Operations on decimals — all four operations: Worked examples

Before anything else, one word we must earn: a decimal place is simply how many digits sit to the right of the point. So $3.5$ has one decimal place, $0.05$ has two, and $7$ (no point) has zero. Keep that meaning in your pocket — every rule below counts these.

The scenario matrix

Every problem on this topic falls into one of these cells. Each worked example below is tagged with the cell it covers.

#	Case class	What makes it tricky	Example
C1	Add, ragged places	one number has more digits after the point	Ex 1
C2	Subtract crossing whole part / borrow	borrowing across the point	Ex 2
C3	Subtract giving a zero-ish small result	answer smaller than either input, watch the leading zero	Ex 3
C3z	Subtract giving an exact zero	equal inputs cancel — degenerate case	Ex 3b
C4	Multiply, leading-zero factor	$0.0 \dots$ eats extra places	Ex 4
C5	Multiply by a power of ten (limiting/shift)	the point just slides — no long multiply	Ex 5
Cn	Negative decimal input (all four ops)	sign rules on top of the point bookkeeping	Ex 5b
C6	Divide by a decimal (both shift)	must scale top and bottom	Ex 6
C7	Divide small ÷ big (answer $< 1$ )	quotient starts $0. \dots$	Ex 7
C8	Divide that never ends (repeating)	when to stop, how to round	Ex 8
C9	Word problem (money, real units)	translate words → operation	Ex 9
C10	Exam twist (mixed operations, order)	do the right operation first	Ex 10

See Place Value and Powers of Ten for why the columns carry the weights they do.

Ex 1 — Addition with ragged places (C1)

Worked example

$6.7 + 12.345$ Forecast: guess the answer to the nearest whole number before reading on. (Around $19$ ?)

Step 1 — Pad to equal places. Write $6.700 + 12.345$ . Why this step? $6.7$ has $1$ place, $12.345$ has $3$ . To add like-for-like (tenths to tenths, thousandths to thousandths) both must be over the same denominator, here $\frac{\cdot}{1000}$ (recall the dot just marks "top to be filled in"). The extra zeros do not change the value: $6.7 = 6.700$ .

Step 2 — Line up the points and add as whole numbers. $\frac{6700}{1000} + \frac{12345}{1000} = \frac{6700 + 12345}{1000} = \frac{19045}{1000}$ . Why this step? Same denominator ⇒ we may add the numerators directly (parent rule).

Step 3 — Drop the point straight down. $\frac{19045}{1000} = 19.045$ . Why this step? Three zeros in the denominator ⇒ three decimal places.

Verify: $19.045$ is just above $19$ , matching the forecast. Reverse-check by subtraction: $19.045 - 12.345 = 6.700 = 6.7$ ✓.

Ex 2 — Subtraction with borrowing across the point (C2)

Worked example

$10.2 - 3.86$ Forecast: a little more than $6$ ?

Step 1 — Pad: $10.20 - 3.86 = \frac{1020}{100} - \frac{386}{100}$ . Why? Equalise place value; $10.2$ becomes $10.20$ so both are hundredths.

Step 2 — Subtract the numerators (borrowing happens here): $1020 - 386 = 634$ . Why? Common denominator ⇒ subtract the tops. The borrow you'd do by hand ( $0$ can't give $6$ , so borrow from the $2$ …) is exactly ordinary whole-number borrowing — the point plays no part yet.

Step 3 — Point down: $\frac{634}{100} = 6.34$ . Why may we just drop the point? Because the answer is still over the same denominator $100$ ( $\frac{634}{100}$ ), and "over $100$ " always means two decimal places — so the point sits straight under the aligned points of the inputs. We never re-count places for subtraction.

Verify: $6.34 + 3.86 = 10.20 = 10.2$ ✓. Sits just above $6$ , matching the forecast.

Ex 3 — Subtraction giving a small, leading-zero answer (C3)

Ex 3b — Subtraction giving an exact zero (degenerate case) (C3z)

Ex 4 — Multiplication with a leading-zero factor (C4)

Ex 5 — Multiply by a power of ten (limiting case: the shift) (C5)

Figure — Operations on decimals — all four operations

Figure Ex 5 — The digits $3729$ stay fixed in place while only the decimal point moves: a coral arrow slides it two steps right for $\times 100$ (giving $372.9$ ), and a mint arrow slides it two steps left for $\times 0.01$ (giving $0.03729$ ). The picture shows that multiplying by a power of ten never touches the digits — it only relocates the point.

Worked example

$3.729 \times 100$ and $3.729 \times 0.01$ Forecast: multiplying by $100$ makes it bigger; by $0.01$ makes it smaller. By how many hops?

Step 1 — Recognise the special factor. $100 = 1 0^{2}$ and $0.01 = 1 0^{- 2}$ . Why this matters? When one factor is a pure power of ten, you don't need a long multiply — the point just slides. Look at the figure: the digits stay put; only the point walks.

Step 2 — $\times 1 0^{2}$ : point moves $2$ places RIGHT (number grows): $3.729 \to 372.9$ . Why right, and why exactly $2$ ? Multiplying by $1 0^{2}$ means multiplying by ten twice, and each $\times 10$ makes every digit worth ten times more — which is the same as sliding the point one column right. Two tens ⇒ two columns right.

Step 3 — $\times 1 0^{- 2}$ : point moves $2$ places LEFT (number shrinks), padding a zero if needed: $3.729 \to 0.03729$ . Why left for a negative power? Multiplying by $1 0^{- 2}$ is dividing by $100$ ; each division by ten shifts the point one step left.

Verify (a direct fraction check — no shortcuts): work $3.729 \times 0.01$ straight from the fraction form. $3.729 = \frac{3729}{1000}$ and $0.01 = \frac{1}{100}$ , so $3.729 \times 0.01 = \frac{3729}{1000} \times \frac{1}{100} = \frac{3729}{100000} = 0.03729,$ which has five digits after the point — exactly what the mint arrow in the figure shows (the point ended up five columns from the last digit). The picture and the fraction agree.

Ex 5b — Negative decimals across ALL four operations (Cn)

Worked example Four quick negatives:

$- 5.4 - 2.3$ , $- 2.5 + 0.8$ , $(- 1.2) \times 0.5$ , $- 8.2 \div 0.4$ Forecast: the first is "more negative," the second lands near $- 1.7$ , the third and fourth are both negative (negative with a positive).

Step 1 — Subtraction of two negatives: $- 5.4 - 2.3$ . Subtracting a positive $2.3$ from $- 5.4$ moves further below zero, so add the sizes and keep the minus: $5.4 + 2.3$ → pad $5.4 = 5.40$ , $\frac{540}{100} + \frac{230}{100} = \frac{770}{100} = 7.70 = 7.7$ ⇒ answer $- 7.7$ . Why add here when the symbol is a minus? Because "take $2.3$ away from something already negative" pushes you deeper into the negatives — the magnitudes combine, and the result stays negative.

Step 2 — Addition with mixed signs: $- 2.5 + 0.8$ . Signs differ ⇒ subtract the smaller size from the larger and keep the larger's sign: $2.5 - 0.8 = 1.7$ (pad $2.5 = 2.50$ , $2.50 - 0.80 = 1.70$ ); the larger size $2.5$ was negative ⇒ $- 1.7$ . Why subtract, not add? Adding a positive to a negative moves you toward zero — a subtraction of magnitudes.

Step 3 — Multiplication: $(- 1.2) \times 0.5$ . Sizes: $12 \times 5 = 60$ , places $1 + 1 = 2$ ⇒ $0.60 = 0.6$ . Sign: negative × positive = negative ⇒ $- 0.6$ . Why negative? $(- 1.2) \times 0.5$ is "half of $- 1.2$ ," still on the negative side of zero.

Step 4 — Division: $- 8.2 \div 0.4$ . Sizes first: kill the point in the divisor $0.4$ (one place) by scaling both by $10$ : $\frac{8.2 \times 10}{0.4 \times 10} = \frac{82}{4} = 20.5$ . Sign: negative ÷ positive = negative ⇒ $- 20.5$ . Why does the sign rule for ÷ match ×? Division is multiplication by the reciprocal, so it obeys the same "unlike signs ⇒ negative" rule.

Verify: $- 7.7 + 2.3 = - 5.4$ ✓ (undo Step 1); $- 1.7 + 2.5 = 0.8$ ✓ (undo Step 2); $- 0.6 \div 0.5 = - 1.2$ ✓ (undo Step 3); $- 20.5 \times 0.4 = - 8.2$ ✓ (undo Step 4).

Ex 6 — Divide by a decimal, both shift (C6)

Ex 7 — Small ÷ big, answer below one (C7)

Ex 8 — A division that never ends (C8)

Ex 9 — Money word problem (C9)

Ex 10 — Exam twist: order of operations (C10)

Worked example Evaluate

$2.5 + 0.4 \times 1.5$ Forecast: trap alert — if you add first you'd get one number, if you multiply first, another. Which is correct?

Step 1 — Multiply BEFORE adding. Order of operations says $\times$ outranks $+$ , so compute $0.4 \times 1.5$ first: strip points $4 \times 15 = 60$ ; places $1 + 1 = 2$ ⇒ $0.60 = 0.6$ . Why multiply first? The $+$ sign joins $2.5$ to the product $0.4 \times 1.5$ , not to the raw $0.4$ . The multiplication is "glued tighter" than the addition, so it must be resolved first — otherwise you would be adding the wrong quantity.

Step 2 — Now add the result to $2.5$ . $2.5 + 0.6$ : pad to equal places, then add numerators over the common denominator $10$ : $\frac{25}{10} + \frac{6}{10} = \frac{25 + 6}{10} = \frac{31}{10} = 3.1$ . Why pad and add numerators? Both numbers are now tenths (over $10$ ), so like place values line up and we add the tops directly (the addition rule from the parent note).

Step 3 — State the answer and expose the trap. Correct value $= 3.1$ . The tempting wrong route is to add first: $2.5 + 0.4 = 2.9$ , then $2.9 \times 1.5 = 4.35$ . Why is that wrong? It performs the addition before the multiplication, breaking the ordering rule, so it answers a different expression $(2.5 + 0.4) \times 1.5$ , not the one asked.

Verify: correct $2.5 + (0.4 \times 1.5) = 2.5 + 0.6 = 3.1$ ✓; the wrong route gives $4.35 \neq = 3.1$ , confirming the two are genuinely different and that order matters.

Active Recall

Recall Ex 4: why did

$6 \times 4 = 24$ turn into $0.0024$ ? Because the factors had $3 + 1 = 4$ decimal places total, and $24$ has only two digits, so we pad zeros on the left to fill four places: $\frac{24}{1 0 ^{4}} = 0.0024$ .

Recall Ex 5: which way does the point move when multiplying by

$1 0^{- 2}$ ? Two places to the left (the number shrinks), because $\times 1 0^{- 2}$ means dividing by $100$ .

Recall Ex 5b: what is the sign of a negative decimal

divided by a positive decimal? Negative — division follows the same "unlike signs ⇒ negative" rule as multiplication (it is multiplication by the reciprocal). E.g. $- 8.2 \div 0.4 = - 20.5$ .

Recall Ex 8: why does

$10 \div 3$ never terminate? The remainder $1$ returns every step, so the digit $3$ repeats forever — the denominator $3$ has a prime factor other than $2$ or $5$ .

Connections

Operations on decimals — all four operations (parent)
Place Value and Powers of Ten
Fractions to Decimals conversion
Rounding and Significant Figures
Long Division algorithm
Scientific Notation