Operations With Numbers And Decimals

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes, diagrams for Operations With Numbers And Decimals (Maths [CIE], Core) — use them in your lesson, or run the topic as a live class game.

Notes

Mathematical Symbols

**+** : addition, plus, sum, total
**−** : subtraction, minus, difference, take away
**×** : multiplication, product, times
**÷** : division, quotient, divide, share
**=** : equal to; **≠** : not equal to; **≈** : approximately equal to
**>** : greater than; **<** : less than; **≥** : greater than or equal to; **≤** : less than or equal to
**( )** : brackets for grouping; **^** : powers (indices); **√** : positive square root; **∛** : cube root

Order of Operations (BIDMAS/BODMAS)

Operations must be done in order: **B**rackets, **I**ndices (or **O**rders), **D**ivision and **M**ultiplication (left to right), **A**ddition and **S**ubtraction (left to right)
Fractions have **invisible brackets** around numerator and denominator, e.g. (2+5)/(7-2)
Roots have **invisible brackets** under the root, e.g. $\sqrt{9+16}$
When using a calculator, type the calculation exactly as written; older calculators may need extra brackets
Always check your answer seems sensible (estimate first)

Addition & Subtraction

**Column method**: line up digits by place value; add/subtract each column right to left
Addition: carry tens to the next column if $sum \ge 10$
Subtraction: if top digit is smaller, borrow 10 from the next column to the left
Alternative subtraction: **counting up** from the smaller number to the larger
Estimate by rounding to check your answer $(e.g., 4000+1000=5000$ for 3985+1273)

Multiplication & Division

**Column method**: multiply each digit of bottom number by each digit of top number, using zeros as placeholders for non-units columns
**Lattice method**: draw grid with diagonals, multiply digit pairs, sum along diagonals
**Grid method**: split numbers by place value, multiply each part, sum results
**Repeated addition (chunking)**: build up using simple multiples $(\times 1, \times 2, \times 4, \times 8, etc.)$
**Short division (bus stop method)**: divide digit by digit from left, carry remainders
**Factorising/cancelling**: treat division as a fraction and simplify before dividing
Dividing by powers of 10 shifts digits to the right $(e.g., 380\div 10=38.0)$

Operations with Decimals

**Add/subtract decimals**: line up decimal points and use column method; write zeros as placeholders
**Multiply decimals**: convert to integers by multiplying by powers of 10, multiply, then divide by the same powers; or ignore decimal points and use estimation to place the decimal
**Divide decimals**: write as a fraction, multiply numerator and denominator by powers of 10 to make integers, divide, then undo the changes; or ignore decimals and use estimation
Always estimate the answer first to check reasonableness $(e.g., 1.57\times 0.78$ ≈ $2\times 1=2$ , so answer is 1.2246)

BIDMAS/BODMAS Order of Operations

Column Method for Addition

Short Division (Bus Stop Method)

Multiplying Decimals by Converting to Integers

Practice questions

Free preview — 8 of 40 questions. Sign up to see them all.

1.What is the value of $7 \times (2 \times 4 + 8) \times (2 + 6)$ ?
Easy
- A448
- B336
- C560
- D224
2.A theatre has 1850 seats. If 967 seats have been sold, how many seats are still available?
Easy
- A883
- B893
- C873
- D863
3.Calculate $3 + 2 \times (8 - 6)$ .
Easy
- A7
- B10
- C5
- D13
4.Insert one pair of brackets to make the statement correct: $7 \times 5 - 2 + 3 = 42$ .
Medium
- A $7 \times (5 - 2) + 3$
- B $(7 \times 5) - 2 + 3$
- C $7 \times 5 - (2 + 3)$
- D $7 \times (5 - 2 + 3)$
5.Work out $(9 + 2^{2}) - 4 \times 2 +$ \√{144}.
Medium
- A13
- B21
- C17
- D9
6.Insert one pair of brackets to make the statement correct: $3 + 2 \times 12 - 4 = 19$ .
Medium
- A $3 + 2 \times (12 - 4)$
- B $(3 + 2) \times 12 - 4$
- C $3 + (2 \times 12) - 4$
- D $(3 + 2 \times 12) - 4$
7.Insert one pair of brackets to make the statement correct: $16 \div 8 + 4 \times 2 = 1$ .
Hard
- A $16 \div (8 + 4) \times 2$
- B $(16 \div 8) + 4 \times 2$
- C $16 \div 8 + (4 \times 2)$
- D $16 \div (8 + 4 \times 2)$
8.Calculate $18 \div (10 - 4) - 11 \times 4 + 9 \times (6 + 1)$ .
Hard
- A28
- B34
- C22
- D16

Summary

## Operations with Numbers & Decimals ### Order of Operations (BIDMAS/BODMAS) - **B**rackets first - **I**ndices (powers, roots) - **D**ivision and **M**ultiplication (left to right) - **A**ddition and **S**ubtraction (left to right) - Fractions and roots have invisible brackets: e.g. (2+5)/(7-2) means (2+5)÷(7-2); √(9+16) means √(9+16). ### Addition & Subtraction - Use column method: line up digits by place value (decimal points aligned). - Add right to left, carry tens. - Subtract right to left, borrow from next column if needed. - Key words: sum (add), difference (subtract). ### Multiplication & Division - Methods: column, lattice, grid, repeated addition. - For decimals: multiply by powers of 10 to make integers, then undo. - Division: short division (bus stop) for single-digit divisors; factorising/cancelling for larger. - Dividing by 10,100,... shifts digits right. ### Operations with Decimals - Add/subtract: line up decimal points, use zeros as placeholders. - Multiply: ignore decimals, multiply as integers, then place decimal using estimation. - Divide: write as fraction, multiply numerator and denominator by powers of 10 to make integers, divide, then adjust. ### Key Tips - Estimate answers to check reasonableness. - Use BIDMAS carefully; brackets change order. - For calculator, type exactly as written; use extra brackets if needed.

Key terms

BIDMAS/BODMAS: Order of operations: Brackets, Indices (Orders), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Column method: A written method for addition, subtraction, or multiplication where digits are aligned by place value and operations are performed column by column from right to left.
Short division (bus stop method): A method for division where the divisor is written to the left of the dividend, and digits are processed from left to right, carrying remainders.
Place value: The value of a digit based on its position in a number (e.g., units, tens, hundreds, tenths, hundredths).
Invisible brackets: Brackets implied by fraction bars or root symbols; e.g., the numerator and denominator of a fraction are treated as if in brackets.

Slides

Mathematical Symbols

The four basic operations $are: + (addition), - (subtraction), \times (multiplication), \div (division)$ .
Other important symbols include $= (equals), \ne (not$ equal to), ≈ (approximately equal to), and ≡ (identical to).
Inequality signs: $> (greater than), < (less than), \ge (greater$ than or equal $to), \le (less$ than or equal to).
Brackets ( ) are used to group parts of a calculation.

Which symbol means 'approximately equal to'?

Order of Operations (BIDMAS/BODMAS)

BIDMAS/BODMAS tells you the order to do operations: Brackets, Indices (Orders), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Always calculate inside brackets first.
Then powers or roots, then multiplication and division, then addition and subtraction.
For example, (5 – $3) + 2 \times 7^{2} = 2 + 2 \times 49 = 2 + 98 = 100$ .

What is the correct order of operations?

Addition then Multiplication then Brackets
Brackets then Indices then Division/Multiplication then Addition/Subtraction
Division then Brackets then Addition
Multiplication then Subtraction then Brackets

Addition and Subtraction of Large Numbers

Use the column method: line up digits by place value, add/subtract each column from right to left.
In addition, if a column sum is 10 or more, carry the tens digit to the next column.
In subtraction, if the top digit is smaller, borrow 10 from the next column to the left.
Always estimate your answer first to check it is reasonable.

When adding 3985 and 1273, the sum of the tens column $is ____$ . We write 5 and carry $____ to$ the hundreds column.

Multiplication Methods

Common methods: column method, lattice method, grid method, repeated addition.
In the column method, multiply each digit of the bottom number by each digit of the top number, using zeros as placeholders for digits beyond the ones.
Always estimate your answer to check it is reasonable.

Which of these are true about multiplication?

The product of 725 and 19 is 13,775.
In the column method, you always start multiplying from the leftmost digit.
Estimation can help check if your answer is reasonable.

Division Methods

Short division (bus stop method) is efficient for dividing by a single digit.
For larger numbers, you can use factorising or cancelling (treat division as a fraction and simplify).
Dividing by powers of 10 shifts digits to the right in place value columns.

What is $568 \div 8$ ?

Operations with Decimals

For addition/subtraction, line up decimal points and use column method with place value holders (zeros).
For multiplication, convert decimals to integers by multiplying by powers of 10, then undo at the end.
For division, write as a fraction, multiply numerator and denominator to make integers, then divide and adjust.

Calculate $1.09 + 7.85$

8.94
8.84
9.94
8.04

Brackets and Order of Operations

Brackets change the order of operations: calculate inside brackets first.
Inserting brackets can make a statement true, $e.g., 7 \times (5$ – $2) + 3 = 24$ , but with brackets $(7 \times 5)$ – $2 + 3 = 36$ .
In exam questions, you may need to insert one pair of brackets to make a calculation correct.

Insert one pair of brackets to make $7 \times 5$ – $2 + 3 = 42$ correct: $7 \times (5$ – $2) + 3 = ____$

Working with Fractions and Roots in BIDMAS

Fractions imply division: the numerator and denominator have 'invisible brackets', e.g., (2+5)/(7-2).
Roots are powers: the expression under the root has an $'invisible bracket', e.g., \sqrt{9+16}$ .
Use these rules when entering calculations into a calculator.

What is the value of $(9+22) - 4 \times 2 + \sqrt{144}$ ?

Calculate $3 + 2 \times (8 - 6)$

Decimal Multiplication and Division

To multiply decimals, ignore the decimal point, multiply as integers, then estimate to place the decimal point correctly.
To divide decimals, write as a fraction, multiply numerator and denominator by powers of 10 to make integers, then divide.
Always check your answer with an estimate.

Without a calculator, $1.57 \times 0.78 = ____$

Common Mistakes and Tips

Always use BIDMAS/BODMAS to avoid order errors.
When using a calculator, enter the calculation exactly as written, using brackets where needed.
Estimate your answer before calculating to catch big mistakes.

Which of these are true?

In BIDMAS, multiplication always comes before division.
Fractions have invisible brackets around numerator and denominator.
When subtracting, you should always subtract the smaller number from the larger.

Flashcards

What does BIDMAS stand for?

Brackets, Indices (or powers/orders), Division and Multiplication (left to right), Addition and Subtraction (left to right).

What is the correct order of operations?

Brackets first, then powers/indices, then multiplication and division (left to right), then addition and subtraction (left to right).

Calculate: $48 \div 3 - 5 \times 2$

$48 \div 3 = 16, 5 \times 2 = 10, 16 - 10 = 6$ .

Calculate: $3 + 2 \times (8 - 6)$

Brackets first: $8 - 6 = 2$ . Then multiplication: $2 \times 2 = 4$ . Then addition: $3 + 4 = 7$ .

Calculate: $7 \times (2 \times 4) + 8 \times (2 + 6)$

First brackets: $2 \times 4 = 8, 2 + 6 = 8$ . Then multiplication: $7 \times 8 = 56, 8 \times 8 = 64$ . Then addition: $56 + 64 = 120$ .

Insert one pair of brackets to make $7 \times 5$ – $2 + 3 = 42$ correct.

$7 \times (5$ – $2) + 3 = 7 \times 3 + 3 = 21 + 3 = 24 (not 42)$ . Correct: $7 \times (5$ – $2 + 3) = 7 \times 6 = 42$ .

Insert one pair of brackets to make $8 + 6$ – $2 \times 5 = 28$ correct.

$(8 + 6$ – $2) \times 5 = 12 \times 5 = 60 (not 28)$ . Correct: $(8 + 6)$ – $2 \times 5 = 14$ – $10 = 4 (no)$ . Actually: $8 + (6$ – $2) \times 5 = 8 + 4 \times 5 = 8 + 20 = 28$ .

Insert one pair of brackets to make $8 + 6$ – $2 \times 5 = 60$ correct.

$(8 + 6$ – $2) \times 5 = 12 \times 5 = 60$ .

Insert one pair of brackets to make $4 \times 6 - 2 + 1 = 17$ correct.

$4 \times (6 - 2) + 1 = 4 \times 4 + 1 = 16 + 1 = 17$ .

Insert one pair of brackets to make $3 + 2 \times 12 - 4 = 19$ correct.

$3 + 2 \times (12 - 4) = 3 + 2 \times 8 = 3 + 16 = 19$ .

Insert one pair of brackets to make $9 + 4 - 3 \times 7 = 70$ correct.

$(9 + 4 - 3) \times 7 = 10 \times 7 = 70$ .

Insert one pair of brackets to make $16 \div 8 + 4 \times 2 = 1$ correct.

$16 \div (8 + 4 \times 2) = 16 \div (8 + 8) = 16 \div 16 = 1$ .

Insert one pair of brackets to make $16 \div 8 + 4 \times 2 = 12$ correct.

$(16 \div 8) + 4 \times 2 = 2 + 8 = 10 (not 12)$ . Correct: $16 \div 8 + (4 \times 2) = 2 + 8 = 10 (no)$ . Actually: $(16 \div 8 + 4) \times 2 = (2 + 4) \times 2 = 6 \times 2 = 12$ .

Insert one pair of brackets to make $24 \div 6 + 2 \times 3 = 9$ correct.

$(24 \div 6) + (2 \times 3) = 4 + 6 = 10 (not 9)$ . Correct: $24 \div (6 + 2) \times 3 = 24 \div 8 \times 3 = 3 \times 3 = 9$ .

Insert one pair of brackets to make $24 \div 6 + 2 \times 3 = 2$ correct.

$24 \div (6 + 2 \times 3) = 24 \div (6 + 6) = 24 \div 12 = 2$ .

Insert one pair of brackets to make $5 + 3 \times 10 - 1 = 32$ correct.

$(5 + 3) \times (10 - 1) = 8 \times 9 = 72 (not 32)$ . Correct: $(5 + 3) \times 10 - 1 = 8 \times 10 - 1 = 80 - 1 = 79 (no)$ . Actually: $5 + 3 \times (10 - 1) = 5 + 3 \times 9 = 5 + 27 = 32$ .

Insert one pair of brackets to make $3 \times 2 - 4 - 7 = 9$ correct.

$3 \times (2 - 4) - 7 = 3 \times (-2) - 7 = -6 - 7 = -13 (no)$ . Correct: $3 \times (2 - 4 - 7) = 3 \times (-9) = -27 (no)$ . Actually: $(3 \times 2) - (4 - 7) = 6 - (-3) = 9$ .

Calculate: $1.09 + 7.85$

$1.09 + 7.85 = 8.94$ .

Calculate: 6.21 − 4.37, give answer correct to 1 decimal place.

$6.21 - 4.37 = 1.84$ , correct to 1 decimal place is 1.8.

Subtract 123 from 1 million.

$1,000,000 - 123 = 999,877$ .

Subtract 9 from 2.

$2 - 9 = -7$ .

Calculate: $18 \div (10 - 4) - 11 \times 4 + 9 \times (6 + 1)$

Brackets: $10-4=6, 6+1=7$ . Then division: $18\div 6=3$ . Multiplication: $11\times 4=44, 9\times 7=63$ . Then subtraction and addition left to right: $3-44=-41, -41+63=22$ .

A theatre has 1850 seats. 967 are sold. How many seats are still available?

$1850 - 967 = 883$ seats.

Work out: $6 \times 3 + 24 \div 4$

Multiplication and division first: $6\times 3=18, 24\div 4=6$ . Then addition: $18+6=24$ .

Work out: $4 + (-11)$

$4 + (-11) = -7$ .

Unlock all 40 questions, slides, flashcards & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

🗂️ Coming soon