Multiplying and Dividing Decimals
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
0.4 means 4 tenths, which is the fraction \(\frac{4}{10}\). So \(\frac{4}{10} \times 5 = \frac{20}{10} = 2\). Alternatively, 0.4 + 0.4 + 0.4 + 0.4 + 0.4 = 2.0 — five lots of 0.4 make exactly 2.
This is a good opportunity to challenge the misconception that “multiplying always makes bigger”. Here, 0.4 × 5 = 2: the answer is bigger than 0.4 but smaller than 5. When you multiply a number greater than 1 by a number less than 1, the result always falls between the two factors.
6 ÷ 0.5 asks: “How many halves fit into 6?” Each whole contains 2 halves, so 6 wholes contain 6 × 2 = 12 halves.
This challenges the misconception that “division always makes smaller”. When you divide by a number less than 1, the answer is larger than what you started with.
We are finding 2 tenths of 3 tenths. A tenth of a tenth is a hundredth, so the answer is in hundredths. 3 × 2 = 6, giving 6 hundredths.
The green overlap covers exactly 6 out of 100 squares.
Many students expect the answer to be 0.6. The key insight is that multiplying two decimals each less than 1 gives a result smaller than either factor.
We can write the division as a fraction: \(\frac{1.2}{0.3}\). If we multiply both the numerator and denominator by 10, we get an equivalent fraction: \(\frac{12}{3}\). Since the fraction is equivalent, the answer must be the same.
This is the Equivalence Strategy. It works for any division. For example, to solve 1.5 ÷ 0.05, we can multiply both by 100 to get 150 ÷ 5 = 30. This avoids the messy business of “long division with a decimal point”.
Give an Example Of…
Think carefully — the fourth box is a trap! Give a non-example that looks right but isn’t.
Example: 5 ÷ 0.5 = 10
Another: 3 ÷ 0.1 = 30
Creative: 0.2 ÷ 0.01 = 20 — even a small number divided by an even smaller decimal can produce a large answer.
Trap: 10 ÷ 2 = 5 — a student might offer this thinking “I divided and got a number”, but 5 is smaller than 10, not larger. The key is you must divide by a number between 0 and 1.
Example: 0.8 ÷ 0.4 = 2
Another: 1.5 ÷ 0.3 = 5
Creative: 0.06 ÷ 0.02 = 3 — even tiny decimals can divide to give a whole number. Or 2.5 ÷ 0.5 = 5.
Trap: 0.7 ÷ 0.3 = 2.333… — a student might think “7 ÷ 3 isn’t exact, but with decimals it will sort itself out.” It doesn’t! For the answer to be a whole number, one decimal must be an exact multiple of the other.
Example: 8 × 0.5 = 4, and 8 ÷ 2 = 4. Multiplying by 0.5 is the same as dividing by 2.
Another: 12 × 0.25 = 3, and 12 ÷ 4 = 3. Multiplying by 0.25 is the same as dividing by 4.
Creative: 20 × 0.1 = 2, and 20 ÷ 10 = 2. Multiplying by 0.1 is dividing by 10. Or: 15 × 0.2 = 3, and 15 ÷ 5 = 3.
Trap: 12 × 0.3 = 3.6, and 12 ÷ 3 = 4 — a student might assume that ×0.3 is the same as ÷3, but it isn’t. Multiplying by 0.3 is the same as dividing by \(\frac{10}{3}\), not by 3. The rule only works when the decimal is the reciprocal of the whole number (e.g. 0.5 = \(\frac{1}{2}\), 0.25 = \(\frac{1}{4}\)).
Example: 0.5, because 1 ÷ 0.5 = 2
Another: 0.25, because 1 ÷ 0.25 = 4
Creative: 0.125, because 1 ÷ 0.125 = 8 (since 0.125 = ⅛). Or 0.01, because 1 ÷ 0.01 = 100.
Trap: 0.3 — a student might assume 1 ÷ 0.3 = 3, but actually 1 ÷ 0.3 = 3.333… The decimal must be a unit fraction in disguise (a reciprocal of a whole number, like \(\frac{1}{3}\), \(\frac{1}{4}\)) for the division to produce a whole number.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
True when the decimal is between 0 and 1: for example, 10 × 0.3 = 3, which is smaller than 10. But a decimal can also be greater than 1: 10 × 1.5 = 15, which is larger than 10.
This targets the misconception that “a decimal” always means “a number less than 1”. Decimals like 2.7, 15.3, or even 100.01 are all decimals — and multiplying by any of these makes the number larger, not smaller.
If both numbers are between 0 and 1, you are taking a fraction of a fraction. For example, 0.7 × 0.8 = 0.56, and 0.56 is less than both 0.7 and 0.8. Similarly, 0.1 × 0.9 = 0.09, which is less than both.
This is a powerful result that challenges the misconception “multiplication always makes bigger”. When both factors are proper fractions (between 0 and 1), the product is always smaller than either one.
Dividing a positive number by a value between 0 and 1 always makes it larger. For example, 4 ÷ 0.5 = 8, and 8 > 4. Similarly, 2 ÷ 0.1 = 20.
When you divide by a small positive number, you are asking “how many of these small pieces fit in?” — and the answer is always more than you started with. This is one of the most counterintuitive facts in decimal arithmetic and directly challenges the “division always makes smaller” misconception.
True when neither product digit cancels a trailing zero: 0.4 × 0.7 = 0.28 (each has 1 d.p., the answer has 2 d.p. — more than either). But 0.5 × 0.4 = 0.20 = 0.2, which has only 1 d.p. — the same as each factor, not more.
Students often learn the rule “add the decimal places” and assume the answer always has more places than either factor. The rule gives the maximum number of decimal places, but trailing zeros can reduce the actual count. This is a subtle but important distinction when students check their work.
Odd One Out
Which is the odd one out? The challenge: make a valid mathematical case for each one being the odd one out.
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: 4.2
Reasoning: “6 × 7 = 42, so 0.6 × 0.7 = 4.2. I just put the decimal point back in.”
The student has miscounted the decimal places. They correctly computed 6 × 7 = 42 but only shifted the decimal by one place instead of two. Since 0.6 has one decimal place and 0.7 has one decimal place, the product needs 1 + 1 = 2 decimal places.
The correct answer is 0.42, not 4.2.
Sense Check: Ask the student: “Is 0.6 more or less than 1? Is 0.7 more or less than 1? If I multiply two numbers that are less than 1, can the answer be 4?”
Answer: 0.5 × 8 = 4 is larger ✔
Reasoning: “Multiplying by a decimal always makes a number smaller. 0.5 × 8 = 4 and 0.5 × 6 = 3. Since 8 is bigger it gets made less small, so 4 is larger.”
The answer is correct — 0.5 × 8 = 4 is larger — but the reasoning is dangerously wrong. The student claims “multiplying by a decimal always makes a number smaller”. This is false: 1.5 × 8 = 12, which is larger than 8, not smaller. The rule only holds when the decimal is between 0 and 1.
The correct reasoning is simpler: both calculations multiply by the same factor (0.5), so the one with the larger starting number gives the larger result.
Answer: 0.8
Reasoning: “4 × 0.2 = 0.8. Division is the same sort of thing as multiplication with decimals.”
The student has multiplied instead of dividing. They computed 4 × 0.2 = 0.8 and assumed that was the same as dividing. This reveals a fundamental confusion between the two operations — the belief that “multiplication and division with decimals work the same way”.
The correct answer is 20. Dividing by 0.2 asks: “How many lots of 0.2 fit into 4?” Since 0.2 goes into 1 five times, it goes into 4 twenty times.
Answer: 2
Reasoning: “6 ÷ 3 = 2, so 6 ÷ 0.3 = 2 as well. The decimal point doesn’t change a division.”
The student has ignored the decimal point in the divisor, treating 0.3 as if it were 3.
The correct answer is 20. It is often easiest to write the division as a fraction: \(\frac{6}{0.3}\). If we multiply the top and bottom by 10, we get an equivalent fraction: \(\frac{60}{3}\). And 60 ÷ 3 = 20.