Questions: Dividing Decimals

Question 1

A student solves 7.2 ÷ 0.3 by multiplying the dividend by 10 to get 72, but leaves the divisor as 0.3. They then compute 72 ÷ 0.3 = 240. What went wrong?

• A. Nothing — 240 is the correct answer to 7.2 ÷ 0.3
• B. They should have divided 0.3 by 10 rather than multiplying 7.2 by 10
• C. They only transformed the dividend, not the divisor — both must be multiplied by the same power of 10
• D. Long division cannot be used when the dividend contains a decimal

Answer: (C)

The clearing strategy works because multiplying both dividend and divisor by the same number doesn't change the value of the division (it's equivalent to multiplying the fraction 7.2/0.3 by 10/10 = 1). If you only multiply the dividend, you've changed the problem — 72 ÷ 0.3 is 10 times larger than 7.2 ÷ 0.3. Both numbers must be multiplied by the same power of 10: 7.2 × 10 = 72 and 0.3 × 10 = 3, giving 72 ÷ 3 = 24.

Question 2

Why does multiplying both the dividend and divisor by 10 not change the answer to a division problem?

• A. Division distributes over multiplication, so multiplying either number by 10 always cancels out
• B. Division is a fraction, and multiplying both numerator and denominator by the same number leaves the value of the fraction unchanged
• C. It only works when multiplying by 10 — multiplying by 100 or 1000 would change the answer
• D. It does change the answer — you must divide the result by 10 at the end to compensate

Answer: (B)

7.2 ÷ 0.3 is the same as the fraction 7.2/0.3. Multiplying numerator and denominator by 10 gives 72/3, which equals the same value — you've multiplied by 10/10 = 1. This is the same principle as equivalent fractions: 1/2 = 2/4 = 5/10. The key constraint is that you must multiply both by the same factor. This reasoning also extends to multiplying by 100, 1000, or any power of ten — as long as both numbers are multiplied by the same amount.

Question 3

When dividing 8.4 by 4 (a whole-number divisor), you should multiply 8.4 by 10 to clear the decimal before dividing.

• T. True
• F. False

Answer: False

When the divisor is already a whole number, no transformation is needed. You simply divide as you would with whole numbers and bring the decimal point straight up into the quotient at the same position it appears in the dividend. 8.4 ÷ 4: divide 84 ÷ 4 = 21, then place the decimal directly above its position in 8.4 to get 2.1. The 'clear the decimal' strategy is only necessary when the divisor is itself a decimal.

Question 4

Estimating 7.2 ÷ 0.3 as approximately 20 before calculating helps verify that your answer is in the right range.

• T. True
• F. False

Answer: True

7 ÷ 0.3 ≈ 7 ÷ (1/3) = 21, so the answer should be in the low-to-mid twenties. The correct answer of 24 is consistent with this estimate. If a calculation produced 2.4 or 240, the estimate immediately signals a decimal placement error — off by a factor of 10. Estimation doesn't tell you the exact answer, but it's a powerful check that catches the most dangerous mistake in decimal division: placing the decimal point in the wrong position.

Question 5

Explain in your own words why multiplying both the dividend and divisor by 10 gives the same answer as the original decimal division problem.

Model answer: Division is equivalent to a fraction — 7.2 ÷ 0.3 is the same as 7.2/0.3. When you multiply both the numerator and denominator of a fraction by the same number, you're multiplying by a form of 1 (for example, 10/10 = 1), which doesn't change the value. So 7.2/0.3 × (10/10) = 72/3, which equals 24 — the same as 7.2 ÷ 0.3. The critical requirement is that both numbers must be multiplied by the same power of 10. Multiplying only the dividend would be like multiplying only the numerator of a fraction, which changes its value.

Students who only know the procedure ('move the decimal point') without understanding why often make errors in unusual cases — like when the divisor has two decimal places and they're unsure whether to multiply by 10 or 100. Understanding the fraction equivalence makes the rule self-evident: you need to clear all decimal places from the divisor, so you multiply by whatever power of 10 is needed to do that, and apply the same multiplication to the dividend.