Adding and Subtracting with Decimals

Method 1 (Money): Think of 0.7 as 70p and 0.6 as 60p. Adding 70p + 60p gives 130p, which is £1.30 or 1.3.

Method 2 (Place Value): 7 tenths + 6 tenths = 13 tenths. Since 10 tenths make 1 whole, you have 1 whole and 3 tenths left over.

First, rewrite 0.5 as 0.50 so both numbers have two decimal places. Now add column by column: 0 + 5 = 5 hundredths, 5 + 2 = 7 tenths. The answer is 0.75.

A student who doesn’t align the decimal points might add 5 + 25 = 30 and write 0.30. The key insight is that the 5 in 0.5 represents 5 tenths, so it must be lined up with the 2 in 0.25, not the 5. Alternatively: half plus a quarter equals three-quarters.

Imagine a grid of 100 small squares makes 1 whole. Each small square is one hundredth (0.01). If you remove just one small square, you have 99 squares left, which is 0.99.

You cannot subtract 8 hundredths from nothing! You must rewrite 0.3 as 0.30. Now the subtraction is clear: 0.30 − 0.18.

Borrow 1 tenth to make 10 hundredths: 10 − 8 = 2 hundredths. Then 2 tenths − 1 tenth = 1 tenth. Answer: 0.12.

Example: 0.3 + 0.7 = 1

Another: 0.45 + 0.55 = 1

Creative: 0.001 + 0.999 = 1 — pushing to the extremes. Or 0.123 + 0.877 = 1 — non-obvious digits that still complement each other.

Trap: 0.4 + 0.06 — a student who misaligns the decimal points might calculate 4 + 6 = 10 and think the answer is 1. But 0.4 + 0.06 = 0.46.

Example: 0.8 + 0.5 = 1.3 — 8 tenths + 5 tenths = 13 tenths, carry 1 into the ones column.

Another: 0.36 + 0.48 = 0.84 — 6 hundredths + 8 hundredths = 14 hundredths, carry 1 tenth.

Creative: 0.99 + 0.01 = 1.00 — carrying ripples all the way through every column, like an odometer turning over.

Trap: 0.2 + 0.3 = 0.5 — no carrying is needed here. A student might think all decimal addition involves carrying.

Example: 0.25 + 0.75 = 1 — both numbers have 2 decimal places, but the answer is a whole number with none.

Another: 0.36 + 0.14 = 0.5 — both have 2 decimal places, but the answer has only 1.

Creative: 0.125 + 0.875 = 1 — three decimal places each, but the carrying cascades through every column to produce a whole number.

Trap: 0.5 + 0.5 — students might write “1.0” and assume that because the decimal point is still there, the number of decimal places hasn’t changed. But 1.0 is just 1 (0 decimal places).

Example: 0.3 − 0.8 = −0.5

Another: 1.2 − 3.5 = −2.3

Creative: 0.001 − 1 = −0.999 — subtracting a whole number from a tiny decimal.

Trap: 0.8 − 0.3 = 0.5 — a student might write this thinking the answer is negative because “you’re subtracting decimals.” But here the first number is larger.

True case: 0.3 + 0.4 = 0.7, which is less than 1. False case: 0.6 + 0.8 = 1.4, which is greater than 1.

Students who only work with “friendly” decimals like 0.1, 0.2, 0.3 may think the answer always stays below 1. The key is whether the tenths digits sum to 10 or more.

This works only when both decimals have the same number of decimal places: treating 0.35 + 0.42 as 35 + 42 = 77 gives 0.77, which is correct.

But it fails when the decimal places differ: treating 0.3 + 0.04 as 3 + 4 = 7 gives 0.07, which is wrong (the correct answer is 0.34). This strategy is dangerous because it ignores place value.

If both numbers are less than 0.5, then their sum must be less than 0.5 + 0.5 = 1. For example, 0.499 + 0.499 = 0.998, which is still less than 1.

This challenges students to think about upper bounds. The answer can get very close to 1, but it can never reach it.

Trailing zeros don’t change a number’s value: 0.5 = 0.50 = 0.500. So padding with zeros is always valid and makes column addition or subtraction much easier to carry out correctly.

This is one of the most useful strategies students can learn — it avoids the column misalignment errors that cause so many mistakes.

The student has used the “bring down the decimals” misconception. They assumed that because 5 has no written decimals, there is nothing to subtract .38 from.

The correct approach is to rewrite 5 as 5.00. Then you can see that you need to subtract 38 hundredths from 0 hundredths, which requires borrowing. The correct answer is 2.62.

The answer is correct, but the reasoning is dangerous! The student is just adding the digits like whole numbers.

Why this fails: If the question was 0.3 + 0.04, this student would likely add 3 + 4 = 7 and get 0.7 again. But the correct answer is 0.34. You must add tenths to tenths and hundredths to hundredths.

The student treated the columns as separate difference problems, just finding the difference between 3 and 7 regardless of the order.

This is the “always subtract the smaller digit from the larger” misconception. You must regroup (borrow) from the 5 ones, changing the 3 tenths into 13 tenths. Then 13 − 7 = 6. Correct answer: 2.6.

The student treats the whole-number part and decimal part as separate blocks, then flips the decimal subtraction because “bottom is bigger than top”.

The correct approach: rewrite as 6.10 − 2.45, then subtract column by column with regrouping. The correct answer is 3.65.