Compare the Values of Digits and Explain Why | Grade 5 Math | LevelKite

Look at the number $4.44$ . It uses the digit 4 three times. The three 4s look the same, but they are not worth the same amount.

A digit is one of the symbols 0 to 9. The value of a digit is how much it is worth in the place where it sits, such as the ones place or the tenths place.

You already know that each place is worth 10 times as much as the place to its right. In this lesson you will use that fact to compare the values of the same digit in two places of one number. You will also explain each comparison with words and an equation.

This chart shows what each 4 in $4.44$ is worth.

The value of each 4 in 4.44

Place	Digit	Value in words	Value as a number
Ones	4	4 ones	$4$
Tenths	4	4 tenths	$0.4$
Hundredths	4	4 hundredths	$0.04$

Start with two 4s whose places are next to each other: the 4 in the ones place and the 4 in the tenths place. Their values are $4$ and $0.4$ .

Ten groups of 4 tenths make 40 tenths, and 40 tenths is the same as 4 ones. You can say this comparison in two ways:

Times as much tells you how many of the smaller value make the larger one. $0.4 \times 10 = 4$ , so the 4 in the ones place is worth 10 times as much as the 4 in the tenths place.
A fraction of tells you what part of the larger value the smaller one is. $4 \div 10 = 0.4$ , and dividing by 10 gives one tenth of a value. So the value of the 4 in the tenths place is $\frac{1}{10}$ of the value of the 4 in the ones place.

Both sentences compare the same two values. The first starts from the smaller value. The second starts from the larger value.

The 4 in the tenths place and the 4 in the hundredths place work the same way:

0.04 \times 10 = 0.4 \qquad 0.4 \div 10 = 0.04

Worked example

Two places apart

In $4.44$ , the 4 in the ones place is worth how many times as much as the 4 in the hundredths place?

Write the value of each digit first, because you compare what the digits are worth. The 4 in the ones place is worth 4 ones. The 4 in the hundredths place is worth 4 hundredths.
$4 \text{ ones} = 4 \qquad 4 \text{ hundredths} = 0.04$
Count the places from the hundredths place to the ones place. Hundredths to tenths is 1 place. Tenths to ones is 1 more place. That is 2 places to the left. The decimal point is not a place, so do not count it.
Each place is worth 10 times as much as the place to its right, so each place to the left makes the value 10 times as much. There are 2 places, so multiply by 10 two times, one place at a time.
$0.04 \times 10 = 0.4 \qquad 0.4 \times 10 = 4$
Multiplying by 10 and then by 10 again is the same as multiplying once by 100, because 10 times 10 is 100. Write one equation that uses both values. In words: 100 groups of 4 hundredths make 400 hundredths, and 400 hundredths is 4 ones.
$0.04 \times 100 = 4$
Say it the other way too. Going from 4 back to 0.04 undoes both steps, so you divide by 10 two times. That is the same as dividing once by 100, and dividing by 100 gives one hundredth of a value.
$4 \div 100 = 0.04$

Answer

The 4 in the ones place is worth $100$ times as much as the 4 in the hundredths place. Said the other way, the value of the 4 in the hundredths place is $\frac{1}{100}$ of the value of the 4 in the ones place.

Key concept

Same digit, different place, different value

A digit from 1 to 9 is worth a different amount in each place. When the same digit sits in two places of one number, compare the two values.

Go from one digit's place to the other, one place at a time:

Each place to the left makes the value 10 times as much.
Each place to the right makes the value $\frac{1}{10}$ of what it was.

So you multiply or divide by 10 once for each place moved.

How the values compare when the same digit, from 1 to 9, is 1, 2 or 3 places apart

Places apart	Multiply or divide by	Value of the digit further left	Value of the digit further right
1 place	$10$	$10$ times as much as the other	$\frac{1}{10}$ of the other
2 places	$10 \times 10 = 100$	$100$ times as much as the other	$\frac{1}{100}$ of the other
3 places	$10 \times 10 \times 10 = 1{,}000$	$1{,}000$ times as much as the other	$\frac{1}{1{,}000}$ of the other

Worked example

Three places apart, as a fraction

In $1{,}352.3$ , the digit 3 appears twice. The value of the 3 in the tenths place is what fraction of the value of the 3 in the hundreds place?

Write the two values. The 3 in the hundreds place is worth 3 hundreds. The 3 in the tenths place is worth 3 tenths.
$3 \text{ hundreds} = 300 \qquad 3 \text{ tenths} = 0.3$
Count the places from the hundreds place to the tenths place: hundreds to tens, tens to ones, ones to tenths. That is 3 places to the right. Each place to the right makes the value one tenth of what it was, so divide by 10 three times.
$300 \div 10 = 30 \qquad 30 \div 10 = 3 \qquad 3 \div 10 = 0.3$
Dividing by 10 three times is the same as dividing once by 1,000, because 10 times 10 times 10 is 1,000. Dividing by 1,000 gives one thousandth of a value.
$300 \div 1{,}000 = 0.3$
Check by multiplying. If the fraction is right, 1,000 times the smaller value gives the larger value.
$0.3 \times 1{,}000 = 300$

Answer

The value of the 3 in the tenths place is $\frac{1}{1{,}000}$ of the value of the 3 in the hundreds place. Said the other way, the 3 in the hundreds place is worth $1{,}000$ times as much as the 3 in the tenths place.

Finding the answer is half the job. You also need to explain it, so that someone else can see why it is true. A strong explanation uses the values of the digits, not only where the digits sit. Give it three parts:

Values. Say what each digit is worth.
Places. Say how many places apart the digits are, and what that does to the value.
Equation. Write an equation that uses the two values.

Here are two explanations for the 4 in the ones place and the 4 in the tenths place of $4.44$ :

Positions only: The 4 in the ones place is worth more because it is further to the left.
Values: The 4 in the ones place is worth $4$ . The 4 in the tenths place is worth $0.4$ . They are one place apart, so the 4 in the ones place is worth 10 times as much: $0.4 \times 10 = 4$ .

The first explanation is true, but it does not say how much more. The second one does, and its equation backs it up.

Worked example

Explain with words and an equation

Explain how the value of the 8 in the tens place of $80.08$ compares with the value of the 8 in the hundredths place. Use words and an equation.

Values. The 8 in the tens place is worth 8 tens. The 8 in the hundredths place is worth 8 hundredths.
$8 \text{ tens} = 80 \qquad 8 \text{ hundredths} = 0.08$
Places. From hundredths to tenths, tenths to ones, and ones to tens is 3 places to the left. So the value becomes 10 times as much, three times over.
$0.08 \times 10 = 0.8 \qquad 0.8 \times 10 = 8 \qquad 8 \times 10 = 80$
Equation. Three multiplications by 10 are the same as one multiplication by 1,000, because 10 times 10 times 10 is 1,000. Write it with the two values.
$0.08 \times 1{,}000 = 80$

Answer

The 8 in the tens place is worth $80$ and the 8 in the hundredths place is worth $0.08$ . The tens place is 3 places to the left, so multiply by 10 three times: $0.08 \times 1{,}000 = 80$ . The 8 in the tens place is worth $1{,}000$ times as much.

Remember: a digit's value comes from both the digit and its place. To compare the same digit in two places, write the two values. Then count the places you move to get from one to the other, and multiply or divide by 10 once for each place.

For the same digit from 1 to 9, the one further left is worth 10 times as much when the places are 1 apart, 100 times as much when they are 2 apart, and 1,000 times as much when they are 3 apart. The one further right is worth $\frac{1}{10}$ , $\frac{1}{100}$ or $\frac{1}{1{,}000}$ as much.

Always explain with the two values and an equation, such as $0.04 \times 100 = 4$ .