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Decimals: tenths and hundredths

Numbers · level 3, lesson 7 of 7

A question to start

Which is bigger, 0.5 or 0.45?

Lots of people, and not only children, say 0.45, "because 45 is bigger than 5". But 0.5 is bigger. Half a dollar (50 cents) is more than 45 cents. By the end of this lesson you will know exactly why, and you will never be fooled by this again.

The idea

Going past the ones place

You know that each place is worth ten times the place to its right: a ten is ten ones, a hundred is ten tens. Read it the other way and each place is worth one tenth of the place to its left: a ten is a tenth of a hundred, a one is a tenth of a ten.

What is a tenth of a one? We need a new place, to the right of the ones. Cut 1 into 10 equal parts: each part is a tenth, \(\tfrac{1}{10}\). To show where the ones end and the new places begin, we put a dot called the decimal point.

\[0.1 = \frac{1}{10} \qquad 0.3 = \frac{3}{10} \qquad 2.7 = 2\tfrac{7}{10}\]

And a tenth of a tenth? Cut a tenth into 10 parts, and each part is \(\tfrac{1}{100}\) of the whole: a hundredth. It gets the next place to the right.

Tens Ones . Tenths Hundredths
4 3 . 2 5

So 43.25 means 4 tens, 3 ones, 2 tenths and 5 hundredths. The word decimal comes from the Latin for "ten", and the whole system is built on tens, in both directions.

Seeing hundredths: the hundred square

Take a square and call it 1 whole. Cut it into 10 columns: each column is a tenth. Cut every column into 10 small squares: there are 100 small squares, so each is a hundredth.

0.45 (45 hundredths)0.5 = 0.50 (50 hundredths)

Now the opening question answers itself. 0.45 is 4 columns and 5 small squares. 0.5 is 5 full columns, which is 50 small squares. 0.5 is bigger, by 5 hundredths.

The trap was reading "0.45" as if it were "45" and "0.5" as if it were "5". But the 5 in 0.5 is in the tenths place: it is worth 5 tenths, which is 50 hundredths.

Zeros at the end change nothing

\(0.5 = 0.50\). Five tenths is the same amount as fifty hundredths (five columns is fifty small squares). Adding a zero at the end of a decimal, after the point, changes nothing, because it only says "and no hundredths". This is a useful trick for comparing: write both numbers with the same number of decimal places, and then compare them like whole numbers. 0.50 against 0.45: 50 hundredths beats 45 hundredths.

But a zero between the point and a digit matters a lot: 0.05 (five hundredths) is ten times smaller than 0.5.

Decimals on the number line

Cut the gap from 0 to 1 into ten equal steps. The marks are 0.1, 0.2, ..., 0.9. Now zoom in on the gap from 0.4 to 0.5 and cut it into ten again: the new marks are hundredths, 0.41, 0.42, ..., 0.49.

00.40.51 0.400.410.420.430.44 0.450.460.470.480.490.50

0.45 sits exactly halfway between 0.4 and 0.5, so it is less than 0.5. Every decimal has its own point on the line, and we can always zoom in further: between 0.45 and 0.46 there are thousandths, 0.451, 0.452, and so on.

Decimals are fractions with 10 and 100 on the bottom

A decimal is a fraction in disguise, one whose denominator is 10 or 100 (or 1000, ...):

\[0.7 = \frac{7}{10} \qquad 0.07 = \frac{7}{100} \qquad 0.45 = \frac{45}{100} \qquad 3.25 = 3\tfrac{25}{100}\]

Using equivalent fractions from the last lesson, some well-known fractions become decimals easily:

  • \(\tfrac12 = \tfrac{5}{10} = 0.5\)
  • \(\tfrac14 = \tfrac{25}{100} = 0.25\) and \(\tfrac34 = \tfrac{75}{100} = 0.75\)
  • \(\tfrac15 = \tfrac{2}{10} = 0.2\)
  • \(\tfrac{3}{20} = \tfrac{15}{100} = 0.15\)

Money is hundredths

A dollar is 100 cents, so one cent is a hundredth of a dollar. That is why prices have two decimal places: 3.45 dollars means 3 dollars, 4 dimes (tenths of a dollar) and 5 cents (hundredths). A quarter is 25 cents, which is 0.25 dollars, which is \(\tfrac14\) of a dollar: the coin's name tells you the fraction!

Comparing decimals

To compare two decimals:

  1. Line up the decimal points.
  2. If it helps, add zeros at the end so they have the same number of places.
  3. Compare place by place from the left, just like whole numbers.

Compare 2.307 and 2.37: write 2.307 and 2.370. Ones: 2 and 2. Tenths: 3 and 3. Hundredths: 0 and 7. So 2.37 is bigger.

Worked examples

1. Read a decimal. What is the 6 worth in 12.36? In 6.21?

In 12.36 the 6 is in the hundredths place: \(\tfrac{6}{100}\), six hundredths. In 6.21 it is in the ones place: 6.

2. Fraction to decimal. Write \(\tfrac{9}{100}\), \(\tfrac{3}{5}\) and \(2\tfrac{7}{10}\) as decimals.

\(\tfrac{9}{100} = 0.09\) (nine hundredths, so a 0 in the tenths place). \(\tfrac35 = \tfrac{6}{10} = 0.6\). \(2\tfrac{7}{10} = 2.7\).

3. Order. Put in order, smallest first: 0.3, 0.27, 0.303, 0.03.

Write each with three places: 0.300, 0.270, 0.303, 0.030. As thousandths: 300, 270, 303, 30. Order: 0.03, 0.27, 0.3, 0.303.

4. Gravity on Mars. Gravity on Mars pulls about 0.38 times as hard as on Earth. Is that more or less than a third? Than a half?

0.38 is 38 hundredths. A half is 50 hundredths, so 0.38 is less than a half. A third is a bit more than 33 hundredths (\(\tfrac13 = 0.333\ldots\)), so 0.38 is a bit more than a third. On Mars you would weigh a bit more than a third of your Earth weight.

5. Money. Which costs more: 4.09 dollars or 4.1 dollars?

4.1 = 4.10. Ten cents beats nine cents, so 4.1 dollars costs more.

Try it

  1. Write as decimals: (a) \(\tfrac{4}{10}\) (b) \(\tfrac{37}{100}\) (c) \(\tfrac{5}{100}\) (d) \(6\tfrac{3}{10}\)
  2. Write as fractions: (a) 0.9 (b) 0.13 (c) 0.08
  3. What is the value of the 3 in: (a) 3.52 (b) 5.32 (c) 5.23?
  4. Which is bigger? (a) 0.6 or 0.58 (b) 1.09 or 1.1 (c) 0.7 or 0.07
  5. Order from smallest: 0.8, 0.75, 0.805, 0.08.
  6. Write \(\tfrac14\), \(\tfrac25\) and \(\tfrac{7}{20}\) as decimals.
  7. Name a decimal between 0.4 and 0.5. Then name one between 0.45 and 0.46.
  8. Harder: Jo says "0.25 is bigger than 0.3 because it has more digits." Use money to explain why Jo is wrong.
Answers
  1. (a) 0.4 (b) 0.37 (c) 0.05 (d) 6.3
  2. (a) \(\tfrac{9}{10}\) (b) \(\tfrac{13}{100}\) (c) \(\tfrac{8}{100}\), which simplifies to \(\tfrac{2}{25}\).
  3. (a) 3 ones. (b) 3 tenths (\(\tfrac{3}{10}\)). (c) 3 hundredths (\(\tfrac{3}{100}\)).
  4. (a) 0.60 against 0.58: 0.6. (b) 1.10 against 1.09: 1.1. (c) 0.70 against 0.07: 0.7.
  5. With three places: 0.800, 0.750, 0.805, 0.080. Order: 0.08, 0.75, 0.8, 0.805.
  6. \(\tfrac14 = \tfrac{25}{100} = 0.25\). \(\tfrac25 = \tfrac{4}{10} = 0.4\). \(\tfrac{7}{20} = \tfrac{35}{100} = 0.35\).
  7. For example 0.43 (any of 0.41 to 0.49 works). Between 0.45 and 0.46: for example 0.455 (zoom in to thousandths: 0.451 to 0.459 all work).
  8. 0.25 dollars is 25 cents. 0.3 dollars is 0.30 dollars, which is 30 cents. 30 cents is more than 25 cents, so 0.3 is bigger. The number of digits does not tell you the size of a decimal; the place of each digit does.

Common mistake

"Longer decimals are bigger." 0.125 is not bigger than 0.2. Compare place by place from the left: tenths 1 against 2, so 0.2 is bigger. Or add zeros: 0.125 against 0.200.

Forgetting the zero placeholder. Seven hundredths is 0.07, not 0.7. The 7 must sit in the hundredths place, so the tenths place needs a 0 to hold it open.

Where it leads