Problem 12 - Calculus Single Variable

Question

Use long division to convert the rational fraction to a (possibly nonterminating) decimal with a repeating block. Identify the repeating block. \(31 / 14\)

Step-by-Step Solution

Answer

The repeating block is '214285', so 31/14 = 2.214285... with '214285' repeating.

Step 1: Set Up the Division

Start by writing the division problem as a long division: divide 31 by 14. Write 31 inside the division bracket and 14 outside at the left.

Step 2: Initial Division

Determine how many times 14 goes into 31. It goes in 2 times, because 2 × 14 = 28. Write 2 above the division bracket.

Step 3: Subtraction

Subtract 28 from 31 to find the remainder. 31 - 28 = 3. Write this remainder of 3 below.

Step 4: Decimal Point and Bring Down

Since 14 does not divide into 3, place a decimal point in the quotient and bring down a 0, making the new number 30.

Step 5: Continue Division

Determine how many times 14 goes into 30. It goes in 2 times (2 × 14 = 28). Write 2 after the decimal point in the quotient.

Step 6: Subtract Again

Subtract 28 from 30 to get a remainder of 2. Write 2 as the remainder.

Step 7: Bring Down Next Zero

Bring down another 0, turning the remainder into 20.

Step 8: Continue the Process

Determine how many times 14 goes into 20. It goes in 1 time (1 × 14 = 14). Write 1 in the quotient.

Step 9: Final Subtraction

Subtract 14 from 20 to get a remainder of 6.

Step 10: Identify the Repeating Block

Bring down another 0 to make the remainder 60. Divide: 14 into 60 goes 4 times (4 × 14 = 56), remainder 4. Bring down another 0 to make 40. Divide: 14 into 40 goes 2 times, remainder 12. Bring down another 0 to make 120. Divide: 14 into 120 goes 8 times, remainder 8. Bring down another 0 to make 80. Divide: 14 into 80 goes 5 times, remainder 10. Bring down another 0 to make 100. Divide: 14 into 100 goes 7 times, remainder 2. Bring down another 0 to return to 20. This forms a repeating sequence, 0.214285, where the block '214285' repeats.

Key Concepts

Rational Fraction

A rational fraction is simply a fraction that consists of two integers: a numerator and a denominator. These integers are whole numbers, and what's important to remember is that the denominator cannot be zero, because division by zero is undefined.

• For example: In the fraction \(\frac{31}{14}\), 31 is the numerator and 14 is the denominator.
• This type of fraction represents a rational number, which means it can be expressed as the quotient of these two integers.

Understanding rational fractions is key to many areas of mathematics, as they can be converted into decimals, sometimes terminating and sometimes repeating. By using long division, these fractions can be converted into a decimal form, helping us to understand their value more clearly.

Repeating Decimal

A repeating decimal is a decimal number where digits repeat in a predictable pattern, continuing indefinitely. When converting a rational fraction like \(\frac{31}{14}\) to its decimal form using long division, you might encounter such a repeating pattern.

• A repeating decimal is often written with a bar over the digits that repeat. For example, 0.214285 has a repeating block "214285".
• The repeating pattern helps us understand and check the results of our long division.

This is indicated by placing a bar over the digits or sometimes writing them multiple times to show repetition. It's important to identify the repeating block correctly, as it provides a precise description of the decimal's behavior.

Remainder

The remainder is what is left over after division is attempted and completed as much as possible using whole numbers. In the long division process of converting the fraction \(\frac{31}{14}\), the concept of a remainder is crucial since it helps to perform the division repeatedly to reach a more precise result.

• Each time you divide, multiply, and subtract in long division, the left over part until the next step is called the remainder.
• The remainder guides what happens next in the division process as it dictates if we need to bring down another digit.

Understanding the remainder helps us in evaluating incomplete divisions, which ultimately assists in identifying repeating decimals when performing long division.

Division Process

The division process in long division is all about breaking down a complex division problem into smaller, more manageable parts. This is done step-by-step, allowing us to progressively carry out each simple division that would lead us towards our result.

• To perform the division for \(\frac{31}{14}\), we begin by seeing how many times 14 fits into 31, writing down the initial result.
• We continue by subtracting the product of the division and repeat the process as numbers are brought down one by one.

By repeating this sequence of dividing, multiplying, subtracting, and bringing down digits (if necessary), we obtain the decimal equivalent of a rational fraction. When a pattern begins to repeat, we know we've encountered a repeating decimal. This systematic approach assists us in accurately transforming fractions into decimals, ensuring our understanding is both thorough and reliable.