Problem 37 - Prealgebra

Question

Divide the decimals. \(\frac{6.278}{8.6}\)

Step-by-Step Solution

Answer

The result of dividing 6.278 by 8.6 is 0.73.

Step 1: Adjust the Decimals

To simplify the division \( \frac{6.278}{8.6} \), move the decimal in the divisor (8.6) to the right one place to make it a whole number (86). Do the same with the dividend (6.278), moving its decimal one place to the right to make it 62.78. The division becomes \( \frac{62.78}{86} \).

Step 2: Set Up Long Division

Write 86 outside the long division bracket and 62.78 inside it. Prepare to divide as you would with whole numbers.

Step 3: Divide the Whole Numbers

Divide 627 by 86. 86 goes into 627 seven times (since 86 \( \times \) 7 = 602). Subtract 602 from 627 to get 25.

Step 4: Bring Down the Next Digit

Bring down the next digit (8) to make it 258.

Step 5: Continue Division

Divide 258 by 86, which equals 3 with a remainder (since 86 \( \times \) 3 = 258 exactly). Subtract to get a remainder of 0.

Step 6: Calculate the Decimal Places

Since we moved the decimal one place to the right at the start (in both the divisor and dividend), the decimal in the quotient needs to be one place from the right.

Step 7: Finalize the Result

Since our quotient from step 5 is 0.73 after inserting the decimal place, our final answer is 0.73.

Key Concepts

Long Division Simplified

Long division might sound intimidating, but it's essentially breaking a bigger problem into smaller, more manageable parts. Think of it as a step-by-step journey to find the answer. In our exercise, we're working with the division problem of \( \frac{62.78}{86} \). Here's how the process goes:

• Firstly, set up the long division by placing 86 outside the division bracket, known as the divisor.
• Place 62.78 inside the bracket as the dividend.
• We need to divide each part of the dividend following the rules of long division until all parts are handled.

By working through the digits of the dividend one at a time, we find how many times 86 can fit into these sections, providing our answer gradually and systematically.

Adjusting Decimals for Easy Division

Decimals can seem tricky, especially during division, but adjusting them makes the process simpler. When dealing with decimals in division, the goal is to turn the divisor into a whole number. Here's how it works for our task:

• The original division was \( \frac{6.278}{8.6} \).
• To eliminate the decimal from the divisor (8.6), move the decimal one place to the right. This turns 8.6 into 86.
• To keep the division correct, do the same with the dividend (6.278), moving it one place right to become 62.78.

This adjustment gives us \( \frac{62.78}{86} \), a much easier problem to manage without decimals in the divisor.

Understanding Division with Remainders

Sometimes division doesn't result in a neat number, and that's where remainders come in. When dividing 62.78 by 86, the numbers didn't perfectly even out with each step. Here's what happened:

• First, we found 86 goes seven times into 627, leaving a remainder when subtracted.
• Next, we brought down the next digit to continue the division, converting the remainder into a new dividend section.
• Since 86 fitted precisely three times into the new section, we ended with a remainder of zero.

Handling remainders is about continuing the process until the result is accurate or further simplifications yield no more parts to bring down. Then, reflecting the initial decimal adjustment, we place the decimal in the quotient where needed to ensure the solution, like our final result of 0.73.