Adding tenths and hundredths as fractions

When you reach for it

Leyla and Tendai help at a research station in a rainforest. Each month they read a rain gauge, a tall tube marked in meters, and write down how deep the water stood. Some months they write the depth in tenths of a meter, such as \displaystyle \frac{3}{10} meter. Other months they read it more finely, in hundredths of a meter, such as \displaystyle \frac{45}{100} meter. At the end of April they want the total for March and April together.

The trouble is that the two amounts count pieces of different sizes. A tenth is a whole cut into 10 parts; a hundredth is a whole cut into 100 parts, so it is much smaller. Three of one and forty-five of the other are not 48 of anything. You can only add counts when they count the same piece: that is adding fractions with like denominators, where \displaystyle \frac{30}{100} + \frac{45}{100} is just 30 + 45 hundredths.

So you reach for this skill whenever an amount in tenths meets an amount in hundredths and the two must be put together: rainfall, lengths of rope or vine, and money, where a dime is a tenth of a dollar and a penny is a hundredth. It rests on tenths and hundredths, which showed that \displaystyle \frac{7}{10} = \frac{70}{100}, and on equivalent fractions in pictures: cutting every piece of a shaded shape into smaller equal pieces changes the numbers but not the amount.

The procedure

The lesson is taught in two parts, one a day.

Part 1: write the tenths as hundredths, then add

Take a square as one whole. Cut it into 10 columns: each column is one tenth. Now cut every column into 10 small squares. The whole is a 10 by 10 grid of 100 small squares, and each small square is one hundredth (a picture of a fraction as part of a shape is the area model).

Every column holds 10 small squares, so every tenth is 10 hundredths. Three tenths are three columns, which is 3 \times 10 = 30 small squares:

Multiplying the top and the bottom by 10 is the same as cutting each piece into 10. The bottom says the pieces are 10 times as many; the top says the count of shaded pieces is 10 times as large. The shaded amount does not change.

Now both amounts count hundredths, and counts of the same piece add like whole numbers:

The bottom number stays 100. Adding 30 small squares to 45 small squares gives 75 small squares, and they are still hundredths. Adding the bottoms as well would say the squares had changed size, and they have not.

The answer can be written as a decimal too. Seventy-five hundredths is 7 full columns and 5 more squares, so it is 7 tenths and 5 hundredths:

Three things make this work every time.

• A tenth always splits into hundredths exactly. Since 10 \times 10 = 100, any number of tenths is ten times as many hundredths. Going the other way does not always work: 45 hundredths is 4 columns and 5 loose squares, not a whole number of tenths. That is why the tenths are rewritten, not the hundredths.
• Order does not matter. \displaystyle \frac{45}{100} + \frac{3}{10} gives the same \displaystyle \frac{75}{100}, because of the commutative property of addition. With three amounts you may also add any two first (the associative property), for example the two tenths before the hundredths.
• A decimal is already a count of tenths or hundredths. 0.6 is \displaystyle \frac{6}{10} and 0.27 is \displaystyle \frac{27}{100}. To add amounts given as decimals, write each as a fraction and add as above.

Part 2: sums past one whole, and word problems

A grid holds 100 small squares, so 100 hundredths make one whole:

When a sum of hundredths reaches 100 or more, the extra squares spill onto a second grid. In October and November the rainforest gauge read \displaystyle \frac{8}{10} meter and \displaystyle \frac{56}{100} meter. The total is

Split the 136 squares into one full grid and the rest:

The same total has three names. As a fraction it is \displaystyle \frac{136}{100}. As a mixed number, a whole number and a fraction together (see Mixed numbers and improper fractions), it is \displaystyle 1\frac{36}{100}. As a decimal it is 1.36, read “one and thirty-six hundredths”.

When the amounts already have whole parts, add the wholes and add the parts. The parts may pass one whole again, and then that whole joins the others. Every amount in a sum must be of the same whole: meters with meters, dollars with dollars.

Worked examples

Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.

Boundaries

• The pieces must be the same size before they are added. Tenths are rewritten as hundredths, and only then are the tops added. The bottom of the answer is 100, not 10 + 100.
• Tenths go to hundredths, not always back. Every number of tenths is a whole number of hundredths, because 10 \times 10 = 100. A number of hundredths is a whole number of tenths only when it ends in 0: \displaystyle \frac{40}{100} = \frac{4}{10}, but \displaystyle \frac{45}{100} is not any whole number of tenths.
• Tenths and hundredths only. This move works because 10 goes into 100 exactly. Adding, say, thirds and fourths needs a different shared piece; that is adding fractions with unlike denominators.
• The same whole. \displaystyle \frac{3}{10} of a meter and \displaystyle \frac{45}{100} of a dollar cannot be added: the fractions are parts of different wholes, so their sum names nothing.
• More than one whole. When the top reaches 100 hundredths or more, the sum is at least one whole and is often easier to read as a mixed number or a decimal with a whole part, such as \displaystyle 1\frac{36}{100} = 1.36.

Where this goes next

Adding by renaming pieces is the start of all fraction addition with different bottoms: the parent idea is adding fractions after renaming them in the same pieces, and the general case is adding fractions with unlike denominators.

For decimals, the same idea becomes a column method. Lining up the decimal points puts tenths under tenths and hundredths under hundredths, so the pieces match without writing any fractions; that is adding and subtracting decimals. Seeing where a sum such as 1.36 sits between 1 and 2 is decimals on a number line, and deciding whether 1.55 dollars is more than 1.50 dollars is comparing decimals. Sums of prices are worked further in money written as decimals.

Exercises

Exercises 1 to 12 go with Part 1, and 13 to 22 with Part 2. The reasoning group draws on both.

Part 1: write the tenths as hundredths, then add

1. Write as hundredths: (a) \displaystyle \frac{4}{10}, (b) \displaystyle \frac{9}{10}.
2. Add: \displaystyle \frac{2}{10} + \frac{35}{100}. Give the answer as a fraction and a decimal.
3. Add: \displaystyle \frac{5}{10} + \frac{41}{100}. Give the answer as a fraction and a decimal.
4. Add: \displaystyle \frac{13}{100} + \frac{7}{10}.
5. Add: \displaystyle \frac{1}{10} + \frac{9}{100}. Write the answer as a decimal.
6. Write 0.4 and 0.29 as fractions, then add them.
7. Leyla’s gauge read 0.5 meter in January and 0.08 meter in February. Find the total as a fraction and a decimal of a meter, and in centimeters.
8. Add: \displaystyle \frac{3}{10} + \frac{12}{100} + \frac{4}{10}.
9. Find the missing number: \displaystyle \frac{6}{10} + \frac{?}{100} = \frac{71}{100}.
10. A vine near the station grows \displaystyle \frac{2}{10} meter one week and \displaystyle \frac{23}{100} meter the next. How much did it grow in the two weeks?
11. Write 0.7 and 0.14 as fractions with bottom 100, then add them.
12. Find the missing number of tenths: \displaystyle \frac{?}{10} + \frac{5}{100} = \frac{65}{100}.

Part 2: sums past one whole, and word problems

13. Add: \displaystyle \frac{9}{10} + \frac{24}{100}. Write the answer as a fraction, a mixed number and a decimal.
14. Add: \displaystyle \frac{7}{10} + \frac{63}{100}. Write the answer as a mixed number and a decimal.
15. Add: \displaystyle 1\frac{3}{10} + 2\frac{41}{100}.
16. Add: \displaystyle 4\frac{6}{10} + \frac{75}{100}. Write the answer as a mixed number and a decimal.
17. Add: \displaystyle \frac{5}{10} + \frac{5}{100} + \frac{5}{10}. Write the answer as a decimal.
18. Tendai has 6 dimes and 72 pennies. How many dollars is that?
19. Deniz has 0.8 dollar in one pocket and 0.65 dollar in another. A map of the trails costs 1.50 dollars. Does she have enough?
20. The two wettest months gave \displaystyle \frac{9}{10} meter and \displaystyle \frac{79}{100} meter of rain. Find the total as a mixed number and a decimal.
21. Leyla joins two ropes end to end, without overlap: one is \displaystyle 1\frac{4}{10} meters long and the other is \displaystyle \frac{85}{100} meter. How long is the joined rope?
22. Over three months the gauge read \displaystyle \frac{6}{10} meter, \displaystyle \frac{47}{100} meter and \displaystyle \frac{5}{10} meter. Find the total.

Reasoning

23. Tendai writes \displaystyle \frac{4}{10} + \frac{25}{100} = \frac{29}{110}. Without working the right answer first, explain why his answer cannot be right. Then find the right sum.
24. To add \displaystyle \frac{3}{10} + \frac{45}{100}, Deniz wants to write the hundredths as tenths instead of the tenths as hundredths. Explain why that does not work here, and say when hundredths can be written as tenths.
25. Without finding the exact sum, explain why \displaystyle \frac{7}{10} + \frac{46}{100} is more than one whole.