Decimal place value and rounding

The same whole can have smaller units

Split one whole into ten equal parts: each is one tenth, 1/10 = 0.1. Split each tenth into ten equal parts: each is one hundredth, 1/100 = 0.01. Splitting each hundredth into ten gives thousandths: 1/1000 = 0.001.

The number 2.307 has two ones, three tenths, no hundredths and seven thousandths:

The zero keeps the hundredths place. 2.37 is 2 + 3/10 + 7/100, a different number.

Check your understanding 1 Read places, then build a number

For 2.608, write the whole-number part and the counts of tenths, hundredths and thousandths, including any zero. Then write a decimal made from 3 whole units, 4 tenths, 0 hundredths and 5 thousandths.

Your four place counts and new decimal

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2.608 has 2 whole units, 6 tenths, 0 hundredths and 8 thousandths. The second number is 3.405, not 3.45: the zero keeps the hundredths place while the five counts thousandths.

Compare matching places

Write 0.6 as 0.600 to compare it with 0.58 as 0.580. Six tenths is greater than five tenths, so 0.6 > 0.58. More written digits do not make a decimal larger.

Adding zeros at the right end of a decimal fraction does not change its value: 0.5 = 0.50 = 0.500. Putting a zero between the dot and the five does change its place: 0.05 is five hundredths.

Check your understanding 2 Order values with different written lengths

Write 0.7, 0.68 and 0.705 in increasing order, then decreasing order. Explain why writing more digits does not settle the comparison.

Your two orders and comparison

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Compare 0.700, 0.680 and 0.705. Increasing order is 0.68 < 0.7 < 0.705; decreasing order is 0.705 > 0.7 > 0.68. Matching the places lets you compare equal units. Also, 0.7 = 0.70 = 0.700: a different written length need not change the value.

Convert fractions through tenths or hundredths

The denominators here are factors of 10 or 100, so a simple equivalent fraction gives the decimal. To reverse 0.125, write 125/1000 and simplify if needed: it is 1/8.

Check your understanding 3 Use the requested notation

Express 0.035 as a fraction in simplest form. Then express 7/20 as a decimal. Check that the two answers name different amounts.

Your fraction and decimal

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0.035 = 35/1000 = 7/200, while 7/20 = 35/100 = 0.35. The first is thirty-five thousandths; the second is thirty-five hundredths. The denominator 20 is a factor of 100, so hundredths give the second decimal directly.

Round to the requested place

For 4.376, the nearest whole number is 4. To one decimal place it is 4.4; to two decimal places it is 4.38. Each answer locates neighbouring multiples of the chosen unit: ones, tenths or hundredths.

Keep the original number for later calculations unless the question explicitly asks you to use a rounded value.

Check your understanding 4 Keep two decimal places

Round 6.284 to two decimal places.

Rounded number

Check answer

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6.28. This field checks the number; when writing on paper, show the two decimal places requested.

Check your understanding 5 Carry across a whole number

Round 9.995 to the nearest whole number, to one decimal place and to two decimal places. Show the requested places in each answer.

Your three rounded forms

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The answers are 10, 10.0 and 10.00. To two places, 9.995 is exactly halfway between 9.99 and 10.00. Choose the larger nonnegative value at a halfway point. The carry passes through the tenths and into the whole-number part. The zeros in 10.0 and 10.00 record the requested decimal places, although all three answers have the same value.

Check your understanding 6 A number line decision

Which is nearer to 1: 0.98 or 1.03? Find the two gaps before deciding.

Your answer or drawing on paper

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The gaps are 1-0.98 = 0.02 and 1.03-1 = 0.03. The smaller gap is 0.02, so 0.98 is nearer. Being above or below one does not alone decide the distance.