Decimals and Percentages · Grade 6

How do you compare two decimal numbers?

Compare place by place from left to right and stop at the first place where the digits differ. A longer number is not necessarily a larger one, because length is not size: 0.5 is greater than 0.45 even though it has fewer digits. You may add zeros at the end to line the two up.

Learning objectives

Left to right, stopping at the first difference

The comparison begins at the place with the highest value and works down. The moment you find a digit that differs, it decides, and there is no point looking at anything after it.

The reason is that each place is worth ten times the one after it. Even if every remaining digit of the smaller number were larger, together they could not make up one place.

Trailing zeros change nothing

0.5 is exactly equal to 0.50 and to 0.500. Adding zeros at the end only spells out places that were empty anyway, so it is allowed and it does not change the value.

That is useful for comparing: write both numbers with the same number of digits and the comparison becomes a comparison of two whole numbers. 0.5 against 0.45 becomes 50 against 45.

Note that this applies only at the end. A zero at the front, as in 0.05, does change things — it pushes every digit into a lower place.

Rounding

Rounding to the nearest tenth means keeping one digit after the point. Look at the digit immediately after it: if it is five or more, round up; otherwise leave it.

The same rule works for any place. Round according to the single digit immediately after the place you want, not according to the whole tail behind it.

A number line makes both ideas visible at once. Between zero and one there are ten parts, the tenths, and each of those divides again into ten, the hundredths. Rounding means picking the nearest mark; comparing means asking which number sits further along.

Worked examples

  1. Which is larger: 0.7 or 0.65?

    1. Compare the tenths place: 7 against 6
    2. 7 is greater than 6, and the decision is already made
    3. Check: 0.70 against 0.65

    Answer: 0.7 is larger, despite having fewer digits

  2. Order from smallest to largest: 0.4, 0.39, 0.402

    1. Write all to the same length: 0.400, 0.390, 0.402
    2. Now compare as whole numbers: 400, 390, 402
    3. Order them: 390, then 400, then 402

    Answer: 0.39, then 0.4, then 0.402

  3. Round 3.472 to the nearest hundredth

    1. The hundredths digit is the second after the point, which is 7
    2. Look at the digit after it: 2
    3. 2 is less than 5, so the hundredths digit stays as it is

    Answer: 3.47

Common mistakes

Assuming more digits means a larger number
Length is not size. 0.45 is longer than 0.5 and smaller than it, because the tenths place decides before you ever reach the hundredths.
Comparing from the last digit
The comparison starts at the place with the highest value. Starting from the end gives every digit equal weight, and they are not equal at all.
Rounding in stages
To round to a tenth, look only at the hundredths digit. Rounding first to a thousandth, then a hundredth, then a tenth can push the number up without justification.

What to remember

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