Place Value After the Point
The places keep dividing by ten past the decimal point: tenths, then hundredths. Lining up the points lines up the places.
A decimal is a fraction with a hidden denominator of , , or . The digit right after the point counts tenths, the next counts hundredths, so means tenths and hundredths — . Money is the friendliest example: is cents out of .
Line up the points. Hundredths: . Tenths: . Wholes: . The sum is .
In a place-value chart: ones, tenths, hundredths. Say it as "three and seven hundredths". The matters: without it, would mean ones and tenths, ten times as much as hundredths.
Reading as smaller than because . Line the decimals up by place value: versus — fifty hundredths is more than thirty-five hundredths.
Write "four and six hundredths" as a decimal. Which digit is in the tenths place of ?
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Four ones, tenths, hundredths: . In the first digit after the point is the tenths digit: (it is worth tenths, ).
In Plain Terms & More Examples
Line up the decimal points so tenths sit under tenths and hundredths under hundredths, then add place by place.
Hundredths , tenths , wholes : the sum is .
Hundredths , tenths : the sum is .
Lining up the last digits instead of the decimal points. Stacking over with the two s together treats as if it were , and the sum comes out as . Line up the points (write as ) and the sum is .
Find .
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Line up the points. Hundredths: ; write , carry tenth. Tenths: . Ones: . Sum: .
Comparing and Rounding
To compare and , write and — fifty hundredths beats thirty-five. Rounding to tenths looks at the hundredths digit: or more rounds up.
As hundredths: . Least to greatest: .
lies between the tenths and . Halfway between them is . The hundredths digit is , so is past halfway and closer to : it rounds up to .
Chopping off the extra digit instead of rounding, or losing the carry. Round to the nearest tenth: the is or more, so the tenths go up to tenths, which is one more whole. So , not (chopped) and not .
Round to the nearest tenth, and write as a decimal.
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(the rounds the up). — a holds the tenths place.
Word Problems
A dollar is cents, so one cent is , one hundredth of a dollar. Prices are decimals with two places: is dollar, dimes and pennies.
Dimes are tenths of a dollar and pennies are hundredths. So lining up the decimal points lines up dollars with dollars, dimes with dimes and pennies with pennies, and every money problem is a decimal problem you already know how to do.
A pencil costs . Paying with : , so the change is cents.
Another way, the way a shopkeeper counts: count up from to . Five cents reaches , and more cents reaches : .
A pencil costs , a ruler and an eraser . Line up the points and add. Hundredths: ; write , carry . Tenths: ; write , carry . Ones: . Total: .
Subtracting the smaller digit from the bigger in every column. For that gives hundredths and tenths , so , more than the dollar you started with! You have to take from , which needs a trade: is hundredths, and hundredths, or .
A notebook costs and a pen costs . What is the total?
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Line up the points: .
Tenths and Hundredths as Fractions
A decimal is a fraction with or on the bottom: , and . To add tenths and hundredths, change the tenths into hundredths first.
One column of a hundredths grid is one tenth, and it holds small squares, so . That is the equivalent-fraction move again, multiplying top and bottom by . It is also why and are equal: columns are small squares.
is tenths: . Multiply top and bottom by : , which is written . So .
Change the tenths to hundredths: . Then , which is .
: seven whole columns. : seven small squares. is ten times as much, because .
Writing as . The first place after the point is tenths, so means . Seven hundredths needs a in the tenths place: .
Write and as fractions. Then write as a decimal.
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and . For the sum, , so .
Why It Works
Our number system gives each place a value ten times the one to its right: thousands, hundreds, tens, ones. Continuing the pattern to the right of the ones gives tenths, hundredths, thousandths — each place is a tenth of the last. So . Decimals are not a new kind of number; they are the place-value system extended below one.
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.