A school sports day committee reviews the results from many previous years, going back through the archived scorebooks, and calculates that the proportion o...
Question 1 Report
A school sports day committee reviews the results from many previous years, going back through the archived scorebooks, and calculates that the proportion of races won by the home team is the recurring decimal \(0.\dot7\dot2\), that is \(0.727272\ldots\). Show that this proportion is equal to \(\dfrac{8}{11}\). (3)
Answer Details
Converting a recurring decimal to a fraction uses a standard algebraic trick: multiplying by a power of \(10\) that shifts the decimal point exactly one full repeating block, then subtracting the original value so the infinitely repeating part cancels out completely.
Let \(x=0.727272\ldots\), so multiplying by \(100\) shifts the decimal point past one full repeating block: \[ 100x=72.727272\ldots \] [1 mark]
Subtracting the original equation removes the repeating part entirely, since both sides still recur identically after the decimal point: \[ 100x-x=72.727272\ldots-0.727272\ldots \] \[ 99x=72 \] [1 mark]
\[ x=\frac{72}{99}=\frac{8}{11} \text{ as required} \] [1 mark]
Multiplying by \(100\) rather than \(10\) is essential here because the repeating block is two digits long ("\(72\)"); multiplying by \(10\) would shift the decimal point by only one digit and leave a mismatched repeating pattern when subtracted.
Download The App On Google Playstore
Everything you need to excel in your exams
