Transfer Test Maths — Revision Space

Place Value AQE & GL ▶

Place value tells us what each digit in a number is worth, based on its position. This is the foundation of all number work!

Reading and Writing Numbers

You need to be able to read and write numbers up to millions. Each group of three digits has a name:

What is Each Digit Worth?

The value of a digit depends on where it sits in the number. The digit 5 in 500 is worth five hundred, but in 50 it is only worth fifty.

In the number 47,235, the 4 is worth 40,000
The 7 is worth 7,000
The 2 is worth 200
The 3 is worth 30
The 5 is worth 5

Partitioning

Partitioning means breaking a number into parts based on place value. This is very useful for mental maths.

Partitioning Example 3,456 = 3,000 + 400 + 50 + 6
28,195 = 20,000 + 8,000 + 100 + 90 + 5

Worked Example: What is the value of the 7 in 372,461?

Look at the position of the 7 in the number: 3 7 2 , 4 6 1

The 7 is in the ten-thousands column.

So the 7 is worth 7 x 10,000 = 70,000

The value of the 7 is 70,000 (seventy thousand)

When reading big numbers, break them into groups of three from the right. Use commas to separate: 2,345,678. Read each group then say its name (millions, thousands).

Don't confuse the digit with its value! If the question asks "What is the value of the 6 in 16,482?" the answer is 6 6,000.

Try These!

What is the value of the 3 in 435,201?
Write 2,056,400 in words.
Partition 67,892 into thousands, hundreds, tens and ones.
Which digit is in the hundred-thousands place in 823,456?
Write five hundred and twelve thousand, three hundred and seven in figures.

1. 30,000 (thirty thousand)
2. Two million, fifty-six thousand, four hundred
3. 60,000 + 7,000 + 800 + 90 + 2
4. 8
5. 512,307

Ordering & Comparing Numbers AQE & GL ▶

You need to compare numbers using the correct symbols and put numbers in order from smallest to largest (ascending) or largest to smallest (descending).

The Symbols

Comparison Symbols < means "is less than" e.g. 34 < 56
> means "is greater than" e.g. 89 > 72
= means "is equal to" e.g. 50 = 50

Think of the symbol as a crocodile's mouth — the crocodile always eats the bigger number!

How to Compare Numbers

Step 1: First, check how many digits each number has. More digits = bigger number.
Step 2: If they have the same number of digits, compare from left to right.
Step 3: The first digit that is different tells you which number is bigger.

Worked Example: Put these in order — 4,521 | 4,512 | 4,251

All three numbers have 4 digits, so compare from left to right.

They all start with 4. Next digit: 5, 5, 2. The number starting 42... is smallest: 4,251

Now compare 4,521 and 4,512. First two digits match (4,5). Third digit: 2 vs 1. 4,512 < 4,521

Ascending order: 4,251, 4,512, 4,521

Negative Numbers

Negative numbers are less than zero. Think of a thermometer: -5 is colder (less) than -2. On a number line, numbers get bigger as you go right.

Number Line ...-5 -4 -3 -2 -1 0 1 2 3 4 5...
<--- colder / smaller warmer / bigger --->

-7 is less than -3 (further from zero on the negative side)
-1 is greater than -10
Any positive number is greater than any negative number

With negative numbers, the bigger the digit, the smaller the number! -8 > -3 -8 < -3. Think: -8 degrees is colder than -3 degrees!

Try These!

Put in order (smallest first): 3,782 3,728 3,872
Use < or > to compare: 45,621 ___ 45,612
Put in order (smallest first): -4, 2, -7, 0, -1
What temperature is colder: -6°C or -2°C?

1. 3,728, 3,782, 3,872
2. 45,621 > 45,612
3. -7, -4, -1, 0, 2
4. -6°C is colder

Rounding AQE & GL ▶

Rounding makes numbers simpler to work with. You round to the nearest 10, 100, or 1,000.

The Rule

Rounding Rules 1. Find the digit you are rounding TO.
2. Look at the digit ONE PLACE to the RIGHT of it.
3. If that digit is 0, 1, 2, 3, or 4 → round DOWN (keep the same).
4. If that digit is 5, 6, 7, 8, or 9 → round UP (add one).
5. Replace all digits to the right with zeros.

Worked Example: Round 4,673 to the nearest 100

Find the hundreds digit: it's the 6 (in 4,673).

Look at the digit to its right: it's 7.

7 is 5 or more, so round UP: the 6 becomes 7.

Replace digits to the right with zeros.

4,673 rounded to the nearest 100 = 4,700

Worked Example: Round 23,451 to the nearest 1,000

Find the thousands digit: it's the 3 (in 23,451).

Look at the digit to its right: it's 4.

4 is less than 5, so round DOWN: the 3 stays the same.

23,451 rounded to the nearest 1,000 = 23,000

Remember the rhyme: "5 or more, raise the score. 4 or less, let it rest."

When rounding 4,950 to the nearest 1,000: the thousands digit is 4, the next digit is 9. Since 9 ≥ 5, round UP to 5,000, NOT 4,000!

Try These!

Round 847 to the nearest 10.
Round 3,562 to the nearest 100.
Round 16,499 to the nearest 1,000.
Round 7,851 to the nearest 10.
A school has 1,276 pupils. Round this to the nearest 100.

1. 850
2. 3,600
3. 16,000
4. 7,850
5. 1,300

Addition & Subtraction AQE & GL ▶

You must be confident with adding and subtracting numbers, both using written methods and mental strategies.

Column Addition (Carrying)

Line up the numbers by place value. Add each column from right to left. If a column adds up to 10 or more, carry the tens digit to the next column.

Column Addition Example: 3,487 + 2,956 3 4 8 7
+ 2 9 5 6
--------
6 4 4 3
1 1 1 ← carried digits

Column Subtraction (Borrowing)

Line up by place value. Subtract each column from right to left. If the top digit is smaller, borrow from the next column.

Column Subtraction Example: 5,032 - 1,478 5 0 3 2
- 1 4 7 8
--------
3 5 5 4

Mental Strategies

Near doubles: 35 + 36 = 35 + 35 + 1 = 71
Rounding and adjusting: 67 + 29 = 67 + 30 - 1 = 96
Partitioning: 156 + 243 = 100 + 200 + 50 + 40 + 6 + 3 = 399
Counting on: For subtraction, count up from the smaller number. 503 - 287: from 287 to 300 = 13, from 300 to 500 = 200, from 500 to 503 = 3. Total = 216.

Worked Example: Word Problem

Question: A school collected 1,245 cans in Week 1 and 987 cans in Week 2. They gave away 450 cans to a food bank. How many cans do they have left?

Add Week 1 and Week 2: 1,245 + 987 = 2,232

Subtract the cans given away: 2,232 - 450 = 1,782

They have 1,782 cans left.

Always estimate first! Round the numbers and work out a rough answer. This helps you check if your final answer makes sense. 1,245 + 987 is roughly 1,200 + 1,000 = 2,200. Our answer of 2,232 is close, so it's probably right.

When borrowing in subtraction, remember to reduce the digit you borrowed from! If you forget to cross it out and reduce it by 1, your answer will be wrong.

Try These!

4,567 + 3,845 = ?
8,003 - 2,756 = ?
A shop sold 1,384 comics in January and 2,197 in February. How many comics were sold altogether?
Tom had £50. He spent £23.75. How much does he have left?

1. 8,412
2. 5,247
3. 3,581 comics
4. £26.25

Multiplication AQE & GL ▶

Multiplication is one of the most important skills for the transfer test. You MUST know your times tables up to 12 x 12 by heart!

Times Tables (Must Know!)

If you don't know your tables instantly, practise them every single day. The Quick Reference tab has a full 12x12 grid. Some tricky ones to remember:

7 x 8 = 56
6 x 7 = 42
8 x 9 = 72
7 x 9 = 63
12 x 12 = 144

Multiplying by 10, 100, 1000

Rules x 10 → digits move ONE place to the LEFT (add one zero)
x 100 → digits move TWO places to the LEFT (add two zeros)
x 1000 → digits move THREE places to the LEFT (add three zeros)

Examples: 34 x 10 = 340 | 34 x 100 = 3,400 | 34 x 1,000 = 34,000

Grid Method (Partitioning Method)

Break the numbers into parts, multiply each pair, then add them all up.

Worked Example: 47 x 36 using Grid Method

Partition both numbers: 47 = 40 + 7 and 36 = 30 + 6

Draw the grid and multiply each pair:

| 40 | 7
----|-------|------
30 | 1,200 | 210
6 | 240 | 42

Add all the parts: 1,200 + 210 + 240 + 42 = 1,692

47 x 36 = 1,692

Long Multiplication (Column Method)

Worked Example: 47 x 36 using Column Method

Write the numbers one above the other: 4 7
x 3 6

Multiply 47 by 6 (the ones digit): 47 x 6 = 282

Multiply 47 by 30 (the tens digit): 47 x 30 = 1,410

Add the two results: 282 + 1,410 = 1,692

47 x 36 = 1,692

Use whichever method you feel most confident with. The grid method is easier to set out and less likely to cause mistakes. The column method is faster once you're good at it. Always double check by estimating: 47 x 36 is roughly 50 x 40 = 2,000, so 1,692 seems about right!

In long multiplication, when you multiply by the tens digit, don't forget to put a zero on the right first! If you multiply 47 x 3 instead of 47 x 30, your answer will be way too small.

Try These!

56 x 10 = ?
234 x 100 = ?
38 x 24 = ? (use either method)
145 x 37 = ?
Each ticket to a school play costs £6.50. If 48 tickets are sold, how much money is collected?

1. 560
2. 23,400
3. 912
4. 5,365
5. £312.00

Division AQE & GL ▶

Division means splitting a number into equal groups. It's the opposite (inverse) of multiplication. You need to know short division (bus stop method) and how to deal with remainders.

Dividing by 10, 100, 1000

Rules ÷ 10 → digits move ONE place to the RIGHT
÷ 100 → digits move TWO places to the RIGHT
÷ 1000 → digits move THREE places to the RIGHT

Examples: 340 ÷ 10 = 34 | 5,600 ÷ 100 = 56 | 72,000 ÷ 1,000 = 72

Short Division (Bus Stop Method)

This is the method you'll use most often. Write the number inside the "bus stop" and the divisor outside.

Worked Example: 847 ÷ 3

How many 3s in 8? 2 remainder 2. Write 2 above the 8, carry the 2 to the next column.

How many 3s in 24? 8 exactly. Write 8 above the 4.

How many 3s in 7? 2 remainder 1. Write 2 above the 7.

847 ÷ 3 = 282 remainder 1 (or 282 r1)

Long Division

Use long division when dividing by a two-digit number (like 15, 23, etc.).

Worked Example: 756 ÷ 12

How many 12s in 7? None. So look at 75. 12 x 6 = 72. Write 6 above.

Subtract: 75 - 72 = 3. Bring down the 6 to make 36.

How many 12s in 36? 12 x 3 = 36 exactly. Write 3 above.

756 ÷ 12 = 63

What To Do With Remainders

Leave it as a remainder: 17 ÷ 5 = 3 r2
Write as a fraction: 17 ÷ 5 = 3 2/5
Round up: "How many 5-seat cars needed for 17 people?" Answer: 4 (you can't leave people behind!)
Round down: "You have £17. How many £5 books can you buy?" Answer: 3 (you can't spend more than you have!)

Read the question carefully to decide what to do with the remainder! The question will guide you — "How many can you fill completely?" means round down. "How many do you need?" means round up.

When carrying in short division, make sure you carry the remainder to the front of the next digit, not after it. If 8 ÷ 3 = 2 r2, the 2 goes in front of the next digit to make twenty-something, not 2 + the next digit.

Try These!

936 ÷ 4 = ?
5,175 ÷ 5 = ?
853 ÷ 6 = ? (give your answer with a remainder)
624 ÷ 12 = ?
There are 350 sweets to share equally among 8 children. How many does each child get, and how many are left over?

1. 234
2. 1,035
3. 142 r1
4. 52
5. 43 sweets each, 6 left over (350 ÷ 8 = 43 r6)

Order of Operations (BODMAS) AQE & GL ▶

When a calculation has more than one operation, there's a special order you must follow. You can't just go left to right!

BODMAS (the order of operations) B — Brackets (do these first)
O — Orders (squares, cubes, roots)
D — Division
M — Multiplication
A — Addition
S — Subtraction

Division & Multiplication are equal (do left to right)
Addition & Subtraction are equal (do left to right)

Worked Example: 3 + 4 x 2

There are no brackets, so look for multiplication/division first.

Do the multiplication: 4 x 2 = 8

Now do the addition: 3 + 8 = 11

3 + 4 x 2 = 11 (NOT 14!)

Worked Example: (6 + 4) x 3 - 2

Brackets first: (6 + 4) = 10

Multiplication next: 10 x 3 = 30

Finally subtraction: 30 - 2 = 28

(6 + 4) x 3 - 2 = 28

The most common mistake is doing addition before multiplication! 3 + 4 x 2 = 14 3 + 4 x 2 = 11. Always do x and ÷ before + and -.

Try These!

5 + 3 x 6 = ?
20 - 12 ÷ 4 = ?
(8 + 2) x (7 - 3) = ?
4 x 5 + 6 x 2 = ?
100 - 3 x (12 + 8) = ?

1. 23 (not 48)
2. 17 (not 2)
3. 40
4. 32
5. 40

Estimation AQE & GL ▶

Estimation means working out a rough answer by rounding the numbers first. This is very useful for checking whether your answer makes sense.

How to Estimate

Step 1: Round each number to something easy to work with (usually the nearest 10 or 100).
Step 2: Do the calculation with the rounded numbers.
Step 3: Compare your estimate with your actual answer. Are they close?

Worked Example: Estimate 487 x 23

Round 487 to 500 and round 23 to 20.

Calculate: 500 x 20 = 10,000

The actual answer (11,201) should be close to 10,000. It is, so the answer seems right!

Estimate: approximately 10,000

Always ask yourself: "Is my answer sensible?" If you're asked the cost of 3 footballs at £12.99 each and you get £389.70, something has gone wrong! A quick estimate: 3 x 13 = about £39.

Try These!

Estimate 312 + 489 by rounding to the nearest 100.
Estimate 78 x 42 by rounding to the nearest 10.
Estimate 891 ÷ 29 by rounding.
Josh calculated 34 x 56 = 19,040. Use estimation to explain why this is wrong.

1. 300 + 500 = 800 (actual: 801)
2. 80 x 40 = 3,200 (actual: 3,276)
3. 900 ÷ 30 = 30 (actual: 30.7...)
4. 30 x 60 = 1,800, so 19,040 is way too large. The actual answer is 1,904.

Special Numbers AQE & GL ▶

There are several special types of numbers you need to know for the transfer test. Being able to recognise and work with these will help you across many topics.

Square Numbers

A square number is the result of multiplying a number by itself. We write this using the ² symbol.

Calculation	Square Number
1 x 1 = 1²	1
2 x 2 = 2²	4
3 x 3 = 3²	9
4 x 4 = 4²	16
5 x 5 = 5²	25
6 x 6 = 6²	36
7 x 7 = 7²	49
8 x 8 = 8²	64
9 x 9 = 9²	81
10 x 10 = 10²	100
11 x 11 = 11²	121
12 x 12 = 12²	144

Example What is 7²?
7² = 7 x 7 = 49

Cube Numbers

A cube number is the result of multiplying a number by itself three times. We write this using the ³ symbol.

Cube Numbers to Know 1³ = 1 x 1 x 1 = 1
2³ = 2 x 2 x 2 = 8
3³ = 3 x 3 x 3 = 27
4³ = 4 x 4 x 4 = 64
5³ = 5 x 5 x 5 = 125

Notice that 64 is both a square number (8²) and a cube number (4³)!

Prime Numbers

A prime number is a number that has exactly 2 factors: 1 and itself. It can only be divided evenly by 1 and by itself.

Prime Numbers to 50 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

1 is NOT a prime number! It only has one factor (itself), but primes need exactly two factors.
2 is the only even prime number. Every other even number can be divided by 2, so they have more than 2 factors.

Factors

Factors are numbers that divide exactly into another number with no remainder. To find all the factors, work in pairs.

Worked Example: Find all the factors of 12

Start with 1: 1 x 12 = 12 → factors: 1, 12

Try 2: 2 x 6 = 12 → factors: 2, 6

Try 3: 3 x 4 = 12 → factors: 3, 4

Try 4: we already have 4 from the pair above. The pairs have met in the middle, so we are done.

Factors of 12 = 1, 2, 3, 4, 6, 12

Multiples

Multiples are the times table of a number. They go on forever!

Example Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70...

HCF (Highest Common Factor)

The HCF is the largest number that divides exactly into two (or more) numbers.

Worked Example: Find the HCF of 12 and 18

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Common factors (in both lists): 1, 2, 3, 6

HCF = 6 (the highest of the common factors)

LCM (Lowest Common Multiple)

The LCM is the smallest number that appears in both times tables.

Worked Example: Find the LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20, 24...

Multiples of 6: 6, 12, 18, 24, 30...

The first number that appears in both lists is 12.

LCM = 12

To find factors, always work in pairs starting from 1. When your pairs meet in the middle, you've found them all!

1 is NOT a prime number! It only has one factor (itself), but primes need exactly TWO factors.

Try These!

List all the factors of 24.
What are the first five multiples of 9?
Is 27 a prime number? Explain why or why not.
Find the HCF of 16 and 24.
Find the LCM of 5 and 8.

1. 1, 2, 3, 4, 6, 8, 12, 24
2. 9, 18, 27, 36, 45
3. No — 27 = 3 x 9, so it has factors other than 1 and itself.
4. HCF = 8 (factors of 16: 1,2,4,8,16 / factors of 24: 1,2,3,4,6,8,12,24 / common: 1,2,4,8)
5. LCM = 40 (multiples of 5: 5,10,15,20,25,30,35,40 / multiples of 8: 8,16,24,32,40)

Roman Numerals AQE & GL ▶

Roman numerals use letters to represent numbers. You'll see them on clocks, dates on buildings, film credits, book chapters, and Super Bowl numbers!

The Symbols

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

The Rules

Larger before smaller = ADD: VI = 5 + 1 = 6, XI = 10 + 1 = 11, LX = 50 + 10 = 60
Smaller before larger = SUBTRACT: IV = 5 - 1 = 4, IX = 10 - 1 = 9, XL = 50 - 10 = 40, XC = 100 - 10 = 90

Common Examples

Key Conversions XIV = 10 + 4 = 14
XXIX = 10 + 10 + 9 = 29
XLIV = 40 + 4 = 44
XCIX = 90 + 9 = 99
MCMXCIX = 1000 + 900 + 90 + 9 = 1999
MMXXVI = 1000 + 1000 + 10 + 10 + 5 + 1 = 2026

Worked Example: Convert MCMXCIX to a number

Break it into chunks: M + CM + XC + IX

M = 1000

CM = 1000 - 100 = 900

XC = 100 - 10 = 90

IX = 10 - 1 = 9

MCMXCIX = 1000 + 900 + 90 + 9 = 1999

Worked Example: Convert 47 to Roman numerals

Break 47 into parts: 40 + 7

40 = XL (50 - 10)

7 = VII (5 + 1 + 1)

47 = XLVII

Break Roman numerals into chunks: MCMXCIX = M + CM + XC + IX = 1000 + 900 + 90 + 9 = 1999. Work from left to right, and look for subtraction pairs (IV, IX, XL, XC, CD, CM) first.

Try These!

Convert XLII to a number.
Convert DCCCXIV to a number.
Write 29 in Roman numerals.
Write 94 in Roman numerals.
Convert MMXXVI to a number.

1. 42 (40 + 2)
2. 814 (500 + 300 + 10 + 4)
3. XXIX (10 + 10 + 9)
4. XCIV (90 + 4)
5. 2026 (1000 + 1000 + 10 + 10 + 5 + 1)

Understanding Fractions AQE & GL ▶

A fraction is a part of a whole. Think of a pizza cut into slices!

Parts of a Fraction

Numerator and Denominator 3 ← NUMERATOR (how many parts you have)
---
4 ← DENOMINATOR (how many equal parts the whole is split into)

Types of Fractions

Proper fraction: numerator < denominator (e.g. 3/4). The fraction is less than 1 whole.
Improper fraction: numerator ≥ denominator (e.g. 7/4). The fraction is 1 whole or more.
Mixed number: a whole number and a fraction together (e.g. 1 3/4).

Converting Between Improper Fractions and Mixed Numbers

Worked Example: Convert 11/4 to a mixed number

Divide 11 by 4: 11 ÷ 4 = 2 remainder 3

The whole number part is 2, and the remainder 3 goes over the denominator 4.

11/4 = 2 3/4

Equivalent Fractions

Equivalent fractions look different but have the same value. You can find them by multiplying or dividing the numerator AND denominator by the same number.

Equivalent Fractions 1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 50/100
1/3 = 2/6 = 3/9 = 4/12
2/5 = 4/10 = 6/15 = 8/20

To check if two fractions are equivalent, cross-multiply! If 2/3 and 8/12 are equivalent, then 2 x 12 should equal 3 x 8. Both give 24, so yes, they are equivalent!

Try These!

What fraction is shaded if 3 out of 8 equal parts are coloured?
Convert 17/5 to a mixed number.
Convert 3 2/3 to an improper fraction.
Find the missing number: 3/4 = ?/20
Are 6/8 and 3/4 equivalent? Explain how you know.

1. 3/8
2. 3 2/5
3. 11/3
4. 15 (multiply top and bottom of 3/4 by 5)
5. Yes. 6/8 simplifies to 3/4 (divide both by 2). Or: 6 x 4 = 24 and 8 x 3 = 24.

Simplifying Fractions AQE & GL ▶

Simplifying (or cancelling) a fraction means making it as small as possible while keeping the same value. Divide the top and bottom by the same number.

Finding the Highest Common Factor (HCF)

The fastest way to simplify is to divide both parts by their HCF — the biggest number that goes into both.

Worked Example: Simplify 12/18

Find factors of 12: 1, 2, 3, 4, 6, 12

Find factors of 18: 1, 2, 3, 6, 9, 18

The HCF is 6.

Divide both by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3

12/18 = 2/3

If you can't spot the HCF right away, just keep simplifying step by step. Divide both by 2, then check again. Keep going until you can't simplify any further.

Try These!

Simplify 8/12
Simplify 15/25
Simplify 24/36
Simplify 9/27

1. 2/3
2. 3/5
3. 2/3
4. 1/3

Comparing & Ordering Fractions AQE & GL ▶

To compare fractions, you need to give them the same denominator (a common denominator). Then just compare the numerators.

Worked Example: Which is bigger, 3/4 or 5/6?

Find a common denominator. The LCM of 4 and 6 is 12.

Convert 3/4: 3/4 = 9/12 (multiply both by 3)

Convert 5/6: 5/6 = 10/12 (multiply both by 2)

Compare: 10/12 > 9/12

5/6 is bigger than 3/4

Another way: convert both fractions to decimals. 3/4 = 0.75 and 5/6 = 0.833... So 5/6 is bigger.

Try These!

Which is bigger: 2/3 or 3/5?
Put in order (smallest first): 1/2, 3/8, 5/6
Which is bigger: 4/7 or 5/9?

1. 2/3 is bigger (2/3 = 10/15, 3/5 = 9/15)
2. 3/8, 1/2, 5/6 (= 9/24, 12/24, 20/24)
3. 4/7 is bigger (4/7 = 36/63, 5/9 = 35/63)

Adding & Subtracting Fractions AQE & GL ▶

Same Denominator (Easy!)

If the denominators are the same, just add or subtract the numerators. Keep the denominator the same.

Same Denominator 3/7 + 2/7 = 5/7
5/8 - 1/8 = 4/8 = 1/2

Different Denominators

If the denominators are different, you must find a common denominator first.

Worked Example: 2/3 + 1/4

Find the LCM of 3 and 4. The LCM is 12.

Convert 2/3: multiply top and bottom by 4. 2/3 = 8/12

Convert 1/4: multiply top and bottom by 3. 1/4 = 3/12

Add the numerators: 8/12 + 3/12 = 11/12

2/3 + 1/4 = 11/12

Adding Mixed Numbers

Worked Example: 2 1/3 + 1 3/4

Add the whole numbers: 2 + 1 = 3

Add the fractions: find common denominator of 3 and 4 → 12

1/3 = 4/12 and 3/4 = 9/12

4/12 + 9/12 = 13/12 = 1 1/12

Add to the whole numbers: 3 + 1 1/12 = 4 1/12

2 1/3 + 1 3/4 = 4 1/12

NEVER add the denominators! 1/3 + 1/4 = 2/7 1/3 + 1/4 = 7/12. You must find a common denominator first.

Try These!

3/5 + 1/5 = ?
1/2 + 2/3 = ?
5/6 - 1/4 = ?
3 2/5 + 1 4/5 = ?
4 1/3 - 1 2/3 = ?

1. 4/5
2. 7/6 = 1 1/6
3. 7/12
4. 5 1/5
5. 2 2/3

Multiplying & Dividing Fractions AQE & GL ▶

Multiplying Fractions

Multiplying fractions is actually easier than adding them! Just multiply straight across.

Multiplying Fractions numerator x numerator
----------------------
denominator x denominator

Example: 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15

Dividing Fractions — KFC!

To divide by a fraction, use KFC: Keep the first fraction, Flip the second fraction, Change the ÷ to x.

Worked Example: 3/4 ÷ 2/5

Keep the first fraction: 3/4

Flip the second fraction: 2/5 becomes 5/2

Change ÷ to x and multiply: 3/4 x 5/2 = 15/8

Convert to mixed number: 15/8 = 1 7/8

3/4 ÷ 2/5 = 1 7/8

Before multiplying, see if you can simplify diagonally (cross-cancel). In 2/3 x 3/4, the 3s cancel to give 2/1 x 1/4 = 2/4 = 1/2. This keeps numbers small!

Try These!

1/3 x 2/5 = ?
3/4 x 2/3 = ?
5/6 ÷ 1/3 = ?
2/3 ÷ 4/5 = ?

1. 2/15
2. 6/12 = 1/2
3. 15/6 = 2 1/2
4. 10/12 = 5/6

Fraction of an Amount AQE & GL ▶

To find a fraction of an amount, divide by the denominator then multiply by the numerator.

Method To find a/b of a number:
1. DIVIDE by the denominator (b)
2. MULTIPLY by the numerator (a)

Think: "divide by the bottom, times by the top"

Worked Example: Find 3/4 of 60

Divide by the denominator: 60 ÷ 4 = 15

Multiply by the numerator: 15 x 3 = 45

3/4 of 60 = 45

Worked Example: Find 2/5 of £35

Divide by the denominator: £35 ÷ 5 = £7

Multiply by the numerator: £7 x 2 = £14

2/5 of £35 = £14

If asked "What fraction of 40 is 10?" — put the part over the whole: 10/40 = 1/4.

Try These!

Find 1/3 of 27
Find 2/5 of 45
Find 3/8 of 64
Sam scored 3/4 of the 80 goals his team scored. How many did Sam score?
A class has 30 children. 2/5 of them walk to school. How many walk?

1. 9
2. 18
3. 24
4. 60 goals
5. 12 children

Decimals AQE & GL ▶

Decimals are another way of writing fractions. The decimal point separates the whole number from the fractional part.

Decimal Place Value

Decimal Places Ones . Tenths Hundredths Thousandths
O . t h th

3.456 = 3 ones + 4 tenths + 5 hundredths + 6 thousandths
= 3 + 4/10 + 5/100 + 6/1000

Converting Fractions to Decimals

Divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75
Or use known equivalents (see Percentages section for the key table)

Ordering Decimals

Compare digit by digit from left to right, just like whole numbers. Make them the same length by adding trailing zeros if needed.

Worked Example: Order 0.45, 0.405, 0.5

Make them the same length: 0.450, 0.405, 0.500

Compare: 405 < 450 < 500

Order: 0.405, 0.45, 0.5

Adding & Subtracting Decimals

The most important rule: line up the decimal points! Then add or subtract as normal.

When ordering decimals, 0.5 is NOT less than 0.45 just because 5 < 45! 0.5 < 0.45 0.5 > 0.45 (because 0.50 > 0.45).

Try These!

Convert 3/8 to a decimal.
Order from smallest: 0.7, 0.72, 0.07, 0.702
3.45 + 12.8 = ?
10 - 3.67 = ?

1. 0.375
2. 0.07, 0.7, 0.702, 0.72
3. 16.25
4. 6.33

Percentages AQE & GL ▶

"Percent" means "out of 100." So 50% means 50 out of 100, which is the same as a half.

Key Conversions (LEARN THESE!)

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/10	0.1	10%
1/5	0.2	20%
2/5	0.4	40%
3/5	0.6	60%
1/3	0.333...	33.3%
2/3	0.666...	66.7%
1/100	0.01	1%

Finding Percentages Mentally

Quick Mental Methods 50% → divide by 2
25% → divide by 4 (or halve then halve again)
10% → divide by 10
5% → find 10% then halve it
1% → divide by 100

Build up any percentage from these!
e.g. 35% = 25% + 10% or 15% = 10% + 5%

Worked Example: Find 35% of £240

Find 10%: £240 ÷ 10 = £24

Find 25%: £240 ÷ 4 = £60

Add them: 35% = 25% + 10%: £60 + £24 = £84

35% of £240 = £84

Converting Between Forms

Fraction → Decimal: divide numerator by denominator
Decimal → Percentage: multiply by 100 (move digits 2 places left)
Percentage → Decimal: divide by 100 (move digits 2 places right)
Percentage → Fraction: write over 100 and simplify. 45% = 45/100 = 9/20

To find any percentage: find 1% first (divide by 100), then multiply by the percentage you need. 17% of 300: 1% = 3, so 17% = 3 x 17 = 51.

Try These!

Find 25% of 80.
Find 15% of £60.
Convert 0.35 to a percentage.
Convert 45% to a fraction in its simplest form.
In a class of 30 pupils, 40% are boys. How many girls are there?

1. 20
2. £9
3. 35%
4. 9/20
5. 18 girls (40% = 12 boys, so 30 - 12 = 18 girls)

2D Shapes AQE & GL ▶

2D shapes are flat shapes with length and width but no depth. You need to know their names, properties, and how many sides and lines of symmetry they have.

Triangles (3 sides)

Type	Properties	Lines of Symmetry
Equilateral	All 3 sides equal, all angles 60°	3
Isosceles	2 sides equal, 2 angles equal	1
Scalene	All sides different, all angles different	0
Right-angle	One angle is exactly 90°	0 (or 1 if also isosceles)

Quadrilaterals (4 sides)

Shape	Properties	Lines of Symmetry
Square	4 equal sides, 4 right angles	4
Rectangle	2 pairs of equal sides, 4 right angles	2
Parallelogram	2 pairs of parallel sides, opposite sides & angles equal	0
Rhombus	4 equal sides, opposite angles equal (like a pushed square)	2
Trapezium	Exactly 1 pair of parallel sides	0 (or 1 if isosceles)
Kite	2 pairs of adjacent sides equal, 1 pair of opposite angles equal	1

Other Polygons

Pentagon: 5 sides (regular pentagon has 5 lines of symmetry)
Hexagon: 6 sides (regular hexagon has 6 lines of symmetry)
Heptagon: 7 sides
Octagon: 8 sides (think of a STOP sign)

Regular means all sides are equal AND all angles are equal. Irregular means they are not.

A square IS a special type of rectangle (it has 4 right angles and 2 pairs of equal sides). A square is also a rhombus (4 equal sides). So a square is the most special quadrilateral!

Try These!

How many lines of symmetry does a regular hexagon have?
Name a quadrilateral with exactly one pair of parallel sides.
What type of triangle has all angles equal?
I have 4 equal sides but my angles are not 90°. What shape am I?
How many sides does an octagon have?

1. 6
2. Trapezium
3. Equilateral triangle
4. Rhombus
5. 8

3D Shapes AQE & GL ▶

3D shapes have three dimensions: length, width, and height. You need to know their names and properties — faces, edges, and vertices (corners).

Shape	Faces	Edges	Vertices
Cube	6 (all squares)	12	8
Cuboid	6 (rectangles)	12	8
Triangular prism	5 (2 triangles, 3 rectangles)	9	6
Square-based pyramid	5 (1 square, 4 triangles)	8	5
Triangular pyramid (tetrahedron)	4 (all triangles)	6	4
Cylinder	3 (2 circles, 1 curved)	2	0
Cone	2 (1 circle, 1 curved)	1	1 (the apex/point)
Sphere	1 (curved)	0	0

Euler's Formula Faces + Vertices - Edges = 2

Check with a cube: 6 + 8 - 12 = 2 ✓
Check with a triangular prism: 5 + 6 - 9 = 2 ✓

This works for any shape with flat faces (not for cylinders, cones, or spheres).

Nets of 3D Shapes

A net is a flat shape that folds up to make a 3D shape. If you "unfold" a 3D shape and lay it flat, you get its net.

Cube: 6 connected squares. There are actually 11 different possible arrangements that fold into a cube!
Cuboid: 6 rectangles (3 pairs of matching rectangles).
Triangular prism: 2 triangles + 3 rectangles.
Square-based pyramid: 1 square + 4 triangles.

To check if a flat shape is a valid net, imagine folding it up. Every face must connect to the right neighbours, and no faces should overlap.

Remember: a face is a flat surface, an edge is where two faces meet (a line), and a vertex is where edges meet (a corner). Vertices is the plural of vertex.

Use Euler's formula to check your answers: Faces + Vertices - Edges always equals 2!

Try These!

How many faces does a square-based pyramid have?
How many edges does a cuboid have?
A 3D shape has 5 faces, 8 edges, and 5 vertices. What is it?
Using Euler's formula, if a shape has 7 faces and 10 vertices, how many edges does it have?
What shapes make up the net of a triangular prism?
A net is made of 1 square and 4 identical triangles. What 3D shape does it fold into?
A cube has a net made of 6 squares. True or false: any arrangement of 6 connected squares will fold into a cube?

1. 5
2. 12
3. Square-based pyramid
4. F + V - E = 2, so 7 + 10 - E = 2, so E = 15
5. 2 triangles and 3 rectangles
6. Square-based pyramid
7. False — only 11 of the possible arrangements actually fold into a cube. Some arrangements would cause faces to overlap.

Angles AQE & GL ▶

An angle measures how much something has turned. Angles are measured in degrees (°).

Types of Angles

Type	Size	Description
Acute	< 90°	Smaller than a right angle
Right angle	= 90°	Exactly a quarter turn (marked with a little square)
Obtuse	90° to 180°	Between a right angle and a straight line
Straight	= 180°	A straight line (half turn)
Reflex	> 180°	More than a straight line
Full turn	= 360°	A complete turn back to where you started

Angle Rules

Key Angle Facts Angles in a triangle = 180°
Angles on a straight line = 180°
Angles at a point (full turn) = 360°
Angles in a quadrilateral = 360°

Worked Example: Find the missing angle in a triangle with angles 65° and 48°

Angles in a triangle add up to 180°.

Add the known angles: 65 + 48 = 113

Subtract from 180: 180 - 113 = 67

The missing angle is 67°

Worked Example: Two angles on a straight line are x and 125°. Find x.

Angles on a straight line add up to 180°.

x + 125 = 180

x = 180 - 125 = 55°

x = 55°

When using a protractor, make sure you read from the correct scale! There are two scales — one starting from 0° on the left and one from 0° on the right. Check which one starts at 0 on the baseline you're measuring from.

Try These!

A triangle has angles of 90° and 35°. Find the third angle.
Two angles on a straight line are 72° and x. Find x.
Three angles around a point are 120°, 95°, and y. Find y.
Is an angle of 150° acute, obtuse, or reflex?
An isosceles triangle has one angle of 40° at the top. Find the two base angles.

1. 55° (180 - 90 - 35)
2. 108° (180 - 72)
3. 145° (360 - 120 - 95)
4. Obtuse (between 90° and 180°)
5. Each base angle = (180 - 40) ÷ 2 = 70°

Symmetry AQE & GL ▶

A shape has symmetry if you can fold it or turn it and it looks exactly the same.

Lines of Symmetry (Reflective Symmetry)

A line of symmetry divides a shape into two identical halves that are mirror images of each other. Think of folding the shape along the line — both sides would match perfectly.

A square has 4 lines of symmetry
A rectangle has 2 lines of symmetry
An equilateral triangle has 3 lines of symmetry
A circle has infinite lines of symmetry
A parallelogram has 0 lines of symmetry

Rotational Symmetry

A shape has rotational symmetry if it looks the same when you rotate it less than a full turn. The order of rotational symmetry is how many times it looks the same in a full 360° turn.

A square has rotational symmetry of order 4
An equilateral triangle has order 3
A rectangle has order 2
Every shape has at least order 1 (a full turn always looks the same)

To find lines of symmetry, imagine folding the shape. If both halves match up perfectly, that fold line is a line of symmetry. Try horizontal, vertical, and diagonal folds.

Try These!

How many lines of symmetry does a regular pentagon have?
What is the order of rotational symmetry of a regular hexagon?
Name a shape with exactly 1 line of symmetry.
How many lines of symmetry does an isosceles triangle have?

1. 5
2. 6
3. Isosceles triangle, kite, or some letters like A, B, etc.
4. 1

Coordinates AQE & GL ▶

Coordinates tell us the exact position of a point on a grid. They are always written in brackets as (x, y).

How Coordinates Work (x, y)
x = how far ACROSS (left/right) → read first
y = how far UP or DOWN → read second

Remember: "Along the corridor, up the stairs"
Or: "x is a cross, so x goes across"

Four Quadrants

When a grid has negative numbers, it splits into 4 quadrants:

Quadrant 1: x positive, y positive (top right) e.g. (3, 4)
Quadrant 2: x negative, y positive (top left) e.g. (-2, 5)
Quadrant 3: x negative, y negative (bottom left) e.g. (-3, -1)
Quadrant 4: x positive, y negative (bottom right) e.g. (4, -2)

Finding the Midpoint

Midpoint Formula Midpoint of (x1, y1) and (x2, y2):
= ( (x1 + x2) ÷ 2 , (y1 + y2) ÷ 2 )

Just find the average of the x-coordinates and the average of the y-coordinates!

Worked Example: Find the midpoint of (2, 6) and (8, 10)

Average x: (2 + 8) ÷ 2 = 10 ÷ 2 = 5

Average y: (6 + 10) ÷ 2 = 16 ÷ 2 = 8

Midpoint = (5, 8)

Don't mix up x and y! The x-coordinate ALWAYS comes first. (y, x) (x, y). "Along the corridor first, THEN up the stairs."

Try These!

What are the coordinates of a point 3 right and 5 up from the origin?
Plot these in your mind: (2, 4), (6, 4), (6, 1), (2, 1). What shape do they make?
Find the midpoint of (1, 3) and (7, 9).
A point at (-3, 4) is in which quadrant?

1. (3, 5)
2. A rectangle
3. (4, 6)
4. Quadrant 2 (x negative, y positive)

Transformations GL Focus ▶

A transformation changes the position, size, or orientation of a shape. At transfer test level, you need to know reflection, translation, and rotation.

Reflection

A reflection flips a shape over a mirror line. Each point of the shape is the same distance from the mirror line on the other side.

Reflecting in the x-axis: the y-coordinate changes sign (e.g. (3, 4) → (3, -4))
Reflecting in the y-axis: the x-coordinate changes sign (e.g. (3, 4) → (-3, 4))
The reflected shape is the same size and shape, but flipped (like looking in a mirror)

Translation

A translation slides a shape without rotating or flipping it. Every point moves the same distance in the same direction.

Described as moving "x units right/left and y units up/down"
Example: translating (2, 3) by "4 right and 2 down" gives (6, 1)
The shape stays exactly the same size and orientation — it just moves!

Rotation

A rotation turns a shape around a fixed point (the centre of rotation).

Quarter turn = 90°
Half turn = 180°
Three-quarter turn = 270°
Rotation can be clockwise or anticlockwise

When reflecting, use tracing paper or count squares from the mirror line. Each point must be the SAME distance from the line on both sides.

Try These!

Reflect the point (4, 2) in the y-axis.
Reflect the point (-1, 5) in the x-axis.
Translate the point (3, 7) by "5 left and 2 down".
What type of turn is 180°?

1. (-4, 2)
2. (-1, -5)
3. (-2, 5)
4. A half turn

Length AQE & GL ▶

Length measures how long or how far something is. You need to know the metric units and how to convert between them.

Length Conversions 10 mm = 1 cm
100 cm = 1 m
1,000 m = 1 km

To go from BIG to SMALL: MULTIPLY
To go from SMALL to BIG: DIVIDE

Worked Example: Convert 3.5 km to metres

Going from km (big) to m (small), so multiply.

3.5 x 1,000 = 3,500 m

3.5 km = 3,500 m

A useful way to remember: 1 cm is about the width of your little fingernail. 1 m is about the length of a big step. 1 km is about a 12-minute walk.

Try These!

Convert 450 cm to metres.
Convert 2.7 m to cm.
Convert 85 mm to cm.
A path is 1.2 km long. Sarah has walked 850 m. How much further does she have to go?

1. 4.5 m
2. 270 cm
3. 8.5 cm
4. 350 m (1,200 m - 850 m)

Mass (Weight) AQE & GL ▶

Mass measures how heavy something is. In everyday language, we often say "weight" instead of "mass."

Mass Conversions 1,000 g = 1 kg
1,000 kg = 1 tonne

BIG to SMALL: MULTIPLY
SMALL to BIG: DIVIDE

Reading Scales

When reading a scale (like a kitchen scale or a ruler), first work out what each small division is worth. Count the total divisions between two labelled marks and divide.

Worked Example: A scale goes from 0 to 500g with 10 divisions. What does each division represent?

500 ÷ 10 = 50

Each small division = 50 g

Each division is worth 50 g. So the 3rd mark = 150 g.

Try These!

Convert 2,500 g to kg.
Convert 3.75 kg to grams.
A bag of flour weighs 1.5 kg. You use 350 g. How much is left?
Three parcels weigh 750 g, 1.2 kg, and 600 g. Find their total weight in kg.

1. 2.5 kg
2. 3,750 g
3. 1,150 g or 1.15 kg
4. 2.55 kg (750g + 1,200g + 600g = 2,550g = 2.55 kg)

Capacity AQE & GL ▶

Capacity measures how much liquid a container can hold.

Capacity Conversions 1,000 ml = 1 litre (l)

A teaspoon is about 5 ml
A can of fizzy drink is 330 ml
A big bottle of water is 2 litres

Try These!

Convert 3.5 litres to ml.
Convert 750 ml to litres.
A jug holds 2 litres. You pour in 450 ml, then another 800 ml. How much more can the jug hold?
Four glasses each hold 250 ml. How many litres is that in total?

1. 3,500 ml
2. 0.75 litres
3. 750 ml (2,000 - 450 - 800)
4. 1 litre (4 x 250 = 1,000 ml)

Time AQE & GL ▶

Time questions come up very often in the transfer test. You need to convert between 12-hour and 24-hour clock, and calculate time intervals.

12-hour and 24-hour Clock

12-hour	24-hour	12-hour	24-hour
12:00 midnight	00:00	12:00 noon	12:00
1:00 am	01:00	1:00 pm	13:00
6:30 am	06:30	6:30 pm	18:30
9:15 am	09:15	9:15 pm	21:15
11:45 am	11:45	11:45 pm	23:45

Tip: For pm times, add 12 to the hour. For 24-hour times after 12:00, subtract 12 to get the pm time.

Calculating Time Intervals

Use the "counting on" method — count forward from the start time to the end time.

Worked Example: How long is it from 9:45 am to 2:20 pm?

From 9:45 am to 10:00 am = 15 minutes

From 10:00 am to 2:00 pm = 4 hours

From 2:00 pm to 2:20 pm = 20 minutes

Add them up: 15 min + 4 hrs + 20 min = 4 hours 35 minutes

4 hours and 35 minutes

Days in Each Month

Days in Each Month Rhyme "Thirty days hath September,
April, June and November.
All the rest have thirty-one,
Except for February alone,
Which has twenty-eight days clear,
And twenty-nine in each leap year."

Time is NOT decimal! 1 hour 30 minutes is NOT 1.3 hours — it's 1.5 hours. There are 60 minutes in an hour, not 100!

Try These!

Convert 3:45 pm to 24-hour clock.
Convert 19:30 to 12-hour clock.
How long is it from 10:50 am to 3:15 pm?
A film starts at 17:40 and lasts 2 hours 25 minutes. What time does it end?
How many days are in March?

1. 15:45
2. 7:30 pm
3. 4 hours 25 minutes
4. 20:05
5. 31

Perimeter AQE & GL ▶

Perimeter is the total distance around the outside of a shape. Imagine walking around the edge — the perimeter is how far you would walk.

Perimeter Formulas Rectangle: P = 2 x (length + width) or P = 2l + 2w
Square: P = 4 x side length or P = 4s
Regular shape: P = number of sides x side length
Any shape: just add up ALL the sides!

Worked Example: Find the perimeter of an L-shape

An L-shaped room has sides of 8m, 3m, 5m, 2m, 3m, and 5m.

Add up all the sides: 8 + 3 + 5 + 2 + 3 + 5 = 26

Perimeter = 26 m

Finding Missing Sides

For compound (L-shaped) perimeters, you often need to work out missing sides first. Look at the shape and work out what the missing side must be by comparing it to the other sides.

Label EVERY side of the shape before adding them up. Draw them on the diagram if they're not labelled. For compound shapes, the missing side usually equals the difference between two other sides.

Try These!

Find the perimeter of a rectangle with length 12 cm and width 7 cm.
A regular hexagon has sides of 5 cm. What is its perimeter?
A square has a perimeter of 36 cm. What is the length of one side?
An equilateral triangle has a perimeter of 27 cm. What is the length of each side?

1. 38 cm (2 x 12 + 2 x 7)
2. 30 cm (6 x 5)
3. 9 cm (36 ÷ 4)
4. 9 cm (27 ÷ 3)

Area AQE & GL ▶

Area measures the amount of surface a shape covers. It is measured in square units (cm², m², km²).

Area Formulas Rectangle: A = length x width    (A = l x w)
Square: A = side x side    (A = s²)
Triangle: A = ½ x base x height    (A = ½bh)
Parallelogram: A = base x height    (A = bh)

Worked Example: Find the area of a triangle with base 8 cm and height 5 cm

Use the formula: A = ½ x base x height

Substitute the values: A = ½ x 8 x 5

Calculate: A = ½ x 40 = 20

Area = 20 cm²

Area of Compound Shapes

Split the shape into rectangles (or triangles), find the area of each part, then add them together.

Worked Example: L-shaped compound area

An L-shape can be split into two rectangles: one is 6m x 4m, the other is 3m x 2m.

Area of rectangle 1: 6 x 4 = 24 m²

Area of rectangle 2: 3 x 2 = 6 m²

Total area: 24 + 6 = 30 m²

Total area = 30 m²

For the triangle formula, the HEIGHT must be perpendicular (at right angles) to the base — not the slanted side! The height goes straight up, not along the slope.

Don't confuse area and perimeter! Area is measured in square units (cm²) and perimeter is measured in linear units (cm). "Area covers, perimeter borders."

Try These!

Find the area of a rectangle 9 cm by 6 cm.
Find the area of a triangle with base 10 cm and height 7 cm.
A square has sides of 8 m. What is its area?
A football pitch is 100 m long and 70 m wide. What is its area?
A parallelogram has base 12 cm and height 5 cm. Find its area.

1. 54 cm²
2. 35 cm²
3. 64 m²
4. 7,000 m²
5. 60 cm²

Volume AQE & GL ▶

Volume measures the amount of space inside a 3D shape. It is measured in cubic units (cm³, m³).

Volume Formula Cuboid: V = length x width x height
V = l x w x h

Cube: V = side x side x side = s³

Worked Example: Find the volume of a cuboid 5 cm x 3 cm x 4 cm

Use the formula: V = l x w x h

Substitute: V = 5 x 3 x 4

Calculate: V = 60

Volume = 60 cm³

Think of volume as "how many small cubes would fit inside the shape?" A volume of 60 cm³ means 60 little 1cm cubes could fit inside.

Try These!

Find the volume of a cube with sides of 4 cm.
Find the volume of a cuboid: length 10 cm, width 6 cm, height 3 cm.
A box is 20 cm x 15 cm x 10 cm. What is its volume?
A swimming pool is 25 m long, 10 m wide, and 2 m deep. What is its volume?

1. 64 cm³ (4 x 4 x 4)
2. 180 cm³
3. 3,000 cm³
4. 500 m³

Imperial & Metric AQE & GL ▶

Most measurements in the UK and Ireland use the metric system (metres, kilograms, litres), but some imperial units are still used in everyday life. You need to be able to convert between them.

Common Imperial Units Still Used

Length: inches, feet, yards, miles
Mass: ounces, pounds, stones
Capacity: pints, gallons

Key Conversions to Learn

Imperial	Metric (approx.)
1 inch	≈ 2.5 cm
1 foot (12 inches)	≈ 30 cm
1 mile	≈ 1.6 km
1 kg	≈ 2.2 pounds
1 litre	≈ 1.75 pints
1 gallon	≈ 4.5 litres

You won't need to memorise all of these — the question will usually give you the conversion. But knowing the rough sizes helps you check if your answer makes sense!

Worked Example: Convert 5 miles to kilometres (1 mile ≈ 1.6 km)

The question tells us 1 mile ≈ 1.6 km.

We want to convert 5 miles, so multiply: 5 x 1.6 = 8

5 miles ≈ 8 km

Worked Example: A recipe needs 2 pints of milk. How many litres is that? (1 litre ≈ 1.75 pints)

We need to go from pints to litres, so divide: 2 ÷ 1.75 = 1.14...

2 pints ≈ 1.14 litres (just over 1 litre)

If a question gives you a conversion like "1 mile = 1.6 km", write it down and use it as a multiplier or divider depending on which way you're converting.

Try These!

Convert 10 miles to kilometres. (1 mile ≈ 1.6 km)
A bag of flour weighs 3 pounds. Roughly how many kilograms is that? (1 kg ≈ 2.2 pounds)
A jug holds 4 pints. How many litres is that approximately? (1 litre ≈ 1.75 pints)
A shelf is 6 feet long. How many centimetres is that? (1 foot ≈ 30 cm)

1. 16 km (10 x 1.6)
2. ≈ 1.36 kg (3 ÷ 2.2 = 1.36...)
3. ≈ 2.29 litres (4 ÷ 1.75 = 2.28...)
4. 180 cm (6 x 30)

Bar Charts AQE & GL ▶

Bar charts use bars to show data. The taller the bar, the bigger the value. They're great for comparing categories.

Reading Bar Charts

Check the title — what is the chart about?
Read the axis labels — what do the axes represent?
Check the scale — what does each square or line represent?
Read values carefully — if a bar is between two lines, estimate the value.

Drawing Bar Charts

Give the chart a title
Label both axes
Choose an appropriate scale (gaps between numbers should be equal)
Draw bars with equal widths and equal gaps
Use a ruler!

When choosing a scale, look at the largest value in your data. Your scale should go a bit higher than that. If the largest value is 45, a scale going up to 50 in steps of 5 would work well.

Try These!

A bar chart shows: Maths = 25 pupils, English = 18, Science = 22, Art = 30. Which subject is most popular?
How many more pupils chose Art than English?
How many pupils were surveyed in total?

1. Art (30 pupils)
2. 12 more (30 - 18)
3. 95 (25 + 18 + 22 + 30)

Pie Charts AQE & GL ▶

A pie chart is a circle divided into slices (sectors). Each slice represents a proportion of the whole. The bigger the slice, the bigger the share.

Reading Pie Charts

The whole pie = 100% = 360°
A quarter of the pie = 25% = 90°
A half of the pie = 50% = 180°
If a slice shows 1/4, and the total is 80 people, then that slice = 80 ÷ 4 = 20 people

Worked Example: A pie chart shows how 120 pupils travel to school. The "walk" slice is 90°. How many walk?

90° out of 360° = 90/360 = 1/4

1/4 of 120 pupils = 120 ÷ 4 = 30

30 pupils walk to school

Try These!

A pie chart represents 200 people. A sector of 72° represents "bus." How many people take the bus?
What fraction of the pie chart is 120°?
If 25% of 60 pupils like football, how many is that?

1. 40 people (72/360 = 1/5, and 1/5 of 200 = 40)
2. 1/3 (120/360)
3. 15 pupils

Line Graphs AQE & GL ▶

Line graphs show how something changes over time. Points are plotted and connected with lines. They're great for spotting trends (going up, going down, staying the same).

Reading Line Graphs

Going up = increasing (e.g. temperature rising)
Going down = decreasing (e.g. water level falling)
Flat/horizontal = staying the same (no change)
Steeper line = faster change

Reading Between Points

You can estimate values between plotted points. This is called interpolation. Just read across to the line and then down to the axis.

Always check both axes carefully. Read the scale first — don't assume each square is worth 1. It might be worth 2, 5, 10, or even 100!

Try These!

A temperature line graph shows: 9am = 8°C, 12pm = 14°C, 3pm = 16°C, 6pm = 11°C. Between which two times did the temperature rise the most?
What was the temperature at its highest?
Between which times did the temperature fall?

1. 9am to 12pm (rose by 6°C)
2. 16°C (at 3pm)
3. Between 3pm and 6pm

Tables & Timetables AQE & GL ▶

You need to be able to read and interpret information from tables, including bus/train timetables and two-way tables.

Reading Timetables

Timetables usually use the 24-hour clock
Read across the row for one service/journey and down the column for a particular stop
To find journey time, subtract the departure time from the arrival time

Worked Example: Using a bus timetable

A bus leaves the Town Centre at 09:15 and arrives at the Hospital at 09:48. How long is the journey?

From 09:15 to 09:48: count on.

09:15 to 09:48 = 33 minutes

The journey takes 33 minutes.

Two-Way Tables

Two-way tables organise data into rows and columns. You can find totals by adding across rows or down columns.

Try These!

A bus leaves at 14:35 and arrives at 15:12. How long is the journey?
You need to arrive by 10:00. The journey takes 45 minutes. What is the latest bus you can catch?
In a two-way table, Boys who like Maths = 12, Girls who like Maths = 15. Total who like Maths = ?

1. 37 minutes
2. 09:15 at the latest
3. 27

Mean, Median, Mode & Range AQE & GL ▶

These are the main ways to describe and summarise a set of data. Learn what each one means!

The Four Key Measures Mean = add all values ÷ number of values (the "average")
Median = middle value when put in ORDER (the "middle")
Mode = most common value (the "most")
Range = highest - lowest (the "spread")

Worked Example: Test scores — 7, 4, 8, 7, 9, 5, 7

Mean: Add them all: 7 + 4 + 8 + 7 + 9 + 5 + 7 = 47. Divide by 7: 47 ÷ 7 = 6.7 (1dp)

Median: Put in order: 4, 5, 7, 7, 7, 8, 9. The middle value (4th of 7) is 7.

Mode: The most common value is 7 (it appears 3 times).

Range: 9 - 4 = 5

Mean = 6.7, Median = 7, Mode = 7, Range = 5

To find the median, ALWAYS put the numbers in order first! If there's an even number of values, the median is halfway between the two middle values. For example, for 3, 5, 7, 9: median = (5+7)÷2 = 6.

The range is NOT a type of average — it measures how spread out the data is. Also, don't forget to divide by the number of values when finding the mean, not by the total!

Try These!

Find the mean of: 12, 8, 15, 10, 5
Find the median of: 3, 9, 1, 7, 5
Find the mode of: 2, 5, 3, 5, 8, 5, 1
Find the range of: 14, 7, 23, 11, 19
The mean of 4 numbers is 8. Three of the numbers are 6, 9, and 7. What is the fourth number?

1. 10 (total = 50, 50 ÷ 5 = 10)
2. 5 (in order: 1, 3, 5, 7, 9)
3. 5 (appears 3 times)
4. 16 (23 - 7)
5. 10 (total must be 32, so 32 - 6 - 9 - 7 = 10)

Probability Basics AQE & GL ▶

Probability measures how likely something is to happen. It goes from 0 (impossible) to 1 (certain).

The Probability Scale

Probability Scale 0 ————— 0.5 ————— 1
Impossible Unlikely Even chance Likely Certain

Calculating Probability

Probability Formula Probability = number of favourable outcomes
-----------------------------------
total number of possible outcomes

Worked Example: A bag has 3 red, 5 blue, and 2 green sweets. What is the probability of picking a blue sweet?

Total sweets: 3 + 5 + 2 = 10

Blue sweets: 5

P(blue) = 5/10 = 1/2

Probability of picking blue = 1/2 (or 0.5 or 50%)

All probabilities of possible outcomes add up to 1 (or 100%). So if P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7.

Try These!

A dice has 6 sides. What is the probability of rolling a 4?
A spinner has 8 equal sections: 3 red, 2 blue, 1 green, 2 yellow. What is P(red)?
Is the probability of rolling a 7 on a normal dice: impossible, unlikely, or even?
A coin is flipped. What is the probability of getting heads?
The probability of picking a chocolate biscuit from a tin is 3/8. What is the probability of NOT picking a chocolate biscuit?

1. 1/6
2. 3/8
3. Impossible (there is no 7 on a normal dice)
4. 1/2
5. 5/8 (1 - 3/8 = 5/8)

Word Problem Strategy — RUCSAC AQE & GL ▶

Word problems are the heart of the transfer test. They test whether you can USE your maths skills in real situations. Follow this strategy every time:

RUCSAC Strategy R — READ the question carefully (read it twice!)
U — UNDERSTAND what you are being asked to find
C — CHOOSE the right operation(s) to use
S — SOLVE — show your working step by step
A — ANSWER — write a clear answer with units
C — CHECK — does your answer make sense? Estimate to check.

Spotting the Operation

Key Words	Operation
total, altogether, sum, combined, plus, increase	Addition (+)
difference, how many more, left over, decrease, fewer, minus, change	Subtraction (-)
times, product, groups of, each, every, double, triple	Multiplication (x)
share, split, divide, each, per, half, quarter	Division (÷)

Underline the key numbers and the question in the problem. Cross out any information you don't need. Many transfer test questions include extra information to distract you!

Try These!

A baker makes 248 rolls on Monday and 375 on Tuesday. She sells 590 rolls during the week. How many are left?
Pencils cost 35p each. How much would 12 pencils cost? Give your answer in pounds.
A school trip costs £8.50 per child. If 32 children go, what is the total cost?

1. 33 rolls (248 + 375 = 623, then 623 - 590 = 33)
2. £4.20 (35 x 12 = 420p = £4.20)
3. £272 (£8.50 x 32)

Multi-step Problems AQE & GL ▶

Multi-step problems need two or more calculations to solve. Take it step by step — don't try to do everything at once!

Worked Example: Shopping Problem

Question: Emma buys 3 notebooks at £2.45 each and 2 pens at £1.30 each. She pays with a £20 note. How much change does she get?

Cost of notebooks: 3 x £2.45 = £7.35

Cost of pens: 2 x £1.30 = £2.60

Total cost: £7.35 + £2.60 = £9.95

Change: £20.00 - £9.95 = £10.05

Emma gets £10.05 change.

Worked Example: Journey Problem

Question: A family drives 45 km to a theme park, then drives 12 km to a restaurant, then drives home by the most direct route (38 km). What is the total distance travelled?

Add all three legs of the journey: 45 + 12 + 38

45 + 12 = 57

57 + 38 = 95

Total distance = 95 km

Worked Example: Sharing Problem

Question: 144 football stickers are shared equally among 6 friends. Each friend then gives away 8 stickers. How many does each friend have left?

Share equally: 144 ÷ 6 = 24 stickers each

Each gives away 8: 24 - 8 = 16

Each friend has 16 stickers left.

Try These!

Jack has £15. He buys 2 magazines at £3.25 each and a drink for £1.80. How much does he have left?
A school orders 15 boxes of pencils. Each box contains 24 pencils. The pencils are shared equally among 8 classes. How many pencils does each class get?
A recipe for 4 people uses 300g of flour. How much flour is needed for 10 people?

1. £6.70 (2 x 3.25 = 6.50, + 1.80 = 8.30, 15 - 8.30 = 6.70)
2. 45 pencils (15 x 24 = 360, 360 ÷ 8 = 45)
3. 750g (300 ÷ 4 = 75g per person, 75 x 10 = 750g)

Money Problems AQE & GL ▶

Money questions are very common in the transfer test. You need to calculate with pounds and pence, find change, compare prices, and work out best value.

Best Value / Best Deal

Worked Example: Which is better value — 4 apples for £1.20 or 6 apples for £1.50?

Find the price per apple for each deal.

Deal 1: £1.20 ÷ 4 = 30p per apple

Deal 2: £1.50 ÷ 6 = 25p per apple

6 for £1.50 is better value (cheaper per apple).

Profit and Loss

Profit & Loss Profit = selling price - cost price (when you sell for MORE than you paid)
Loss = cost price - selling price (when you sell for LESS than you paid)

Always write money answers with two decimal places for pence. Write £3.50 not £3.5. And always include the £ sign!

Try These!

A toy costs £14.99. You pay with a £20 note. How much change do you get?
Which is better value: a 500ml bottle for £1.20 or a 750ml bottle for £1.65?
A boy buys a bike for £85 and sells it for £120. What is his profit?
Mum gives you £5 pocket money. You spend £1.45 on a comic, £0.85 on sweets, and £1.20 on a drink. How much have you left?

1. £5.01
2. 500ml: 1.20 ÷ 500 = 0.24p/ml. 750ml: 1.65 ÷ 750 = 0.22p/ml. The 750ml is better value.
3. £35 profit
4. £1.50 (1.45 + 0.85 + 1.20 = 3.50, then 5.00 - 3.50 = 1.50)

Ratio & Proportion AQE & GL ▶

A ratio compares two or more quantities. It tells you how much of one thing there is compared to another. Ratios use the ":" symbol.

Sharing in a Ratio

Worked Example: Share £40 in the ratio 3:5

Add the parts: 3 + 5 = 8 parts

Find the value of 1 part: £40 ÷ 8 = £5

First share: 3 x £5 = £15

Second share: 5 x £5 = £25

Check: £15 + £25 = £40 ✓

£40 shared in ratio 3:5 = £15 and £25

Scaling Recipes

Worked Example: A recipe for 6 people uses 240g flour, 3 eggs, and 150ml milk. How much for 9 people?

Scale factor: 9 ÷ 6 = 1.5 (or multiply by 3/2)

Flour: 240 x 1.5 = 360g

Eggs: 3 x 1.5 = 4.5 eggs (round to 5 in practice)

Milk: 150 x 1.5 = 225ml

For 9 people: 360g flour, 4-5 eggs, 225ml milk

When sharing in a ratio, always check your answer by adding the shares back together. They should equal the original total!

Try These!

Share 45 sweets in the ratio 2:3.
Share £100 in the ratio 1:3:6.
A recipe for 4 uses 200g of sugar. How much sugar for 6 people?
In a class, the ratio of boys to girls is 3:2. There are 30 children. How many boys?

1. 18 and 27 (5 parts, each worth 9)
2. £10, £30, £60 (10 parts, each worth £10)
3. 300g (200 ÷ 4 = 50 per person, 50 x 6 = 300)
4. 18 boys (5 parts, each = 6, boys = 3 x 6 = 18)

Patterns & Sequences AQE & GL ▶

A sequence is a list of numbers that follows a pattern or rule. You need to find the rule and use it to continue the pattern.

Finding the Rule

Look at the difference between consecutive terms. If it's the same each time, the rule is "add _" or "subtract _".

Worked Example: Find the next two terms: 3, 7, 11, 15, ...

Find the differences: 7-3=4, 11-7=4, 15-11=4

The rule is: add 4 each time.

Next terms: 15+4=19, 19+4=23

The next two terms are 19 and 23.

Function Machines

A function machine takes an input, applies a rule, and gives an output.

Function Machine INPUT → [ rule ] → OUTPUT

Example: Input 5 → [ x3 then +2 ] → Output 17
(5 x 3 = 15, 15 + 2 = 17)

If the differences between terms are not the same, look for other patterns: doubling, halving, multiply then add, square numbers, triangular numbers, etc.

Try These!

Find the next two terms: 5, 9, 13, 17, ...
Find the next two terms: 2, 6, 18, 54, ...
Find the rule: 1, 4, 9, 16, 25, ...
Function machine: Input → [x4 - 1] → Output. If input = 7, what is the output?
Function machine: Input → [?] → Output. If input = 3 and output = 11, and input = 5 and output = 17, what is the rule?

1. 21, 25 (add 4)
2. 162, 486 (multiply by 3)
3. Square numbers (1², 2², 3², 4², 5²...)
4. 27 (7 x 4 = 28, 28 - 1 = 27)
5. x3 + 2 (3x3+2=11, 5x3+2=17)

Logic & Reasoning GL Focus ▶

Some transfer test questions need logical thinking rather than just calculation. Here are some strategies to help.

Process of Elimination

In multiple-choice questions, you can often rule out wrong answers. Estimate the answer first, then eliminate options that are clearly too big or too small.

Working Backwards

If you know the answer but need to find the starting number, work backwards using opposite (inverse) operations.

Worked Example: I think of a number, multiply by 4, then add 6. The answer is 30. What was my number?

Start with the answer: 30

Undo "add 6" by subtracting: 30 - 6 = 24

Undo "multiply by 4" by dividing: 24 ÷ 4 = 6

Check: 6 x 4 + 6 = 24 + 6 = 30 ✓

The number was 6.

Trial & Improvement

Sometimes you need to try different values to find the answer. Start with a sensible guess, check if it's too high or too low, then adjust.

Don't be afraid to use rough work and jot down attempts. In the test, use the margins or spare paper to try ideas. Crossing out wrong attempts shows good mathematical thinking!

Try These!

I think of a number, divide by 3, then subtract 5. The answer is 7. What was my number?
Fill in the missing number: ? x 6 - 4 = 32
Two numbers add up to 15 and multiply to give 56. What are the two numbers?
In a class, everyone likes either football or tennis (or both). 18 like football, 12 like tennis, and 5 like both. How many children are in the class?

1. 36 (7 + 5 = 12, 12 x 3 = 36)
2. 6 (32 + 4 = 36, 36 ÷ 6 = 6)
3. 7 and 8
4. 25 children (18 + 12 - 5 = 25, because 5 are counted twice)

Simple Algebra & Equations AQE & GL ▶

Algebra uses letters to represent unknown numbers. Don't be scared of the letters — they're just standing in for numbers you need to find!

Using Letters for Unknowns

When we don't know a number, we use a letter (like n, x, or a) to represent it. Then we can write equations to find out what it is.

Worked Example: n + 5 = 12. What is n?

We need to find what number, added to 5, gives 12.

Subtract 5 from both sides: n = 12 - 5

n = 7

Balancing Equations

Think of the equals sign as a balance — both sides must always weigh the same. Whatever you do to one side, you must do to the other.

Worked Example: 3x - 4 = 14. Find x.

Add 4 to both sides: 3x = 14 + 4 = 18

Divide both sides by 3: x = 18 ÷ 3

x = 6

Function Machines

A function machine takes an input, applies operations in order, and gives an output.

Function Machine Input → [ x3 ] → [ +2 ] → Output

If input is 4: 4 x 3 = 12, then 12 + 2 = 14
If input is 7: 7 x 3 = 21, then 21 + 2 = 23

Writing Expressions

You can turn words into algebra:

"I think of a number, double it and add 3" = 2n + 3
"I think of a number, multiply by 5 then subtract 1" = 5n - 1
"I think of a number, add 4, then multiply by 2" = 2(n + 4)

Substitution

Substitution means replacing letters with numbers and working out the answer.

Worked Example: If a = 5 and b = 3, what is 2a + b?

Replace a with 5: 2 x 5 + b

Replace b with 3: 2 x 5 + 3

Calculate: 10 + 3 = 13

2a + b = 13

Think of the equals sign as a balance — both sides must always weigh the same!

In algebra, we write 2n not 2×n, and n² not n×n. Don't be confused by the shorthand!

Try These!

Solve: n + 8 = 15
Solve: 4x = 28
A function machine does: Input → [x2] → [+5] → Output. What is the output when the input is 6?
"I think of a number, multiply by 3 and subtract 7. The answer is 11." What was the number?
If p = 4 and q = 6, what is 3p + 2q?
Write an expression for: "I think of a number, halve it and add 10."

1. n = 7 (15 - 8)
2. x = 7 (28 ÷ 4)
3. 17 (6 x 2 = 12, 12 + 5 = 17)
4. 6 (3n - 7 = 11, so 3n = 18, so n = 6)
5. 24 (3 x 4 + 2 x 6 = 12 + 12 = 24)
6. n/2 + 10 (or n ÷ 2 + 10)

Times Tables Grid (12x12) AQE & GL ▶

You MUST know all of these by heart. Practise the ones you find tricky every day! The highlighted diagonal shows the square numbers.

x	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

The green diagonal shows the square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. These come up a LOT in the transfer test. Learn them!

Key Formulas AQE & GL ▶

Perimeter Rectangle: P = 2(l + w)
Square: P = 4s
Triangle: P = a + b + c (add all sides)
Regular polygon: P = n x s (number of sides x side length)

Area Rectangle: A = l x w
Square: A = s x s = s²
Triangle: A = ½ x b x h
Parallelogram: A = b x h

Volume Cuboid: V = l x w x h
Cube: V = s x s x s = s³

Averages Mean = total of all values ÷ number of values
Median = middle value (in order)
Mode = most common value
Range = highest - lowest

Angles Triangle angles = 180°
Straight line = 180°
Full turn = 360°
Quadrilateral angles = 360°

Probability P(event) = favourable outcomes ÷ total outcomes
P(not happening) = 1 - P(happening)

Fractions of Amounts a/b of n = n ÷ b x a
"Divide by the bottom, times by the top"

Fraction / Decimal / Percentage Chart AQE & GL ▶

Learn these conversions — they come up in nearly every test paper!

Fraction	Decimal	Percentage
1/100	0.01	1%
1/10	0.1	10%
1/5	0.2	20%
1/4	0.25	25%
3/10	0.3	30%
1/3	0.333...	33.3%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
2/3	0.666...	66.7%
7/10	0.7	70%
3/4	0.75	75%
4/5	0.8	80%
9/10	0.9	90%
1/1	1.0	100%

The most important ones to memorise: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.33 = 33.3%, 1/5 = 0.2 = 20%, and 1/10 = 0.1 = 10%.

Metric Conversions Chart AQE & GL ▶

Measurement	Conversion
Length
mm → cm	÷ 10
cm → m	÷ 100
m → km	÷ 1,000
cm → mm	x 10
m → cm	x 100
km → m	x 1,000
Mass
g → kg	÷ 1,000
kg → g	x 1,000
kg → tonnes	÷ 1,000
Capacity
ml → l	÷ 1,000
l → ml	x 1,000
Time
seconds → minutes	÷ 60
minutes → hours	÷ 60
hours → days	÷ 24

Remember: going from a SMALL unit to a BIG unit means DIVIDING. Going from a BIG unit to a SMALL unit means MULTIPLYING. "Small to Big = Divide. Big to Small = Multiply."

Top 10 Common Mistakes to Avoid AQE & GL ▶

These are the mistakes that catch out the most students in the transfer test. Read them carefully and make sure YOU don't make them!

Mistake 1: Forgetting BODMAS. Doing 3 + 4 x 2 = 14 instead of 3 + 4 x 2 = 11. Always do multiplication and division before addition and subtraction!

Mistake 2: Adding denominators in fractions. 1/3 + 1/4 = 2/7. The answer is 1/3 + 1/4 = 7/12. You must find a common denominator first!

Mistake 3: Confusing area and perimeter. Area = the surface inside (measured in cm²). Perimeter = the distance around the outside (measured in cm). They are NOT the same!

Mistake 4: Thinking 0.5 is less than 0.45 because "5 is less than 45." 0.5 < 0.45. Remember: 0.50 > 0.45. Add a zero to compare properly!

Mistake 5: Treating time like money. 1 hour 30 mins is 1.30 hours. It's actually 1.5 hours (because 30 mins = half an hour). There are 60 minutes in an hour, not 100!

Mistake 6: Giving the digit instead of its value. "What is the value of 6 in 16,482?" 6 6,000. The value depends on the position!

Mistake 7: Not reading the question properly. If it asks for the answer in metres and you give centimetres, you'll lose marks even if your calculation is correct. Always check what units are required!

Mistake 8: Forgetting to include all sides when finding perimeter of compound shapes. Draw the shape, label every single side, then add them ALL up. Don't leave any out!

Mistake 9: With negative numbers, thinking -8 > -3. On a number line, -8 is further LEFT, so it's SMALLER. -3 > -8. Think about temperatures!

Mistake 10: Not checking your answer. ALWAYS do a quick estimate or check. If a question about buying 3 items that cost about £5 each gives you £150, something is clearly wrong!

The transfer test is multiple choice (AQE) or has short answer sections (GL). Use the answer choices to help — if you're not sure, estimate and eliminate obviously wrong answers. Never leave a multiple-choice question blank!

Special Numbers Reference AQE & GL ▶

Square Numbers (1² to 12²) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Cube Numbers (1³ to 5³) 1, 8, 27, 64, 125

Prime Numbers to 50 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Remember: 1 is NOT prime. 2 is the only even prime.

64 appears in both the square numbers (8²) and cube numbers (4³). This is a favourite trick question!

Imperial / Metric Conversion Chart AQE & GL ▶

Approximate conversions between imperial and metric units. Questions will usually give you the conversion, but knowing these rough values helps you check your answers.

Type	Imperial	Metric (approx.)
Length	1 inch	≈ 2.5 cm
	1 foot (12 inches)	≈ 30 cm
	1 yard (3 feet)	≈ 91 cm
	1 mile	≈ 1.6 km
Mass	1 pound (lb)	≈ 0.45 kg
	1 kg	≈ 2.2 pounds
	1 stone (14 lbs)	≈ 6.35 kg
Capacity	1 pint	≈ 0.57 litres
	1 litre	≈ 1.75 pints
	1 gallon (8 pints)	≈ 4.5 litres

A quick way to roughly convert miles to km: multiply by 8 then divide by 5. For km to miles: multiply by 5 then divide by 8.

x	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

x	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

x	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144