Rounding to Decimal Places

3.45 sits exactly halfway between 3.4 and 3.5. To decide whether to round up or down, we look at the hundredths digit: it’s 5. By the convention “5 or more — round up,” we round to 3.5.

Alternatively, picture a number line from 3.4 to 3.5. The midpoint is 3.45. Because it’s on the boundary, we follow the agreed rule and round up. Without this convention, we’d be stuck — so the “round up at 5” rule exists to break the tie consistently.

When we round to 2 decimal places, the answer must show exactly 2 digits after the decimal point. The thousandths digit of 2.304 is 4, which is less than 5, so we round down: 2.304 becomes 2.30.

Although 2.30 and 2.3 have the same value, writing “2.3” implies we rounded to 1 decimal place, not 2. The trailing zero in 2.30 communicates the precision of our rounding — it says “I checked the hundredths column and it really is a zero.” In science and maths, this distinction matters.

To round to 1 decimal place, we look only at the hundredths digit. In 7.349, the hundredths digit is 4. Since 4 < 5, we round down, keeping the tenths digit as 3. The answer is 7.3.

If we look at a number line, 7.349 is clearly less than the midpoint 7.35, so it is closer to 7.3 than 7.4. A common mistake is to “chain round” (9 rounds 4 to 5, then 5 rounds 3 to 4), but rounding is about distance to the nearest benchmark, not a sequence of steps.

The thousandths digit of 4.998 is 8, which is ≥ 5, so we round up. Rounding up from 4.99 means increasing the hundredths digit — but 9 can’t become 10 in that column, so it carries to the tenths. The tenths 9 also can’t become 10, so it carries again to the ones: 4.99 becomes 5.00.

This surprises students because the whole number part changes — and in this case, every digit changes. On a number line, 4.998 is very close to 5 — only two thousandths away. We write 5.00 (not just 5) to show we rounded to 2 decimal places.

Example: 3.82

Another: 3.76

Creative: 3.8 itself — it already has 1 decimal place and “rounds” to 3.8. Or 3.84999 — pushing as close to 3.85 as possible without reaching it.

Trap: 3.85 — a student might think “it starts with 3.8 so it must round to 3.8.” But the hundredths digit is 5, so we round up: 3.85 rounds to 3.9, not 3.8. This tests whether students understand the rounding boundary.

Example: 6.96 — rounds to 7.0, so the ones digit changes from 6 to 7.

Another: 9.95 — rounds to 10.0, changing the ones digit from 9 to 0 (and creating a tens digit!).

Creative: 99.96 — rounds to 100.0. The ones, tens, and hundreds digits all change. The number of digits increases!

Trap: 5.94 — a student might think “the 9 will carry over and change the 5.” But the hundredths digit is 4, which is less than 5, so we round down to 5.9. The ones digit stays as 5.

Example: 3.604 — the thousandths digit is 4, so we round down to 3.60.

Another: 8.197 — the thousandths digit is 7, so we round up from 8.19 to 8.20.

Creative: 6.9999 — rounds to 7.00. Also try 3.996 — rounds to 4.00, requiring you to write two zeros to show precision!

Trap: 2.409 — a student might see the 0 in the hundredths place and assume it stays, giving 2.40. But the thousandths digit is 9, which rounds that 0 up to 1. The correct answer is 2.41, which doesn’t end in zero.

Example: 3.02 — rounds to 3.0 at 1 d.p. and to 3 at nearest whole number. Both have the value 3.

Another: 6.97 — rounds to 7.0 at 1 d.p. and to 7 at nearest whole number.

Creative: 4.999 — rounds to 5.0 at 1 d.p. and to 5 at nearest whole number. The original is less than 5, but both roundings push it up to 5.

Note: While 3.0 and 3 have the same numerical value, 3.0 is more precise. In science, they are treated differently!

Trap: 3.52 — a student might think “it starts with 3 either way.” But 3.52 → 3.5 at 1 d.p., while 3.52 → 4 at nearest whole number. These are different values.

When we round down (e.g. 3.62 → 3.6), the result is smaller. But when we round up (e.g. 3.67 → 3.7), the result is larger.

This is a great chance to discuss negative numbers: -4.62 rounds to -4.6. Is -4.6 smaller or larger than -4.62? (It’s larger, because it’s further to the right on the number line!).

More decimal places always means at least as much accuracy, never less. For example, 3.4567 rounded to 1 d.p. is 3.5 (off by 0.0433), but rounded to 3 d.p. it’s 3.457 (off by only 0.0003).

Each extra decimal place you keep preserves more information about the original number. Rounding to 2 d.p. can never be less accurate than rounding to 1 d.p., because the 2 d.p. answer is always at least as close to the original number.

This is exactly the correct method. To round to 1 decimal place, you look at the digit in the very next column — the hundredths — and nothing else. If it’s 0, 1, 2, 3, or 4, round down. If it’s 5, 6, 7, 8, or 9, round up.

Students who cascade-round will disagree: they think you should look at all the digits to the right and “carry” changes from right to left. But that is a misconception — the correct procedure always uses exactly one digit: the first digit being removed.

It’s true most of the time: rounding 3.67 to 1 d.p. gives 3.7 — only the tenths digit changed (6 → 7). Rounding 3.62 gives 3.6 — technically no digit values changed.

But it fails when rounding causes a carry. Take 6.95: the hundredths digit is 5, so we round up — but the tenths is already 9, so it can’t become 10. It carries: the tenths resets to 0, the ones digit increases from 6 to 7, giving 7.0. Both the tenths and the ones digit changed.

The student is cascade rounding (also called sequential rounding) — rounding digit by digit from right to left. This is incorrect.

Visual Proof: Look at where 2.449 sits on a number line. It is clearly to the left of 2.45.

By rounding in stages, the student effectively moved the number from 2.449 to 2.5, making it less accurate! We must look only at the hundredths digit (4), which tells us to round down to 2.4.

The answer is correct — 8.4 — but the reasoning is dangerously wrong. The student is treating decimal digits as a whole number (reading “37” and rounding it to “40”). This is a whole-number strategy that happens to give the right answer here by coincidence.

It fails with different numbers: applying the same logic to 8.137 would treat the digits as “137” — but round 137 to what? To 100 (giving 8.1)? To 140 (giving 8.14)? The method becomes incoherent. The correct reasoning: look at the hundredths digit (7). Since 7 ≥ 5, round the tenths up from 3 to 4, giving 8.4.

The student is truncating (chopping off digits) instead of rounding. Truncating is like using scissors to cut the number — it ignores everything that falls to the floor.

But rounding is about deciding which benchmark is closer. The hundredths digit is 7. Since 7 ≥ 5, we are closer to the next tenth up. We round the tenths digit up from 6 to 7. The correct answer is 5.7.

The student’s rounding logic is actually correct, but they made a mistake by dropping the trailing zero. Writing “8.5” implies the answer is rounded to 1 decimal place, not 2.

When asked to round to 2 decimal places, the answer must show 2 digits after the decimal point: 8.50. The trailing zero is essential — it communicates precision. It says “I checked the hundredths column and it really is zero.”