Questions: Multiplying and Dividing by Powers of Ten

Question 1

What is 4.7 × 100?

• A. 4.700
• B. 47
• C. 470
• D. 0.047

Answer: (C)

Multiplying by 100 = 10² shifts every digit two places to the left. The 4 moves from the ones place to the hundreds place, and the 7 moves from the tenths place to the tens place: the result is 470. Option B (47) is only one shift — that would be 4.7 × 10. Option A (4.700) adds zeros after the decimal but keeps the value identical to 4.7, which is the 'appending zeros' error. Option D is what dividing by 100 gives, not multiplying.

Question 2

A student calculates 36.5 ÷ 10 and writes 36.50. What error did the student make?

• A. No error — 36.50 is correct because you append a zero when dividing by 10
• B. The student shifted the digits the wrong direction: dividing by 10 shifts digits right, giving 3.65
• C. The student should have shifted digits left, giving 365
• D. You cannot divide a decimal by 10

Answer: (B)

Dividing by 10 shifts every digit one place to the RIGHT (toward smaller place values): 36.5 ÷ 10 = 3.65. The 3 moves from tens to ones, the 6 from ones to tenths, the 5 from tenths to hundredths. The student confused dividing (shifts right, makes smaller) with multiplying (shifts left, makes larger). Writing 36.50 doesn't change the value at all — it still equals 36.5.

Question 3

When you multiply a number by 10, each digit moves one place to the left in the place-value chart.

• T. True
• F. False

Answer: True

Multiplying by 10 makes every digit worth 10 times as much, which means each digit moves into the next-higher position: ones become tens, tens become hundreds, tenths become ones. This leftward shift is why the product is larger. The decimal point itself is a fixed marker — it's the digits that move relative to it.

Question 4

Multiplying any number by a power of ten generally produces a larger result.

• T. True
• F. False

Answer: False

This only holds when multiplying by powers of ten greater than 1 (10, 100, 1000...). Multiplying by 10⁰ = 1 leaves the number unchanged. And dividing by a power of ten (which is multiplying by a negative power, like 10⁻¹) makes the number smaller. The correct rule: multiplying by 10ⁿ where n > 0 shifts digits left (larger); dividing by 10ⁿ shifts digits right (smaller).

Question 5

Explain why 4.5 × 10 = 45, not 4.50. What is wrong with the idea of 'just adding a zero'?

Model answer: Adding a zero after the decimal (4.50) doesn't change the value — 4.5 and 4.50 are the same number. The correct operation is to shift each digit one place left: the 4 moves from the ones place to the tens place, and the 5 moves from the tenths place to the ones place. The result is 45. The 'add a zero' shortcut only works for whole numbers (like 5 × 10 = 50) because appending a zero there does shift every digit left. For decimals, you must actually move the digits — not append.

The 'add a zero' rule is a memorized shortcut that breaks down with decimals. Understanding that digits shift position (and why) prevents this error. 4.5 has a 4 in the ones place and a 5 in the tenths place. Multiplying by 10 promotes each digit one place: 4 → tens, 5 → ones. Result: 45. Writing 4.50 leaves the digit positions unchanged and gives the same value as 4.5.