Problem 17 - Prealgebra

Question

In Exercises 17-40, divide the decimals. \(\frac{0.3478}{0.47}\)

Step-by-Step Solution

Answer

The division of the decimals results in approximately 0.7404.

Step 1: Convert to Fraction

Begin by rewriting the division of decimals as a fraction. Given \[ \frac{0.3478}{0.47} \]this can be written as the fraction \( \frac{0.3478}{0.47} \).

Step 2: Align Decimal Places

To divide these decimals, we need to get rid of the decimals by aligning the decimal places. Multiply both the numerator and the denominator by 100 to remove the decimals: \[ \frac{0.3478 \times 100}{0.47 \times 100} = \frac{34.78}{47} \]

Step 3: Perform the Division

Next, perform the division of the fraction: 1. Divide 34.78 by 47 using long division. 2. 47 goes into 347 eight times (47 * 8 = 376, which is too much, so 47 * 7 = 329). 3. Subtract 329 from 347 to get 18. 4. Bring down the next digit, making it 187. 5. 47 goes into 187 three times (47 * 3 = 141). 6. Subtract 141 from 187 to get 46. 7. Bring down the next digit, making it 460. 8. 47 goes into 460 nine times (47 * 9 = 423). 9. Subtract 423 from 460 to get 37. 10. Bring down a zero to continue the division, making it 370. 11. 47 goes into 370 seven times (47 * 7 = 329). 12. Subtract 329 from 370 to get 41, and bring down another zero. 13. 47 goes into 410 eight times (47 * 8 = 376), then 34 again, etc. The quotient is approximately 0.7404, repeating as necessary.

Step 4: Round the Result

If required, you can round the quotient to a suitable number of decimal places. Here, we obtained a quotient of around 0.7404, which can be rounded to 0.74 for simplicity or further decimal places if accuracy is required.

Key Concepts

Long Division

Long division is a method used to divide numbers, especially when the numbers involved are decimals or large values. By breaking down the division process into manageable steps, long division makes it easier to find the quotient.

To perform long division with decimals, you begin as if dividing whole numbers. Note that during the process, you will carefully manage the placement of decimal points in your quotient. Here’s how it typically works:

• Place the divisor outside the division bracket, with the dividend inside.
• Start by determining how many times the divisor fits into the first part of the dividend.
• Multiply, subtract, and bring down the next number, repeating this sequence until the entire dividend is processed.
• Keep note of where and when to place the decimal in the quotient, to ensure accurate results.

Remember, practice makes perfect with long division—especially when handled with decimals.

Fraction Conversion

Before diving into decimal division, transforming the equation into a fraction simplifies the process.

Converting decimals into fractions is helpful when clearing decimal places, allowing for straightforward multiplication. Take \(\frac{0.3478}{0.47}\), for instance, you maintain the structure, but will—most often—multiply to eliminate decimals:

• Identify the position of the decimal in both numbers.
• Multiply the numerator and denominator by ten or any power that clears the decimals, here it's 100.
• This results in \(\frac{34.78}{47}\).

Through fraction conversion, we set the stage for precise long division.

Decimal Alignment

Aligning decimals correctly is crucial for accurate division results. This means ensuring both numbers have the same number of decimal places before dividing.

Here’s how it’s done:

• Count the number of decimal places in each number.
• Multiply each by the appropriate power of ten to remove all decimals—making sure both numbers are integers in the fraction.
• The example follows by turning \(\frac{0.3478}{0.47}\) into \(\frac{34.78}{47}\), simplifying division.

By aligning decimals, we ensure that the division yields an accurate and easy-to-read quotient.

Rounding Decimals

After completing the long division process, you might end up with a long string of digits. Rounding decimals makes the number more manageable and easier to interpret.

Here's how to round decimals accurately:

• Identify how many decimal places you need for your answer (usually specified).
• Look at the digit immediately after your last significant figure.
• If it's 5 or more, round up the last significant digit. If it's less than 5, leave it as is.
• For example, rounding 0.7404 could give 0.74 if two decimal places are desired.

Rounding simplifies values while maintaining a close approximation to the actual result, which is especially useful in reports or when needing easy numbers for further calculations.