2.1 Numbers, Place Value, Decimals & Four Operations

2.1 Numbers, Place Value, Decimals & Four Operations

A secure mastery of place value, written arithmetic algorithms, order of operations, and decimal calculations forms the bedrock of success in the Kent Test 11+ mathematics paper. The GL Assessment exam format demands both high accuracy and speed without the aid of a calculator. This section provides an exhaustive guide to core numerical skills, common exam traps, and efficient step-by-step methods.

Place Value Structure & Reading Large Numbers

Our base-10 numerical system relies on position to determine value. Each column represents a power of $10$. Moving left across a number multiplies its value by $10$; moving right divides its value by $10$.

In the number $3,452,809.716$:

• The digit $4$ represents $4$ Hundred Thousands ($400,000$).
• The digit $8$ represents $8$ Hundreds ($800$).
• The digit $7$ represents $7$ Tenths ($0.7$ or $\frac{7}{10}$).
• The digit $6$ represents $6$ Thousandths ($0.006$ or $\frac{6}{1000}$).

Common 11+ Place Value Question Types

1. Value of a Digit: Identifying the exact quantity represented by a specified digit (e.g., in $5,063,214$, the value of $6$ is $60,000$ or six ten-thousands).
2. Multiplying/Dividing by Powers of $10$: Shifting digits relative to the decimal point.
   • Example: $43.8 \times 100 = 4,380$ (digits move 2 places to the left).
   • Example: $5.09 \div 1000 = 0.00509$ (digits move 3 places to the right).

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Standard Written Algorithms for the Four Operations

1. Column Addition and Subtraction

Column methods require strict vertical alignment by place value. When subtracting across zero digits, regrouping (borrowing) must be executed methodically.

Worked Example (Subtraction across zeros): Calculate $8,002 - 3,485$.

$$\begin{array}{rcccc} & 7 & 9 & 9 & 12 \\ & 8 & 0 & 0 & 2 \\ - & 3 & 4 & 8 & 5 \\ \hline & 4 & 5 & 1 & 7 \end{array}$$

• Units: Cannot do $2 - 5$. Regroup from thousands: $8,000$ becomes $7,000 + 900 + 90 + 10$. Units becomes $12$. $12 - 5 = 7$.
• Tens: $9 - 8 = 1$.
• Hundreds: $9 - 4 = 5$.
• Thousands: $7 - 3 = 4$. Result = $4,517$.

2. Long Multiplication

The standard grid or column method breaks multi-digit multiplication into manageable products.

Worked Example: Calculate $348 \times 27$.

• Step 1 (Multiply by 7 units): $348 \times 7 = 2,436$.
• Step 2 (Multiply by 2 tens): Place a zero in the units column, then $348 \times 20 = 6,960$.
• Step 3 (Add partial products): $2,436 + 6,960 = 9,396$.

3. Short and Long Division

Division tests estimation and remainder handling. Express remainders as whole numbers, fractions, or decimals as required by the question context.

Worked Example: Calculate $4,536 \div 14$.

• $14$ into $45$ goes $3$ times ($3 \times 14 = 42$), remainder $3$.
• $14$ into $33$ goes $2$ times ($2 \times 14 = 28$), remainder $5$.
• $14$ into $56$ goes $4$ times ($4 \times 14 = 56$), remainder $0$.
• Result = $324$.

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Order of Operations (BIDMAS / BODMAS)

Mathematical operations must be executed in a strict hierarchical order to avoid ambiguity:

1. Brackets: Perform calculations inside parentheses ( ) first.
2. Indices: Evaluate powers and square roots ($x^2$, $\sqrt{x}$).
3. Division and Multiplication: Left-to-right in order of appearance (equal priority).
4. Addition and Subtraction: Left-to-right in order of appearance (equal priority).

Critical Exam Caution: Division does NOT take precedence over Multiplication, nor does Addition take precedence over Subtraction! They are evaluated strictly left to right as they appear.

Worked Example: Evaluate $36 - 4 \times (2 + 5) + 8 \div 2$.

1. Brackets: $(2 + 5) = 7$. Expression becomes $36 - 4 \times 7 + 8 \div 2$.
2. Multiplication & Division (left to right):
   • $4 \times 7 = 28$.
   • $8 \div 2 = 4$.
   • Expression becomes $36 - 28 + 4$.
3. Addition & Subtraction (left to right):
   • $36 - 28 = 8$.
   • $8 + 4 = 12$.

• Final Result = $12$.

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Rounding, Estimation & Reasonable Bounds

Rounding simplifies numbers while retaining approximate value.

Rounding Rules

• Locate the target rounding position (e.g., nearest hundred, 2 decimal places).
• Inspect the indicator digit immediately to its right:
  • If the indicator is $5, 6, 7, 8,$ or $9$, round the target digit up.
  • If the indicator is $0, 1, 2, 3,$ or $4$, leave the target digit unchanged.

Example: Round $45,873$ to the nearest hundred. Target is $8$ (hundreds), indicator is $7$ (tens). Round up $8$ to $9 \rightarrow 45,900$.

Estimation Strategies for 11+ Speed

Before performing lengthy calculations, estimate the expected result by rounding each number to 1 significant figure.

• Example: Estimate $48.7 \times 19.2 \approx 50 \times 20 = 1,000$. If your calculated answer is $935.04$, it is reasonable; an answer of $9,350.4$ immediately signals a decimal point displacement error.

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Decimal Operations

Decimal Addition and Subtraction

Always align decimal points vertically and insert trailing zeros as placeholders.

Worked Example: Calculate $14.8 - 3.095$.

$$\begin{array}{rccccccc} & 1 & 4 & . & 8 & 0 & 0 \\ - & & 3 & . & 0 & 9 & 5 \\ \hline & 1 & 1 & . & 7 & 0 & 5 \end{array}$$

Decimal Multiplication

1. Ignore decimal points and multiply as whole numbers.
2. Count the total number of decimal places in all original factors.
3. Place the decimal point in the final product so it has the same total count of decimal places.

Worked Example: Calculate $0.4 \times 0.08$.

• Whole numbers: $4 \times 8 = 32$.
• Decimal places: $0.4$ (1 place) + $0.08$ (2 places) = 3 total decimal places.
• Insert leading zeros: $0.032$.

Decimal Division

Never divide by a decimal! Transform the divisor into a whole number by multiplying both divisor and dividend by the same power of $10$.

Worked Example: Calculate $7.56 \div 0.18$.

• Multiply both numbers by $100$: $7.56 \times 100 = 756$; $0.18 \times 100 = 18$.
• Perform division: $756 \div 18 = 42$.