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All courses Math Foundations · MF.NS.1 1 of 60
Compare, order, and estimate whole numbers and decimals using place value and magnitude sense.

Compare close decimals

Problem
Compare \(6.407\) and \(6.47\).
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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Hint

Rewrite 6.47 as 6.470 so the decimal places line up, then compare from left to right.

For close decimals, the first place value that is different decides which number is greater.

Solution walkthrough

01

Align the place values

\[\begin{aligned} 6.407 \\ 6.470 \end{aligned}\]

The prompt gives 6.407 and 6.47. Add a trailing zero to 6.47 to write it as 6.470; this does not change its value and places ones, tenths, hundredths, and thousandths in matching columns.

02

Find the first unequal place

\[\text{ones}:~6~=~6;~\text{tenths}:~4~=~4;~\text{hundredths}:~0~<~7\]

Compare from the greatest place value toward the right. The ones digits are both 6 and the tenths digits are both 4. The first difference is at the hundredths place: 6.407 has 0 hundredths, while 6.470 has 7 hundredths, and 0 is less than 7.

03

Check with thousandths

\[407~<~470\]

As a check, compare the decimal parts in thousandths. The first number has 407 thousandths and the second has 470 thousandths. Because 407 is less than 470, the comparison found from place value is consistent.

04

State the comparison

\[6.407~<~6.47\]

Therefore, 6.407 is less than 6.47. This answers the comparison prompt and matches choice A.

+

Another way

  1. Subtract to check the order: 6.470 - 6.407 = 0.063, which is positive, so 6.470 is greater than 6.407.

!

Common mistake

Do not compare only the last visible digits. The 7 in 6.407 is in the thousandths place, but the first unequal place is the hundredths place, where 0 is less than 7.

Solution walkthrough video