which continue indefinitely, terminating decimals come to an end. Understanding these is crucial, especially when simplifying expressions or during the conversion process from fractions.Here's why a decimal like 0.75 is considered terminating:- **Finite Digits**: It ends after two places, which is manageable and easy to work with.- **Relation to Fractions**: A fraction with a denominator that has factors of only 2 and/or 5 (like \( \frac{3}{4} \)) will convert to a terminating decimal because these factors are factors of 10 (the base of our number system).It's important in mathematics to distinguish between terminating and non-terminating decimals as it affects how calculations are performed, and, in some cases, it dictates what rounding or precision is required.