Problem 34 - Prealgebra

Question

Divide the decimals. \(\frac{6.375}{0.85}\)

Step-by-Step Solution

Answer

The result of dividing 6.375 by 0.85 is 7.5.

Step 1: Adjust the Division Problem

Convert the division of decimals to whole numbers by eliminating the decimal point. Multiply both the numerator and the denominator by 100 to remove the decimals: \( 6.375 \times 100 = 637.5 \) and \( 0.85 \times 100 = 85 \). The new problem becomes \( \frac{637.5}{85} \).

Step 2: Simplify the Division

Although 637.5 is not yet a whole number, it's useful to realize that this equates to dividing 6375 by 850 \( \left( \frac{6375}{850} \right)\). Simplify this by ensuring the numerator (6375) is divisible by 850. As 6375 is already prepared for division with 850, move to division.

Step 3: Perform Long Division

Divide 6375 by 850 using long division. Determine how many times 850 fits into 6375. Begin dividing: \(850 \times 7 = 5950\), and \(850 \times 8 = 6800\). 7 times fits perfectly, since 6800 exceeds 6375. Subtract \(6375 - 5950 = 425\) and bring down the next digit (which doesn’t exist here). Thus, 7 remain with a difference of 425 to zero.

Step 4: Obtain the Decimal Quotient

The decimal quotient is obtained by dividing 425 by 850. Append decimal places to continue the division. Add a zero after 63750 making it effectively 4250 for calculation. Completing division ends at \(5\), as the whole number portion is \(7.5\). Confirm \(7.5\) as correct and conclude division completing.

Key Concepts

Long Division

Long division provides a structured method of dividing numbers, particularly when working with large values or decimals. The process involves several steps:

• Divide: Determine how many times the divisor fits into the first portion of the dividend.
• Multiply and Subtract: Multiply the result by the divisor, subtract from the current dividend, and bring down the next digit.
• Repeat: Continue the cycle of dividing, multiplying, and subtracting until no numbers are left or if a specific decimal precision is achieved.

For instance, dividing 6375 by 850 involves determining how many complete times 850 fits in the initial digits of 6375, then continuing the process until completion. It's essential in getting non-intuitive results especially when decimals are involved. Practice helps in mastering this technique, allowing for efficient and accurate calculations.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means ensuring the numerator and denominator have no common factors other than 1. In the decimal division, such as \(\frac{6375}{850}\), the aim is to make division easier by eliminating common factors.
An easy method involves:

• Checking for common factors and dividing both the numerator and denominator by them.
• Using prime factorization to identify shared factors.

For our case, evaluating the fraction showed no simple division operations to simplify further without calculations, implying readiness for long division. Simplification ensures the calculations are accurate and less cumbersome, preventing errors and making the division process smoother.

Multiplying Decimals

When working with decimals, multiplying them correctly is key to converting them into easy-to-manage whole numbers. This process clears decimal points, simply the calculations, especially with division.
For instance, in converting \(\frac{6.375}{0.85}\) to a simpler division problem, multiply both by 100 to remove decimals: \(6.375 \times 100 = 637.5\) and \(0.85 \times 100 = 85\). This leaves a cleaner, non-decimal division problem: \(\frac{637.5}{85}\).
It's important to remember:

• Count the total number of decimal places initially.
• Ignore decimals while multiplying.
• Incorporate the same number of decimal places in the product as were originally present.

Using these principles ensures precision and avoids missteps, particularly in operations like division where further manipulation can compound errors.