TL;DR: Keep extra digits until the last line unless a teacher demands rounding each step to match a key. Rounding 1/3 to 0.33 too early, then multiplying by 300, yields 99 instead of 100. Money often wants two decimal places at the end. Percents on homework often want a whole percent or one decimal. The assignment is the standard, not your calculator’s eight-digit display, not “looks nicer.”
What rounding is not
Rounding is not deleting a remainder because it is ugly. 37.5% is a terminating story from 3/8. Rounding it to 38% is a choice that should be labeled ≈. If later you convert back to a fraction, 38% will not return 3/8. Keep 37.5 when the fraction was exact.
Rounding is not the same as significant figures, though they meet in labs. Sig figs care about measurement. Homework percents often care about “nearest percent.” Read which.
Bankers, money, and halves
$1.255 to cents: many people go to $1.26, some banker’s round to even. Registers have a policy. Your worksheet might say “nearest cent.” 5-up is common in schools. If two methods disagree by a cent, it is the half. Show the unrounded $1.255 then the chosen cent. Graders can see you knew.
Tip 18% of $54.20 is $9.756. Cards take $9.76. Cash might take $9.75 or $10. Those are different roundings plus a possible generosity bump. Name generosity separately so 18% remains 18%.
Intermediate traps
Percent increase from 80 to 100 is 25%, exact. No rounding. Inventing 25.00% does not add science. From 80 to 81 is 1.25%. If the key wants a whole percent, 1% is a harsh round of 1.25; 1.3% to one decimal is closer. Ask which place. Silent whole-percent rounding of 1.25 to 1 loses a quarter of the change in relative terms. That might be what they want. It should not be a surprise.
Chained factors: use 1.1 × 1.1 = 1.21, not 1.1 rounded to 1 then squared. Rounding factors to 1 is how compounding disappears. If you must round, round the final net percent, not each 1.10 to a whole number.
Calculator displays vs stored values
A handheld may show 0.33333333 and store more. A website may show 33.33%. If you copy 33.33 into the next step as 0.33, you rounded twice. Copy more digits, or keep the fraction 1/3. Online tools are not obligated to show your teacher’s preferred places. You are obligated to round for the answer blank, not for the internal math.
Phone screenshot rounding is a third display. Do not let JPEG compression of a screenshot become a fourth rounding. Type the digits.
When early rounding is required
Some elementary keys round each step to the nearest whole number to keep arithmetic in reach. Then you must obey even if 99 appears instead of 100. Write “per step per instructions.” That sentence saves you from a clever 100 that misses the key. Clever is not always the assignment.
Tax tables sometimes round tax to a cent per line, then add. Recreating with one final round on the sum can miss a cent. If you are matching a receipt, round as the receipt’s jurisdiction rounds, not as a math purist rounds.
A personal default when nobody spoke
I keep two extra digits internally, round the reported percent to one decimal if it is not whole, and round money to cents. I label ≈ if I rounded a repeating decimal. Defaults are not laws. They are better than eight random digits or zero digits. If a rubric exists, it beats my default. If a legal tax rule exists, it beats both.
Repeating percents and fraction keys
1/3 as 33.3% vs 33.33% vs 33 1/3% changes a 900-point scaled score by more than a rounding shrug. Prefer the fraction until the last box. If the last box wants a percent, 33.3% with a bar or 33 1/3% is closer to honest than 33%. If the key is 33, write 33 and the unrounded 33.3% in parentheses if you are allowed. Parentheses save arguments.
Do not round 66.6% up to 67% in one problem and leave 33.3% down to 33 in the same problem so they fail to sum to 100. Complements should still complement after rounding, or you should round the last piece to make 100 and say you did. Silent 99% and 101% pairs are rounding without a policy.
Gradebooks that round then weight
If a 83.4% category rounds to 83% then weights at 50%, you lost 0.2 points of category before the 0.50 even ran, 0.1 course points. Tiny. If every category rounds down, the tiny piles. Ask whether the LMS stores full precision. If you recreate in a sheet with rounded category percents, you can miss the portal by a tenth that looks like a “wrong weight.” Match the LMS rounding policy, then argue weights if needed, not the other way around.
Three diners, one tip, pennies that will not share
The food is $86.15. An 18% tip is 0.18 times 86.15 = 15.507, and the house wants money to the cent, so the tip becomes $15.51 if you round half up at the end. Split three ways, 15.51 divided by 3 is 5.17 with a penny left over. If each friend rounds their own slice of the unrounded 15.507 to $5.17, the server receives $15.51 only if someone also eats the remainder on purpose. If each friend first rounds the tip rate to 20% “to make it even,” the tip jumps to $17.23 and the rounding decision changed the rate, which is a different policy, not a cleaner version of 18. Label the moment you rounded. Early rounding of 0.18 to 0.2 is not intermediate digit hygiene. It is a new percent.
A poll whose printed percents sum to 101
Four answers print as 34%, 29%, 27%, and 11%. They sum to 101 because each figure was rounded to the nearest whole percent from counts that did not owe you integers. The honest repair is to show the counts or one decimal, not to “fix” 11 down to 10 because 101 feels illegal. Forcing the printed numbers to sum to 100 moves error into a category the pollster may not have chosen. If you must allocate seats by those percents, use the largest-remainder method on the raw counts. Do not feed the already-rounded percents back through a calculator and then wonder why a 200-person sample produced 202 people. Rounding is a display choice. The counts remain the whole.
