KJSEA 2025 Maths Q25 — Converting a Recurring Decimal to a Fraction
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The Question
“A number is written in decimal form as 0.17 recurring, where only the 7 repeats forever (0.1777...). Express this recurring decimal as a fraction in its simplest form.”
Name the decimal
Give the recurring decimal a letter so you can work with it like an algebra term. Because only the 7 repeats, the digit 1 stays fixed and the sevens run on without end.
Multiply to shift the repeating part
Multiply by ten so the first seven moves to just after the decimal point, then multiply by one hundred so the second seven lines up in the same place. The goal is to create two numbers whose endless tails of sevens match exactly.
Subtract to cancel the endless sevens
Take the smaller multiple away from the larger one. Since both have the same never-ending string of sevens after the point, those tails cancel completely and leave a clean whole number, wiping out the recurring part.
Solve and simplify
Divide both sides by 90 to get the fraction, then reduce it by dividing the top and bottom by their common factor of 2 to reach the simplest form.
Final Result
The recurring decimal 0.17 (with the 7 repeating) equals the fraction 8/45.
Why this method works
The multiply-and-subtract method works because a recurring decimal has an infinitely long, perfectly regular tail. By multiplying by powers of ten you slide the decimal point until two versions of the number share the exact same infinite tail. Subtracting one from the other makes those identical tails cancel, turning an endless decimal into a finite whole number. That leaves a simple linear equation you can solve for x, which proves the recurring decimal was really a fraction all along.
Dividing 8 by 45 gives 0.1777..., which matches the original decimal.