Problem 55 - Algebra and Trigonometry

Question

Express the repeating decimal as a fraction. $$ 0.777 \dots $$

Step-by-Step Solution

Answer

The repeating decimal 0.777... as a fraction is \( \frac{7}{9} \).

Step 1: Write the Repeating Decimal as a Variable

Let \( x = 0.777\ldots \). This means that \( x \) is the repeating decimal.

Step 2: Multiply the Variable by 10

Multiply both sides of the equation in Step 1 by 10 to shift the decimal point one place to the right: \( 10x = 7.777\ldots \).

Step 3: Set Up a Subtraction Equation

Subtract the equation from Step 1 from the equation in Step 2: \( 10x - x = 7.777\ldots - 0.777\ldots \), which simplifies to \( 9x = 7 \).

Step 4: Solve for the Variable

To solve for \( x \), divide both sides of the equation from Step 3 by 9: \( x = \frac{7}{9} \).

Step 5: Verify the Result

Convert \( \frac{7}{9} \) back to a decimal to ensure it equals \( 0.777\ldots \). Dividing 7 by 9 yields a repeating decimal of \( 0.777\ldots \), confirming our solution is correct.

Key Concepts

convert repeating decimal

Converting a repeating decimal into a fraction might seem tricky at first, but it's a systematic process that anyone can learn. When you see a repeating decimal like \(0.777\ldots\), the first step is to assign it a variable: let's call it \(x\). This simplifies the approach, as working with equations is often more straightforward than dealing with decimals directly.
Next, you need to shift the decimal point to the right so that the repeating sequence no longer appears as part of a decimal. You achieve this by multiplying the variable by a power of 10. For example, by multiplying \(x = 0.777\ldots\) by 10, you get \(10x = 7.777\ldots\). This step is crucial because it sets the stage for the next move: subtraction, which helps to eliminate the repeating part.
After setting up your equations, you'll subtract the original equation from this new one. This subtraction helps to isolate the variable and makes the repeating decimal disappear. What remains is a simple equation for which you can solve the variable, effectively converting the repeating decimal into a fraction.

algebra

The beauty of algebra lies in its ability to simplify complex numerical relationships. By representing repeating decimals algebraically, we can tap into the power of equations and manipulation to uncover solutions. When dealing with repeating decimals, creating an equation such as \(x = 0.777\ldots\) is a strategic move. It allows us to use algebraic operations to find an equivalent fraction.
Multiplying both sides by 10 is an algebraic technique that simplifies our problem. It strategically progresses us to the point where subtraction reveals the core of our problem: removing the repeating decimal. Through subtraction, we derive a new equation: \(9x = 7\), which leaves us with a solvable and tangible objective.
Solving for \(x\) involves a simple division, requiring not much more than basic algebra. It's rewarding and straightforward: dividing both sides by 9 to find \(x = \frac{7}{9}\). It's always advisable to check if our algebraic work was executed correctly. Verifying this by converting \(\frac{7}{9}\) back into a decimal reassures us that algebra indeed governs numerical relationships effectively.

fraction simplification

Simplifying fractions is often the final and crucial step in mathematical problems. Once you've expressed a repeating decimal like \(0.777\ldots\) as a fraction \(\frac{7}{9}\), the task becomes ensuring it is fully simplified. Fraction simplification ensures that the fraction is in its most reduced form, which makes it easier to work with.
"Is \(\frac{7}{9}\) in simplest form?" is a common question one might ask. A fraction is in its simplest form when the numerator (top number) and the denominator (bottom number) share no common factors other than 1. In this case, both 7 and 9 are prime with respect to each other, meaning they can't be reduced further.
Simplifying fractions not only facilitates ease in arithmetic operations but also provides clarity in understanding mathematical relationships. By adhering to this simplification practice, you're ensuring that your solutions are both accurate and aesthetically trustworthy.