Using the digits 1, 5, 3 and 8 once each, write the largest four-digit decimal number less than 1.
\(0.8531\)
Goal: We need a number less than 1 that uses the digits 1, 5, 3, 8 exactly once, and has four digits after the decimal.
Any number less than 1 looks like:
\(0.\text{(tenths)(hundredths)(thousandths)(ten-thousandths)}\)
Place the biggest digit in the tenths place to make the number as large as possible, but still less than 1.
Biggest digit is \(8\).
So tenths = \(8\), the number starts: \(0.8\dots\)
Next biggest digit in the hundredths place.
Remaining digits: \(5, 3, 1\).
Choose \(5\).
Now we have: \(0.85\dots\)
Next, fill the thousandths place.
Remaining digits: \(3, 1\).
Choose \(3\).
Now: \(0.853\dots\)
Finally, fill the ten-thousandths place with the last digit.
Last digit is \(1\).
Complete number: \(0.8531\).
Why this is largest: A bigger digit earlier (like tenths) increases the value more than a bigger digit later (like thousandths). We placed digits in descending order: \(8\rightarrow5\rightarrow3\rightarrow1\).
Answer: \(0.8531\)