Place Value and Number

Understanding place value is the foundation of everything you do in mathematics. In the FSCE 11+ exam, you will need to work confidently with numbers up to millions, compare and order numbers, round them accurately, handle negative numbers, and recognise Roman numerals. This lesson gives you a thorough grounding in all of these skills, with worked examples that show you exactly how to think through each problem.

What Is Place Value?

Every digit in a number has a value that depends on its position. We call this place value. The same digit can represent completely different amounts depending on where it sits.

Consider the number 3,456,789. Each digit occupies a specific column:

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
3	4	5	6	7	8	9

The digit 3 is worth 3,000,000 (three million), while the digit 3 in the number 230 is worth just 30. Same digit, completely different value — that is the power of place value.

Reading Large Numbers

When you read a large number, group the digits in threes from the right:

• 1,234,567 is read as "one million, two hundred and thirty-four thousand, five hundred and sixty-seven"
• 4,050,200 is read as "four million, fifty thousand, two hundred"

Notice that we do not say "zero hundred" — we simply skip any group that contains zeros.

Ordering and Comparing Numbers

To compare numbers, always start from the leftmost digit and work your way right.

The Comparison Method

1. First, check how many digits each number has — a number with more digits is always larger (when both are positive).
2. If they have the same number of digits, compare the leftmost digit first.
3. If the leftmost digits are the same, move to the next digit.
4. Continue until you find a difference.

We use these symbols:

• > means "is greater than"
• < means "is less than"
• = means "is equal to"

Rounding Numbers

Rounding means replacing a number with a simpler, approximate value. The rule is straightforward:

1. Find the digit in the place you are rounding to.
2. Look at the digit immediately to the right.
3. If that digit is 5 or more, round up.
4. If that digit is 4 or less, round down (keep the rounding digit the same).
5. Replace all digits to the right with zeros.

Rounding to Different Places

Original Number	Rounded to nearest 10	Rounded to nearest 100	Rounded to nearest 1,000
3,847	3,850	3,800	4,000
12,563	12,560	12,600	13,000
99,951	99,950	100,000	100,000

Negative Numbers

Negative numbers are numbers less than zero. We write them with a minus sign: -3, -15, -200.

Think of a number line:

... -5  -4  -3  -2  -1   0   1   2   3   4   5 ...

Key facts about negative numbers:

• The further left on the number line, the smaller the number.
• -10 is less than -3 because -10 is further from zero on the negative side.
• To find the difference between a negative and a positive number, you can count through zero. The difference between -3 and 4 is 7 (count 3 steps from -3 to 0, then 4 steps from 0 to 4).

Temperature Context

Negative numbers appear frequently in temperature questions. If the temperature is -4°C and it rises by 7 degrees, the new temperature is 3°C. If it falls by 5 degrees from 2°C, the new temperature is -3°C.

Roman Numerals

Roman numerals use letters to represent numbers:

Symbol	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1,000

Rules for Roman Numerals

1. Adding rule: When a smaller numeral comes after a larger one, add them. VI = 5 + 1 = 6.
2. Subtracting rule: When a smaller numeral comes before a larger one, subtract. IV = 5 - 1 = 4.
3. Maximum three: You can use a symbol up to three times in a row (III = 3), but not four times (never IIII — use IV instead).

Common Roman numerals to recognise:

Roman	Value	Roman	Value
IV	4	XL	40
IX	9	XC	90
XIV	14	CD	400
XIX	19	CM	900
XXIV	24	MCMXC	1990

Worked Examples

Worked Example 1: Identifying Place Value

Question: In the number 2,705,413, what is the value of the digit 7?

Step-by-step solution:

1. Write the number in a place value table:

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
2	7	0	5	4	1	3

2. The digit 7 is in the hundred thousands column.
3. Therefore the value of the digit 7 is 700,000 (seven hundred thousand).

Answer: 700,000

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Worked Example 2: Ordering Numbers

Question: Put these numbers in order from smallest to largest: 34,567; 304,567; 34,657; 3,457

Step-by-step solution:

1. First, count the digits in each number:
   • 3,457 has 4 digits
   • 34,567 has 5 digits
   • 34,657 has 5 digits
   • 304,567 has 6 digits
2. The 4-digit number (3,457) is the smallest. The 6-digit number (304,567) is the largest.
3. For the two 5-digit numbers, compare from the left: both start with 3, then 4. The third digits are 5 and 6. Since 5 < 6, we have 34,567 < 34,657.
4. Final order: 3,457; 34,567; 34,657; 304,567

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Worked Example 3: Rounding

Question: The population of a town is 247,836. Round this to the nearest ten thousand.

Step-by-step solution:

1. Identify the ten thousands digit: it is 4 (in 247,836).
2. Look at the digit to its right: it is 7.
3. Since 7 is 5 or more, we round up: the 4 becomes 5.
4. Replace all digits to the right of the ten thousands place with zeros.
5. Answer: 250,000

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Worked Example 4: Negative Numbers

Question: At midnight, the temperature in Edinburgh is -6°C. By noon, the temperature has risen by 11 degrees. What is the temperature at noon?

Step-by-step solution:

1. Start at -6°C on the number line.
2. We need to add 11: first go from -6 to 0 (that is +6), then from 0 we have 11 - 6 = 5 degrees left to add.
3. So the temperature at noon is 5°C.
4. Check: the total change is 5 - (-6) = 5 + 6 = 11. Correct!

Answer: 5°C

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Worked Example 5: Roman Numerals

Question: A clock shows the time as half past the hour represented by the Roman numeral IX. A history book was published in the year MCMXLVII. What is the time on the clock, and what year was the book published?

Step-by-step solution:

1. IX: I before X means 10 - 1 = 9. So the time is half past 9, or 9:30.
2. MCMXLVII: Break it down piece by piece:
   • M = 1,000
   • CM = 900 (C before M means 1,000 - 100)
   • XL = 40 (X before L means 50 - 10)
   • VII = 7
   • Total: 1,000 + 900 + 40 + 7 = 1,947

Answer: The time is 9:30. The book was published in 1947.

Place Value Decision Flowchart

flowchart TD
    A[Look at the question] --> B{What are you asked to do?}
    B -->|Find a digit's value| C[Place the number in a place value table]
    C --> D[Read off the column heading for that digit]
    B -->|Compare or order| E[Count the digits in each number]
    E --> F[Compare from the leftmost digit]
    B -->|Round| G[Find the rounding digit]
    G --> H[Look at the next digit to the right]
    H -->|5 or more| I[Round up]
    H -->|4 or less| J[Keep the same]
    B -->|Convert Roman numerals| K[Break into pairs and singles]
    K --> L[Apply adding and subtracting rules]

Common Mistakes

Mistake	Why It Happens	How to Avoid It
Writing 3,847 rounded to the nearest 100 as 3,900	Confusing rounding up the wrong digit	Always identify the exact digit you are rounding and look at the digit immediately to its right
Thinking -3 is greater than -1	Confusing negative numbers with positive ones	Draw a quick number line — numbers get smaller as you go left
Reading IX as 11 instead of 9	Forgetting the subtraction rule	If a smaller numeral comes BEFORE a larger one, subtract it
Saying 450 rounded to the nearest 100 is 400	Unsure what to do when the deciding digit is exactly 5	The rule is: 5 or more rounds UP, always. So 450 rounds to 500
Missing zeros when ordering numbers	Not aligning place values properly	Write numbers in a place value table before comparing

Exam Tips

1. Draw a quick place value table when comparing or ordering large numbers. It takes 10 seconds and prevents errors.
2. Underline the key digit when rounding — underline the digit you are rounding to and circle the digit to its right.
3. Use a number line for negative numbers — even a quick sketch helps you avoid sign errors.
4. Practise reading Roman numerals on clocks — this makes the subtracting rule (IV = 4, IX = 9) automatic.
5. Show your working — in the FSCE exam, writing down your reasoning can earn you marks even if the final answer has a small error.
6. Double-check by reversing — if you round 247,836 to 250,000, ask yourself: is the original number closer to 250,000 or to 240,000? This confirms your answer.

Key Vocabulary

Term	Meaning
Place value	The value of a digit based on its position in a number
Digit	A single symbol used to make numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
Rounding	Replacing a number with a simpler approximate value
Negative number	A number less than zero, written with a minus sign
Integer	A whole number (positive, negative, or zero)
Roman numeral	A number system using letters (I, V, X, L, C, D, M)
Ascending	Going from smallest to largest
Descending	Going from largest to smallest
Approximate	Close to the true value, but not exact
Number line	A straight line on which every point represents a number

Summary

Place value tells you how much each digit in a number is worth. When comparing numbers, always start from the leftmost digit and work right. When rounding, look at the digit to the right of your rounding place: 5 or more means round up, 4 or less means keep the same. Negative numbers sit to the left of zero on the number line — the further from zero, the smaller they are. Roman numerals use an adding and subtracting system with the letters I, V, X, L, C, D, and M.

This content is designed for FSCE 11+ preparation.