Chapter 3 · Place value and decimals
Tenths and Hundredths
Learn what tenths and hundredths are, see them in shaded models and on number lines, and write fractions with denominators of ten or one hundred as decimals.
You already know the ones, tens and hundreds places in whole numbers. A decimal point is a dot written just to the right of the ones place. The places to the right of the decimal point count parts of one whole. A number written with a decimal point is called a decimal.
Picture one whole cut into 10 equal parts. Each part is one tenth of the whole. You can write one tenth as the fraction , which means 1 of 10 equal parts, or as the decimal . The first place to the right of the decimal point is the tenths place. Its digit tells how many tenths there are.
In the bar below, 3 of the 10 equal parts are shaded. That is 3 tenths, so . The 3 sits in the tenths place, and the 0 in the ones place shows that there are no whole ones.
Now picture one whole cut into 100 equal parts. Each part is one hundredth of the whole. You can write it as the fraction or as the decimal . The second place to the right of the decimal point is the hundredths place. Its digit tells how many hundredths there are. In , the 0 in the tenths place shows no tenths, and the 1 in the hundredths place shows one hundredth.
The grid below is one whole cut into 100 equal small squares, so each small square is one hundredth. The squares line up in 10 strips, and each strip is a full line of 10 squares. The 10 strips fill the whole grid, so each strip is one tenth. That means 10 hundredths make 1 tenth.
You can count the shaded part of the grid above in two ways.
- Count strips. There are 3 shaded strips, and each strip is one tenth. So the shaded part is 3 tenths, or .
- Count small squares. There are 30 shaded squares, and each square is one hundredth. So the shaded part is 30 hundredths. Every 10 hundredths make 1 tenth, so these 30 hundredths make exactly 3 tenths, with 0 hundredths left over. Write 3 in the tenths place and 0 in the hundredths place: .
Both counts describe the very same shading. So and cover the same amount of the whole: . The 0 in the hundredths place of only says that there are no extra hundredths. With fractions, the same fact is .
Key concept
Each place after the decimal point names a part of one whole
Where a digit sits after the decimal point tells the size of the parts it counts. The first place, the tenths place, counts parts of one whole cut into 10 equal parts. The second place, the hundredths place, counts parts of one whole cut into 100 equal parts. Ten hundredths make one tenth.
Worked example
Match a shaded grid to a decimal
The grid just above is one whole cut into 100 equal small squares. Which decimal matches its shaded part?
Count the full strips first, because each strip is one tenth of the grid. There are 4 full strips shaded, so the shaded part has 4 tenths. That makes 4 the digit in the tenths place.
Next, count the shaded squares that are not part of a full strip. There are 5. Each square is one hundredth, and 5 squares are too few to fill another strip, so they stay as 5 hundredths. That makes 5 the digit in the hundredths place.
Write the decimal. There are no whole ones, so write 0 in the ones place. Then write the decimal point, 4 in the tenths place and 5 in the hundredths place. To check, the 4 strips hold 40 squares, and 40 squares plus 5 more make 45 of the 100 squares.
Answer
The shaded part is 4 tenths and 5 hundredths, which is 45 hundredths in all, so it matches .
Worked example
Find a point on a number line
What decimal is at Point A on the number line just above?
Read the size of one step first. The short marks across the line are called tick marks, and one step is the distance from one tick mark to the next. From 0 to 1 there are 10 equal steps, so the steps cut one whole into 10 equal parts. That makes each step one tenth.
Count the steps from 0 to Point A. There are 7, and each step is one tenth, so Point A is 7 tenths from 0.
Answer
Point A is at , which is 7 tenths of the way from 0 to 1.
Worked example
Three tenths or three hundredths?
Write and as decimals. Then see where each one sits on the number line below.
In 3/10, the bottom number, called the denominator, is 10, so the parts are tenths. The top number, called the numerator, is 3, so there are 3 tenths. That is less than one whole, so write 0 in the ones place and 3 in the tenths place.
In 3/100, the denominator is 100, so the parts are hundredths, and there are 3 of them. That is less than one whole, so the ones place gets 0. It is also fewer than the 10 hundredths in one tenth, so the tenths place gets 0. The 3 goes in the hundredths place.
Read the step size first, as you did for Point A. The line below shows only the part of the whole from 0 to 0.4. Its labels go up by one tenth, with 10 equal steps between labels. Since 10 hundredths make 1 tenth, each step is one hundredth. So 0.03 is 3 steps from 0, and 0.3, which is 30 hundredths, is 30 steps from 0.
Answer
As decimals, and . On the number line, is just 3 steps from 0, far to the left of .
Takeaway: Where a digit sits after the decimal point tells the size of the parts it counts. The first place counts tenths, the second place counts hundredths, and 10 hundredths make 1 tenth, so . On a number line, read the size of one step before you count steps. In a fraction with a denominator of 10 or 100, the denominator tells you whether the numerator counts tenths or hundredths: , but .