Numbers - Licensurehub

General Professional Knowledge (Essential Professional Skills, Literacy & Numeracy)

The place value system is a positional numeral system where the value of a digit depends on its position in the number. We use base 10 (decimal system), meaning each position represents a power of 10.

1.1 Place Value System – Foundation Explanation

For integers (moving from right to left):

Units (10⁰ = 1) → Tens (10¹ = 10) → Hundreds (10² = 100) → Thousands (10³ = 1000) → Ten thousands (10⁴ = 10,000) → etc.

For decimals (moving from left to right after decimal point):

Tenths (10⁻¹ = 1/10 = 0.1) → Hundredths (10⁻² = 1/100 = 0.01) → Thousandths (10⁻³ = 1/1000 = 0.001) → etc.

Mathematical Representation:

Number: 4,532.768

Expanded form: 4 × 1000 + 5 × 100 + 3 × 10 + 2 × 1 + 7 × 0.1 + 6 × 0.01 + 8 × 0.001

1.2 Key Concepts

Zero as a placeholder: Essential in distinguishing numbers (205 ≠ 25)
Decimal point: Separates whole numbers from fractional parts
Reading decimals: Read digits after decimal individually (3.14 = “three point one four”)

1.3 Working Examples (5 Examples)

Example 1: Write expanded form for 7,049.305

7 × 1000 = 7000
0 × 100 = 0
4 × 10 = 40
9 × 1 = 9
3 × 0.1 = 0.3
0 × 0.01 = 0
5 × 0.001 = 0.005

Answer: 7000 + 40 + 9 + 0.3 + 0.005

Example 2: Which is larger: 0.08 or 0.0800?

Both represent 8/100 = 8 ÷ 100 = 0.08
Trailing zeros after decimal don’t change value

Answer: They are equal (both = 0.08)

Example 3: Write three hundred four and twenty-five thousandths

Three hundred four = 304
Twenty-five thousandths = 25/1000 = 0.025
Combined: 304 + 0.025

Answer: 304.025

Example 4: Arrange in descending order: 5.06, 5.6, 5.006

Align decimals for comparison:
5.6 = 5.600
5.06 = 5.060
5.006 = 5.006
Compare: 5.600 > 5.060 > 5.006

Answer: 5.6 > 5.06 > 5.006

Example 5: What is the value of 7 in 3.75?

3.75: Digits after decimal: 7 (tenths), 5 (hundredths)
7 is in tenths place → 7 × 0.1

Answer: 0.7 (seven tenths)

2. FOUR BASIC OPERATIONS

2.1 Addition (+)

Combining quantities to find total sum.

Commutative Property

a + b = b + a

Associative Property

(a + b) + c = a + (b + c)

Identity Property

a + 0 = a

Decimal addition rule: Align decimal points vertically before adding.

Addition Examples:

Example 1: 345 + 278

345
+ 278
—
623
(Working: 5+8=13, write 3 carry 1; 4+7+1=12, write 2 carry 1; 3+2+1=6)

Answer: 623

Example 2: 23.46 + 7.835

Align decimals:
23.460 ← add zero for alignment
+ 7.835
——-
31.295

Answer: 31.295

Example 3: Word problem – John has £24.50, Mary has £18.75. Total?

24.50 + 18.75 = ?
24.50
+ 18.75
——-
43.25

Answer: £43.25

Example 4: 2/3 + 3/4

Find common denominator (LCM of 3 and 4 = 12)
2/3 = (2×4)/(3×4) = 8/12
3/4 = (3×3)/(4×3) = 9/12
8/12 + 9/12 = 17/12

Answer: 17/12 = 1⁵⁄₁₂

Example 5: ¾ + 0.25

¾ = 0.75
0.75 + 0.25 = 1.00

Answer: 1.00 or 1

2.2 Subtraction (–)

Finding difference between numbers; inverse operation of addition.

NOT commutative: a – b ≠ b – a (unless a = b)

Subtraction Examples:

Example 1: 503 – 267

503
– 267
—
236
(Working: 3-7 can’t do, borrow from 0, but 0 has nothing, borrow from 5: 5→4, 0→10, then 10→9, 13→13; 13-7=6; 9-6=3; 4-2=2)

Answer: 236

Example 2: 12.4 – 3.86

Align decimals:
12.40 ← add zero
– 3.86
——-
8.54
(Working: 0-6 can’t, borrow: 4→3, 10→10; 10-6=4; 3-8 can’t, borrow: 2→1, 13→13; 13-8=5; 1-3 can’t, borrow: 1→0, 11→11; 11-3=8)

Answer: 8.54

Example 3: Temperature drops from 23.5°C to 18.2°C. Difference?

23.5 – 18.2 = ?
23.5
– 18.2
——-
5.3

Answer: 5.3°C

Example 4: 5¼ – 2⅔

Convert to improper fractions:
5¼ = 21/4
2⅔ = 8/3
Common denominator: LCM of 4 and 3 = 12
21/4 = 63/12
8/3 = 32/12
63/12 – 32/12 = 31/12

Answer: 31/12 = 2⁷⁄₁₂

Example 5: 1 – 0.099

1.000 ← add zeros
– 0.099
——-
0.901
(Working: 0-9 can’t, borrow: 0→9, 10→10; 10-9=1; 9-9=0; 9-0=9; 0-0=0)

Answer: 0.901

2.3 Multiplication (×)

Repeated addition; scaling factor. For example, 4 × 3 means 4 groups of 3.

Commutative

a × b = b × a

Associative

(a × b) × c = a × (b × c)

Distributive

a × (b + c) = a×b + a×c

Identity

a × 1 = a

Zero Property

a × 0 = 0

Decimal multiplication: Multiply ignoring decimals, then count total decimal places in factors.

Multiplication Examples:

Example 1: 24 × 13

Using distributive property:
24 × 13 = 24 × (10 + 3)
= (24 × 10) + (24 × 3)
= 240 + 72
= 312

Answer: 312

Example 2: 2.3 × 4.5

Multiply as integers: 23 × 45 = 1035
Count decimal places: 2.3 has 1, 4.5 has 1 → total 2 decimal places
Place decimal: 1035 → 10.35

Answer: 10.35

Example 3: Area of rectangle 3.5m × 2.4m

Area = length × width
3.5 × 2.4 = ?
35 × 24 = 840
Total decimal places = 2 (1+1)
840 → 8.40

Answer: 8.40 m²

Example 4: ¾ × ⅔

Multiply numerators: 3 × 2 = 6
Multiply denominators: 4 × 3 = 12
6/12 = 1/2 (simplify by dividing by 6)

Answer: ½

Example 5: 12 × 0.25

0.25 = 25/100 = 1/4
12 × 1/4 = 12/4 = 3
Or: 12 × 0.25 = 3.00

Answer: 3

2.4 Division (÷)

Splitting into equal parts; inverse of multiplication. a ÷ b means “how many b’s fit into a?”

Types of Division:

Partitive: How many in each group? (12 ÷ 3 = 4 means 12 split into 3 equal groups gives 4 each)

Quotative: How many groups? (12 ÷ 4 = 3 means 12 split into groups of 4 gives 3 groups)

Decimal division: Multiply divisor and dividend by same power of 10 to eliminate decimals.

Division Examples:

Example 1: 144 ÷ 12

12 × 12 = 144, so 144 ÷ 12 = 12
Check: 12 × 12 = 144 ✓

Answer: 12

Example 2: 7.5 ÷ 0.25

Multiply both by 100 to eliminate decimals:
7.5 × 100 = 750
0.25 × 100 = 25
Now: 750 ÷ 25 = 30
Check: 30 × 0.25 = 7.5 ✓

Answer: 30

Example 3: 2⅓ ÷ ½

Convert to improper fraction: 2⅓ = 7/3
Division by fraction = multiply by reciprocal:
7/3 ÷ 1/2 = 7/3 × 2/1 = 14/3

Answer: 14/3 = 4⅔

Example 4: 5 ÷ 0.125

0.125 = 125/1000 = 1/8
5 ÷ 1/8 = 5 × 8/1 = 40
Or: 0.125 × 40 = 5.000 ✓

Answer: 40

Example 5: 8.4 ÷ 4

8.4 ÷ 4 = 2.1
Check: 4 × 2.1 = 8.4 ✓
Or: 84 ÷ 4 = 21, then place decimal: 2.1

Answer: 2.1

3. FRACTIONS AND EQUIVALENTS

3.1 Fraction Fundamentals

A fraction represents a part of a whole, written as a/b where a = numerator (top), b = denominator (bottom, b ≠ 0).

Type	Definition	Example
Proper Fraction	Numerator < Denominator	3/4, 2/5
Improper Fraction	Numerator ≥ Denominator	5/4, 7/3
Mixed Number	Whole number + Proper fraction	1¼, 2½

3.2 Equivalence Principle

Key Rule:

a/b = (a×n)/(b×n) where n ≠ 0

Example: 1/2 = 2/4 = 3/6 = 50/100 = 0.5

Lowest Terms:

Divide numerator and denominator by their Highest Common Factor (HCF).

Example: Simplify 24/36
HCF(24,36) = 12
24÷12 = 2, 36÷12 = 3
So 24/36 = 2/3

3.3 Operations with Fractions

Addition/Subtraction

Same denominator required
a/c ± b/c = (a±b)/c

Multiplication

Multiply numerators, multiply denominators
(a/b) × (c/d) = (a×c)/(b×d)

Division

Multiply by reciprocal
(a/b) ÷ (c/d) = (a/b) × (d/c)

3.4 Fraction ↔ Decimal ↔ Percentage

Conversion	Method	Example
Fraction to Decimal	Divide numerator by denominator	3/8 = 3 ÷ 8 = 0.375
Decimal to Fraction	Write over place value denominator, simplify	0.625 = 625/1000 = 5/8
Fraction/Decimal to Percentage	Multiply by 100, add % sign	0.375 × 100 = 37.5%
Percentage to Fraction/Decimal	Divide by 100, remove % sign	37.5% = 37.5 ÷ 100 = 0.375 = 3/8

3.5 Working Examples (5 Examples)

Example 1: Find three equivalents of 2/3

Multiply numerator and denominator by same number:
2/3 = (2×2)/(3×2) = 4/6
2/3 = (2×3)/(3×3) = 6/9
2/3 = (2×10)/(3×10) = 20/30

Answer: 4/6, 6/9, 20/30 (all equivalent to 2/3)

Example 2: Add 2¼ + 3⅔

Convert to improper fractions:
2¼ = 9/4
3⅔ = 11/3

Find common denominator: LCM of 4 and 3 = 12
9/4 = (9×3)/(4×3) = 27/12
11/3 = (11×4)/(3×4) = 44/12
27/12 + 44/12 = 71/12

Answer: 71/12 = 5¹¹⁄₁₂

Example 3: Multiply 1½ × 2⅓

Convert to improper fractions:
1½ = 3/2
2⅓ = 7/3
Multiply: (3/2) × (7/3) = (3×7)/(2×3) = 21/6
Simplify: 21/6 = 7/2 = 3½

Answer: 3½

Example 4: Convert 0.125 to fraction in lowest terms

0.125 = 125/1000 (thousandths)
Find HCF of 125 and 1000 = 125
125÷125 = 1, 1000÷125 = 8
So 125/1000 = 1/8

Answer: 1/8

Example 5: Arrange 2/3, 3/5, 5/8 in ascending order

Find common denominator: LCM of 3,5,8 = 120
2/3 = (2×40)/(3×40) = 80/120
3/5 = (3×24)/(5×24) = 72/120
5/8 = (5×15)/(8×15) = 75/120
Compare numerators: 72 < 75 < 80

Answer: 3/5 < 5/8 < 2/3

4. RATIOS AND PROPORTIONS

4.1 Ratio Fundamentals

A ratio compares two quantities, written as a:b or a/b. It shows relative size of two values.

Key Properties:

Order matters (a:b ≠ b:a unless a = b)
Ratios can be simplified like fractions (divide by common factor)
Units must be same before comparing
Ratios don’t have units (they cancel out)

Example: Class has 12 boys, 18 girls

Ratio boys:girls = 12:18
Simplify by dividing by HCF(12,18) = 6
12÷6 = 2, 18÷6 = 3

Answer: 2:3 (for every 2 boys, there are 3 girls)

4.2 Proportion Theory

A proportion is an equation stating that two ratios are equal: a:b = c:d or a/b = c/d.

Cross-Multiplication Rule:

If a/b = c/d, then ad = bc

This is used to solve for unknown values in proportions.

Direct Proportion

y = kx (k is constant)
As x increases, y increases proportionally

Inverse Proportion

y = k/x (k is constant)
As x increases, y decreases proportionally

4.3 Unitary Method

Find value of one unit, then multiply to find value of required number of units.

Example: If 5 pens cost £3.50, find cost of 8 pens

Step 1: Find cost of 1 pen
1 pen = £3.50 ÷ 5 = £0.70
Step 2: Multiply for 8 pens
8 pens = £0.70 × 8 = £5.60

Answer: £5.60

4.4 Scale and Maps

Scale Ratio:

1:n means 1 unit on map = n units in reality

Example: Scale 1:50,000 means 1cm on map = 50,000cm in reality = 500m = 0.5km

4.5 Working Examples (5 Examples)

Example 1: Simplify ratio 24:36:60

Find HCF of all three numbers:
Factors of 24: 1,2,3,4,6,8,12,24
Factors of 36: 1,2,3,4,6,9,12,18,36
Factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60
HCF = 12
24÷12 = 2, 36÷12 = 3, 60÷12 = 5

Answer: 2:3:5

Example 2: Divide £120 in ratio 3:5

Total parts = 3 + 5 = 8
Value of 1 part = £120 ÷ 8 = £15
First share (3 parts) = 3 × £15 = £45
Second share (5 parts) = 5 × £15 = £75
Check: £45 + £75 = £120 ✓

Answer: £45 and £75

Example 3: If 8 workers build wall in 6 days, how long for 12 workers?

This is inverse proportion: more workers = less time
Work = workers × days = constant
8 workers × 6 days = 48 worker-days
12 workers × ? days = 48 worker-days
? days = 48 ÷ 12 = 4 days

Answer: 4 days

Example 4: Map scale 1:25,000. Distance on map = 4.5cm. Actual distance?

1cm on map = 25,000cm in reality
4.5cm on map = 4.5 × 25,000cm = 112,500cm
Convert to meters: 112,500 ÷ 100 = 1,125m
Convert to km: 1,125 ÷ 1,000 = 1.125km

Answer: 1.125km or 1,125m

Example 5: Concrete mix cement:sand:gravel = 1:2:4. For 28kg total, how much cement?

Total parts = 1 + 2 + 4 = 7
Value of 1 part = 28kg ÷ 7 = 4kg
Cement (1 part) = 1 × 4kg = 4kg
Sand (2 parts) = 2 × 4kg = 8kg
Gravel (4 parts) = 4 × 4kg = 16kg
Check: 4kg + 8kg + 16kg = 28kg ✓

Answer: 4kg of cement

SUMMARY CONNECTIONS

Place value underpins understanding of all number operations

Four operations apply consistently across integers, decimals, fractions

Fractions, decimals, percentages are different representations of same values

Ratios are comparisons, proportions are equal ratios

All concepts interlink – e.g., ratios can be expressed as fractions, operations with fractions follow same rules as with integers

Pedagogical note: Students should master place value first, as misunderstanding here causes cascading errors. Use concrete materials (base-10 blocks, fraction walls) before moving to abstract work. Real-world contexts enhance understanding and retention.

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