Place Value — Ones, Tens and Hundreds

Place Value — Ones, Tens and Hundreds

Why is 345 different from 534? Because every digit has a place — and that place has a value. The digit 3 in 345 is worth 300, but in 534 it is worth only 30. That is the secret of place value — each position is worth ten times the position to its right.

Background and Basic Definitions

Our number system is decimal — based on 10. We have only ten digits (0–9), and we build larger numbers using place value.

PlaceValueExample (in 583)
Ones (units)\( 1 \)3 ones = 3
Tens\( 10 \)8 tens = 80
Hundreds\( 100 \)5 hundreds = 500

So: \(583 = 500 + 80 + 3 = 5 \times 100 + 8 \times 10 + 3 \times 1\).

Each place is worth 10 times the place to its right:

\[ 1 \longrightarrow 10 \longrightarrow 100 \longrightarrow 1000 \longrightarrow \cdots \]

Solution Steps

  1. Step 1 — Write the number and label each digit: O (ones), T (tens), H (hundreds).
  2. Step 2 — Identify how many hundreds: the third digit from the right (if it exists). Multiply by 100.
  3. Step 3 — Identify how many tens: the second digit from the right. Multiply by 10.
  4. Step 4 — Identify how many ones: the rightmost digit. Its value is simply the digit itself.
  5. Step 5 — Expand the number: \(\text{hundreds} + \text{tens} + \text{ones}\). Check that the sum gives back the original number.

Worked Examples

Example 1: Breaking Down a Three-Digit Number

Problem: Expand the number 472 by place value.

Solution:

  1. The digit 4 is in the hundreds place: \(4 \times 100 = 400\).
  2. The digit 7 is in the tens place: \(7 \times 10 = 70\).
  3. The digit 2 is in the ones place: \(2 \times 1 = 2\).
  4. Check: \(400 + 70 + 2 = 472\). Correct!

Answer: \( 472 = 400 + 70 + 2 \)

Example 2: A Number with Zero in the Tens

Problem: Expand the number 305 by place value.

Solution:

  1. The digit 3 — hundreds: \(3 \times 100 = 300\).
  2. The digit 0 — tens: \(0 \times 10 = 0\). No tens!
  3. The digit 5 — ones: \(5\).
  4. \(305 = 300 + 0 + 5\).

Answer: \( 305 = 300 + 5 \) (no tens — the 0 holds the place)

Example 3: Building a Number from Place Values

Problem: Build the number that has 6 hundreds, 0 tens, and 9 ones.

Solution:

  1. \(6 \times 100 = 600\).
  2. \(0 \times 10 = 0\).
  3. \(9 \times 1 = 9\).
  4. The number: \(600 + 0 + 9 = 609\).

Answer: The number is \(609\).

Example 4: Comparing Numbers Using Place Value

Problem: Which is greater: 529 or 592?

Solution:

  1. Both numbers start with 5 hundreds — tied so far.
  2. Tens: 529 → 2 tens; 592 → 9 tens.
  3. 9 tens > 2 tens, so 592 > 529.

Answer: \( 592 \gt 529 \)

Example 5: Value of a Specific Digit

Problem: What is the value of the digit 7 in the number 174?

Solution:

  1. 174: digits from right to left — 4 (ones), 7 (tens), 1 (hundreds).
  2. The digit 7 is in the tens place.
  3. Its value: \(7 \times 10 = 70\).

Answer: The value of the digit 7 in 174 is \(70\).

Common Mistakes

✗ Common mistake: Thinking the digit itself equals its value: saying the digit 4 in 472 equals 4.

✓ The correct way: The digit 4 in 472 is in the hundreds place, so its value is \(4 \times 100 = 400\). Always ask: "What place is the digit in?"

✗ Common mistake: Confusing the digit with its place name: saying "it has 7 tens, so it is 7".

✓ The correct way: 7 tens = \(7 \times 10 = 70\). The 7 is the number of groups, and each group is worth 10.

✗ Common mistake: When there is a 0 digit (e.g., 308) — skipping it and saying the number is "3 hundreds and 8" forgetting that 0 is a place-holder.

✓ The correct way: The 0 is important! It holds the tens place and shows that 3 is in hundreds, not tens. Without the 0 we would have 38.

Practice Tips

  • Tip — In any number: count from the right: 1st = ones, 2nd = tens, 3rd = hundreds, 4th = thousands. Easy to remember!
  • Tip — To find the greater number: compare hundreds first, then tens, then ones.
  • Tip — Zero is a place-holder — it represents no quantity but makes sure every other digit is in the right position.
  • Tip — You can expand numbers in different ways: 583 = 5 hundreds + 83 = 58 tens + 3. Different break-downs help with addition and subtraction.

Summary and Key Formulas

PlaceValue
Ones\( \times 1 \)
Tens\( \times 10 \)
Hundreds\( \times 100 \)
  • Each place = 10 times the place before it.
  • \( 0 \) = place-holder — don't skip it!
  • Expansion example: \( 472 = 400 + 70 + 2 \).