Express This Decimal As A Fraction 1.21 Repeating

Express This Decimal as a Fraction: 1.21 Repeating

Converting repeating decimals to fractions is a fundamental skill in mathematics that bridges the gap between two different numerical representations. One such example is 1.But 21 repeating, which can be written as **1. In real terms, 21212121... On top of that, ** and requires a systematic approach to transform it into a precise fraction. This article will guide you through the step-by-step process, explain the underlying mathematical principles, and provide examples to solidify your understanding And that's really what it comes down to..

Understanding Repeating Decimals

A repeating decimal is a decimal number that has a digit or a sequence of digits that repeat indefinitely after the decimal point. So for instance, in 1. Think about it: 21 repeating, the digits "21" form the repeating block. To convert this into a fraction, we need to isolate the repeating part and apply algebraic techniques.

The key idea is to use multiplication to shift the decimal point so that the repeating sections align, allowing us to eliminate the repeating portion through subtraction. This method works because multiplying by powers of 10 moves the decimal to the right, effectively positioning the repeating digits in a way that subtraction cancels them out.

Steps to Convert 1.21 Repeating to a Fraction

Step 1: Separate the Integer and Decimal Parts

First, break down 1.212121... into its integer and decimal components:

• Integer part: 1
• Decimal part: 0.212121...

We will focus on converting the decimal part (0.212121...) into a fraction, then add the integer part to the result That's the part that actually makes a difference..

Step 2: Let x Represent the Decimal Part

Let x = 0.212121...

Step 3: Multiply by a Power of 10 to Align the Repeating Block

Since "21" is the two-digit repeating block, multiply both sides of the equation by 100 (which is 10²) to shift the decimal point two places to the right:

• 100x = 21.212121...

Step 4: Subtract the Original Equation from the Multiplied Equation

Subtract the original x = 0.212121... from the multiplied equation to eliminate the repeating part:

• 100x - x = 21.212121... - 0.212121...
• 99x = 21

Step 5: Solve for x

Divide both sides by 99 to isolate x:

• x = 21/99

Step 6: Simplify the Fraction

Simplify 21/99 by finding the greatest common divisor (GCD) of 21 and 99. Both numbers are divisible by 3:

• 21 ÷ 3 = 7
• 99 ÷ 3 = 33
• Simplified fraction: 7/33

Step 7: Add the Integer Part

Combine the simplified decimal fraction with the integer part:

• 1 + 7/33 = (1 × 33 + 7)/33 = (33 + 7)/33 = 40/33

Thus, 1.21 repeating as a fraction is 40/33.

Scientific Explanation: Why This Method Works

The method relies on the principle of algebraic manipulation to eliminate the infinite repetition. 212121...Plus, when we multiply **x = 0. ** by 100, we create a new equation where the repeating decimals align perfectly with the original.

Completingthe Subtraction — Why the Repetition Vanishes

When we write [ 100x = 21.\overline{21}\qquad\text{and}\qquad x = 0.\overline{21}, ]

the infinite strings of “21” after the decimal points are identical. Subtracting the second equation from the first therefore removes the repeating block entirely:

[ \begin{aligned} 100x - x &= 21.In practice, \overline{21} - 0. \overline{21}\ 99x &= 21.

Because the fractional parts cancel, only the integer 21 remains on the right‑hand side. This is why the method works for any repeating decimal: the multiplication aligns the repetend (the repeating digits) with itself, and subtraction eliminates the endless tail, leaving a finite integer that can be solved algebraically.

General Formula for a Two‑Digit Repeating Block

If a decimal has a two‑digit repetend, say (0.\overline{ab}), the same steps give

[ \begin{aligned} x &= 0.\overline{ab},\ 100x &= ab.\overline{ab},\ 100x - x &= ab,\ 99x &= ab,\ x &= \frac{ab}{99} The details matter here..

The fraction can always be reduced by dividing numerator and denominator by their greatest common divisor.

Extending the Technique

1. Repeating Block of Three Digits

Consider (0.[ \begin{aligned} x &= 0.\overline{374}).
This leads to \overline{374},\ 1000x &= 374. \overline{374},\ 1000x - x &= 374,\ 999x &= 374,\ x &= \frac{374}{999}.

2. Mixed Repeating Decimals (Non‑repeating + Repeating)

For a number like (2.16\overline{3}) (the “16” does not repeat, only the “3” does):

1. Let (x = 2.16\overline{3}).
2. Multiply by 10 to move past the non‑repeating part: (10x = 21.\overline{3}).
3. Multiply again by 10 (because the repetend length is 1) to align the repeat: (100x = 213.\overline{3}).
4. Subtract: (100x - 10x = 213.\overline{3} - 21.\overline{3}) → (90x = 192).
5. Solve: (x = \frac{192}{90} = \frac{32}{15}).

3. Longer Repeating Sequences

If the repetend has (k) digits, multiply by (10^{k}) to shift one full block to the left of the decimal point, then subtract the original equation. The denominator will always be a string of (k) nines (e.g., 9, 99, 999, …), which can then be simplified Small thing, real impact. Took long enough..

Why the Process Is Reliable

1. Alignment – Multiplying by a power of 10 moves the decimal point exactly enough to place the next occurrence of the repetend directly after the decimal point.
2. Cancellation – Subtracting the original equation removes the infinite tail because the two copies of the repetend are identical.
3. Integer Numerator – The subtraction leaves only a finite integer, which can be expressed as a ratio of two integers.
4. Reduction – Any common factor can be divided out, yielding the fraction in lowest terms.

Summary of the Procedure

1. Identify the length (k) of the repeating block.
2. Assign a variable (x) to the entire decimal.
3. Multiply (x) by (10^{k}) to shift one full repetend to the left of the decimal point.
4. Subtract the original equation from the multiplied one; the repeating part disappears.
5. Solve for (x) and simplify the resulting fraction.
6. Re‑attach any integer part that was separated at the start.

Final Example: Converting (0.\overline{142857})

The decimal (0.\overline{142857}) is famous because its repetend length is six.

[ \begin{aligned} x &= 0.\overline{142857},\ 10^{6}x &= 1{,}000{,}000x = 142857.\

Continuing from the previous line, we have

[ \begin{aligned} x &= 0.Think about it: \overline{142857},\ 10^{6}x &= 1{,}000{,}000x = 142857. \overline{142857}.

Subtract the original equation from this new one:

[ 10^{6}x - x = 142857.Practically speaking, \overline{142857} - 0. \overline{142857} \quad\Longrightarrow\quad 999,999,x = 142,857 That's the part that actually makes a difference..

Hence

[ x = \frac{142,857}{999,999}. ]

Both numerator and denominator share a factor of 142 857, so the fraction reduces to

[ x = \frac{1}{7}. ]

Thus the repeating decimal (0.\overline{142857}) is exactly the rational number (\frac{1}{7}).

Another illustration: a mixed repeating decimal

Consider (3.12\overline{45}).
Let (y = 3.12\overline{45}).

1. The non‑repeating part has two digits, so multiply by (10^{2}=100):

   [ 100y = 312.\overline{45}. ]

2. The repetend length is two, so multiply again by (10^{2}=100):

   [ 10,000y = 31245.\overline{45}. ]

3. Subtract the first shifted equation from the second:

   [ 10,000y - 100y = 31245.\overline{45} - 312.\overline{45} \quad\Longrightarrow\quad 9,900y = 30,933 Worth knowing..

4. Solve for (y):

   [ y = \frac{30,933}{9,900} = \frac{10311}{3300} = \frac{341}{110}. ]

The mixed decimal (3.12\overline{45}) therefore equals (\frac{341}{110}) Easy to understand, harder to ignore. Simple as that..

Reversing the process

Knowing that (\frac{1}{7}=0.\overline{142857}) shows that the method works in both directions.
If a fraction with a denominator coprime to 10 is expressed in lowest terms, its decimal expansion will be purely periodic; the length of the period equals the smallest (k) for which (10^{k}\equiv1\pmod{d