4th Grade · 4.NF.C.5

Tenths and hundredths in 4th grade

Tenths and hundredths are fractions with 10 or 100 as the bottom number, such as $\frac{9}{10}$ and $\frac{45}{100}$ . In 4th grade, your child learns that 1 tenth is the same amount as 10 hundredths. Your child uses this to add a tenths fraction to a hundredths fraction.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.

What it means, in plain words

A tenth is one of 10 equal pieces of a whole. A hundredth is one of 100 equal pieces, so it is much smaller. Money is the easiest picture: a dime is $\frac{1}{10}$ of a dollar and a penny is $\frac{1}{100}$ of a dollar. A dime is worth 10 pennies, so 1 tenth is the same as 10 hundredths.

Tenths and hundredths in 4th grade start with rewriting tenths as hundredths. Each tenth becomes 10 hundredths, so $\frac{9}{10} = \frac{90}{100}$ , in the same way that 9 dimes are worth 90 cents. On a square cut into 100 small squares, 1 tenth is one full column of 10 small squares.

Adding tenths and hundredths comes next. Fractions can only be added when they count the same size of piece, so your child changes the tenths into hundredths first and then adds the top numbers. In 4th grade, this is the only kind of adding with different denominators (bottom numbers). Sums like $\frac{1}{3} + \frac{1}{4}$ come later.

Your child should be able to:

Explain why 1 tenth is the same as 10 hundredths
Rewrite a fraction with denominator 10 as a fraction with denominator 100
Add a tenths fraction and a hundredths fraction by changing the tenths first
Show tenths and hundredths with dimes and pennies or a 100-square grid

Example

How do you write $\frac{6}{10}$ as a fraction with 100 as the denominator?

Each tenth is 10 hundredths, just as a dime is worth 10 cents.
So 6 tenths is $6 \times 10 = 60$ hundredths.
$\frac{6}{10} = \frac{60}{100}$

Example

What is $\frac{7}{10} + \frac{15}{100}$ ?

Change the tenths into hundredths: $\frac{7}{10} = \frac{70}{100}$ .
Now both fractions count hundredths: $\frac{70}{100} + \frac{15}{100} = \frac{85}{100}$ .
Check with money: 7 dimes and 15 pennies make 85 cents. The sum is $\frac{85}{100}$ .

Where children usually get stuck

Your child writes $\frac{5}{10} + \frac{12}{100} = \frac{17}{110}$ .
The child adds the top numbers and the bottom numbers as if they were separate whole numbers. Tenths and hundredths are different sizes of piece, so they cannot be added until they match.
What helps. Use coins. 5 dimes and 12 pennies make 62 cents, so the sum has to be $\frac{62}{100}$ . Then show how $\frac{5}{10}$ becomes $\frac{50}{100}$ .
Your child says $\frac{1}{100}$ is bigger than $\frac{1}{10}$ , because 100 is bigger than 10.
With whole numbers, bigger numbers mean more. In a fraction, a bigger bottom number means the whole is cut into more pieces, so each piece is smaller.
What helps. Put a dime next to a penny and ask which coin is worth more. The dime is $\frac{1}{10}$ of a dollar and the penny is $\frac{1}{100}$ .
Your child says 6 dimes are $\frac{6}{100}$ of a dollar.
The child counts the coins instead of their value. 6 coins are 6 tenths of a dollar, which is 60 hundredths, not 6.
What helps. Lay out the 6 dimes and count by tens together: 10, 20, 30, 40, 50, 60 cents. That is $\frac{60}{100}$ , or $\frac{6}{10}$ , of a dollar.

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Type the missing numerator in each blank to complete the addition.
$\displaystyle 1/10 + 3/100$
$\displaystyle =$
$\displaystyle /100 + 3/100$
$\displaystyle =$
$\displaystyle /100$
Show the answer and how to solve it
The converted numerator is $10$ because $\frac{1}{10} = \frac{10}{100}$ . Adding the numerators gives $10 + 3 = 13$ , so the sum is $\frac{13}{100}$ .
1. 1Convert $\frac{1}{10}$ to a fraction with denominator $100$ by multiplying the numerator and denominator by $10$ .
2. 2Now both fractions have denominator $100$ . Add the numerators.
3. 3The sum is $\frac{13}{100}$ .
Common mistake. A common mistake is writing $1$ for the first blank instead of $10$ . To convert $\frac{1}{10}$ to hundredths, multiply both the numerator and denominator by $10$ , giving $\frac{10}{100}$ , then add $10 + 3 = 13$ for the final blank.
The idea underneath. To add fractions with different denominators, first rewrite them with the same denominator. Multiplying the numerator and denominator of a fraction by the same number gives an equivalent fraction.
2Warm-up
Find the missing numerator to make the fractions equivalent.
$\displaystyle \frac{1}{10} = \frac{\text{?}}{100}$
Missing numerator =
Show the answer and how to solve it
Answer. 10
The missing numerator is $10$ , because $\frac{1}{10} = \frac{10}{100}$ .
1. 1Look at the denominators in the equation. The fraction on the left has denominator 10, and the fraction on the right has denominator 100.
2. 2To go from 10 to 100, multiply by 10. To keep the fractions equivalent, multiply the numerator by 10 as well.
3. 3So the missing numerator is $10$ .
Common mistake. A common mistake is adding 10 to the numerator instead of multiplying by 10. The denominator changed by a factor of 10 (from 10 to 100), so the numerator must also be multiplied by 10, giving $1 \times 10 = 10$ , not $1 + 10 = 11$ .
The idea underneath. Equivalent fractions are created by multiplying the numerator and denominator by the same number. Since $10 \times 10 = 100$ , the numerator must also be multiplied by 10.
3On level
Liam wants to write $\frac{4}{10}$ with denominator $100$ . His work is shown below.
$\displaystyle \frac{4}{10} = \frac{14}{100}$
Select the statement that best describes Liam's work.
A
The conversion is incorrect because the numerator was left unchanged.
B
The conversion is incorrect because the numerator was multiplied by $100$ instead of $10$ .
C
The conversion is incorrect because $10$ was added to the numerator instead of multiplying by $10$ .
D
The conversion is correct because both the numerator and denominator were multiplied by $10$ .
Show the answer and how to solve it
The conversion is incorrect because $10$ was added to the numerator ( $4 + 10 = 14$ ) instead of multiplying by $10$ ( $4 \times 10 = 40$ ). The correct equivalent fraction is $\frac{40}{100}$ .
1. 1To convert a tenths fraction to hundredths, multiply BOTH the numerator and the denominator by $10$ .
2. 2Compare to Liam's work. Liam wrote $\frac{4}{10} = \frac{14}{100}$ , but the equivalent fraction is $\frac{40}{100}$ . The numerators do not match.
3. 3Look at how Liam got $14$ from $4$ . He added $10$ instead of multiplying by $10$ .
Common mistake. A common mistake is adding $10$ to the numerator instead of multiplying by $10$ . To keep fractions equivalent, the numerator and denominator must be multiplied by the same factor.
The idea underneath. Equivalent fractions are made by multiplying both the numerator and denominator by the SAME factor. Changing only one part — or using addition instead of multiplication — breaks the equivalence.
4On level
Add the fractions. Enter the sum as a fraction with denominator 100.
$\displaystyle \frac{1}{10} + \frac{23}{100}$
Show the answer and how to solve it
Answer. 33
The sum is $\frac{33}{100}$ . To add tenths and hundredths, first rewrite the tenths as hundredths, then add the numerators.
1. 1Look at the two fractions. They have different denominators, $10$ and $100$ . To add them, both fractions need to have the same denominator.
2. 2Rewrite $\frac{1}{10}$ so it has a denominator of $100$ . Multiply both the numerator and denominator by $10$ .
3. 3Now both fractions have the same denominator. Add the numerators and keep the denominator.
4. 4The sum is $\frac{33}{100}$ .
Common mistake. A common mistake is adding the numerators without first making the denominators match. That would give $\frac{1 + 23}{100} = \frac{24}{100}$ , which is wrong. The fractions must have the same denominator before you add the numerators.
The idea underneath. Fractions with denominator $10$ are equivalent to fractions with denominator $100$ when you multiply both the numerator and denominator by $10$ . That is why $\frac{1}{10} = \frac{10}{100}$ .