Prove algebraically that 1.018 (with 18 recurring) = 56/55
How to approach this question
1. Set the recurring decimal equal to a variable, say `x`. So, `x = 1.01818...`
2. Multiply `x` by powers of 10 to create two new equations where the decimal parts after the point are identical. For example, find `10x` and `1000x`.
3. Subtract one equation from the other. This should result in an equation with no recurring decimals.
4. Solve the resulting equation for `x`.
5. Simplify the fraction to its lowest terms to show it equals 56/55.
Full Answer
To prove that a recurring decimal is equal to a certain fraction, we must convert the decimal into a fraction algebraically.
1. Let `x` be the recurring decimal:
`x = 1.0181818...`
2. We want to create two equations where the part after the decimal point is the same, so we can subtract them and eliminate the recurring part. The recurring part is `18`.
3. Multiply `x` by 10 to move the decimal point just before the recurring part begins:
`10x = 10.181818...` (Equation 1)
4. Multiply `x` by 1000 to move the decimal point after the first full cycle of the recurring part:
`1000x = 1018.181818...` (Equation 2)
5. Now, subtract Equation 1 from Equation 2:
`1000x - 10x = 1018.181818... - 10.181818...`
`990x = 1008`
6. Solve for `x` by dividing by 990:
`x = 1008 / 990`
7. Simplify the fraction. We can divide both the numerator and denominator by their common factors.
- Divide by 2: `x = 504 / 495`
- The sum of the digits of 504 is 9, and the sum of the digits of 495 is 18. Both are divisible by 9.
- 504 ÷ 9 = 56
- 495 ÷ 9 = 55
- So, `x = 56/55`.
We have shown that 1.018 (recurring) is equal to 56/55.
Common mistakes
✗ Choosing the wrong powers of 10 to multiply by (e.g., 10x and 100x).
✗ Making a mistake when subtracting the numbers or the algebraic terms.
✗ Failing to simplify the resulting fraction correctly.
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