Questions: Comparing and Ordering Decimals

Question 1

Which decimal is greater: 0.4 or 0.35?

• A. 0.35 — because 35 is greater than 4
• B. 0.4 — because 4 tenths is greater than 3 tenths
• C. They are equal — both have digits in the tenths and hundredths places
• D. 0.35 — because it has more decimal digits

Answer: (B)

Compare digit by digit from left to right. Both numbers have 0 in the ones place (tied). In the tenths place, 0.4 has a 4 and 0.35 has a 3. Since 4 tenths > 3 tenths, 0.4 is larger — the comparison is decided right there, before we even look at the hundredths. Option A is the classic mistake: treating the decimal digits as a whole number (35 > 4) and ignoring place value.

Question 2

A student argues: '0.125 must be bigger than 0.9 because 125 is much greater than 9.' How do you respond?

• A. The student is correct — 125 > 9, so 0.125 > 0.9
• B. The student is wrong — 0.9 is greater because 9 tenths > 1 tenth, and the comparison ends at the tenths place
• C. You need to find a common denominator before you can compare
• D. They are equal because both are less than 1

Answer: (B)

The student is applying the 'longer means bigger' misconception. Compare by place value: in the tenths place, 0.9 has 9 and 0.125 has 1. Since 9 tenths > 1 tenth, 0.9 is larger immediately. Rewriting both with trailing zeros confirms: 0.900 vs. 0.125 — 900 thousandths vs. 125 thousandths. The number of decimal digits is irrelevant; only the place value of the first differing digit determines which is larger.

Question 3

A decimal with more digits after the decimal point is generally greater than a decimal with fewer digits.

• T. True
• F. False

Answer: False

This is one of the two most common decimal misconceptions. 0.9 has one decimal place; 0.125 has three. Yet 0.9 > 0.125, because 9 tenths > 1 tenth. The number of digits after the decimal point tells you the smallest place value represented, not the size of the number. Only the leftmost differing digit determines which decimal is larger.

Question 4

Rewriting 0.5 as 0.50 changes the value of the number.

• T. True
• F. False

Answer: False

Trailing zeros after the last nonzero decimal digit do not change the value. 0.50 means '50 hundredths,' and 50 hundredths = 5 tenths = 0.5. The value is identical. Appending trailing zeros is a useful technique for comparing decimals because it aligns place values and makes the comparison look like a whole-number comparison (e.g., 0.50 vs. 0.38 becomes '50 hundredths vs. 38 hundredths').

Question 5

Why can't you compare decimals simply by treating all the digits after the decimal point as a whole number (e.g., concluding that 0.125 > 0.9 because 125 > 9)?

Model answer: Because the digits after the decimal point represent different place values (tenths, hundredths, thousandths), not a single number. 0.9 means 9 tenths, and 0.125 means 1 tenth + 2 hundredths + 5 thousandths. The first decimal digit represents tenths for both, so you compare those first. Comparing '125 vs. 9' ignores place value entirely — it is like comparing the number 125 to 9 when you should be comparing 900 thousandths to 125 thousandths.

The left-to-right, place-by-place strategy works because each position has a fixed, decreasing value (tenths > hundredths > thousandths). Once you find the first place where the digits differ, the comparison is decided — everything to the right is irrelevant, just as when comparing 847 and 823 you know 847 > 823 as soon as you see the tens digits (4 > 2).