Simplification is finding the exact value of an expression. Approximation is finding a value that is “close enough” to the exact answer — crucial for quick mental math. Both rely heavily on the order of operations and basic arithmetic rules.
The order in which you perform operations changes the result. Always follow this hierarchy:
| Step | Operation | Rule |
|---|---|---|
| B / P | Brackets / Parentheses ( ) { } [ ] |
Solve innermost first |
| O / E |
Orders / Exponents x²
√x
|
Powers and roots |
| D / M | Division and Multiplication | Left to right |
| A / S | Addition and Subtraction | Left to right |
Example: 12 − 4 ÷ 2
(12 − 4) ÷ 2 = 412 − (4 ÷ 2) = 12 − 2 = 10
| Operation | Rule | Example |
|---|---|---|
| Addition / Subtraction | Find a common denominator | 1/2 + 1/3 = (3+2)/6 = 5/6 |
| Multiplication | Multiply numerators together and denominators together | 2/3 × 4/5 = 8/15 |
| Division | Multiply by the reciprocal |
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
|
Memorize standard decimal values to speed up calculation:
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
1/2 |
0.5 | 1/6 |
≈ 0.166 |
1/3 |
≈ 0.33 | 1/7 |
≈ 0.1428 (remember 14-28) |
1/4 |
0.25 | 1/8 |
0.125 |
1/5 |
0.20 | 1/9 |
≈ 0.111 |
1/11 |
≈ 0.0909 | ||
am × an = am+n
am ÷ an = am−n
(am)n = amn
a−n = 1 / an
a0 = 1
We dislike square roots in the bottom of a fraction. To remove
√a from the denominator, multiply top and bottom by the
conjugate.
√x, multiply by √x/√x.
a + √b, multiply by
(a − √b) / (a − √b).
Memorizing these saves huge amounts of time.
| Identity | Expansion / Factorization |
|---|---|
(a + b)2
|
a2 + 2ab + b2
|
(a − b)2
|
a2 − 2ab + b2
|
a2 − b2
|
(a + b)(a − b) |
(a + b)3
|
a3 + b3 + 3ab(a+b)
|
a3 − b3
|
(a − b)(a2 + ab + b2)
|
a3 + b3
|
(a + b)(a2 − ab + b2)
|
In approximation questions (often labeled “What approximate value should come in place of (?)”), do not calculate exact decimals.
49.99 → 5012.01 → 123.94 × 15.02 → 4 × 15 = 60√50 is close to
√49 (= 7), so ≈ 7.1.
log10100 = 2. log1045 is between log 10 (=
1) and log 100 (= 2).
What is 12 − 4 ÷ 2?
What is 2/3 ÷ 4/5?
Which decimal does 1/7 approximate to?
Which law gives am ÷ an?
To rationalize a denominator of the form a + √b, what do you multiply
top and bottom by?
Which identity equals a3 − b3?
Using approximation, what is 3.94 × 15.02 closest to?
What is 5% of 450?
Simplification ⇒ exact value of expression
Approximation ⇒ close-enough value ; mental-math speed
a) B/P ⇒ brackets — innermost first
b) O/E ⇒ orders — powers & roots
c) D/M ⇒ divide/multiply — left to right
d) A/S ⇒ add/subtract — left to right
(eg 12 − 4 ÷ 2 ⇒ 12 − 2 = 10 ; (12−4)÷2 = 4 ×)
+/- ⇒ common denominator (eg 1/2 + 1/3 = 5/6)
× ⇒ tops × tops, bottoms × bottoms (eg 2/3 × 4/5 = 8/15)
÷ ⇒ multiply by reciprocal (eg 2/3 ÷ 4/5 = 5/6)
1/2=0.5 ; 1/3≈0.33 ; 1/4=0.25 ; 1/5=0.20 ; 1/6≈0.166
1/7≈0.1428 (14-28) ; 1/8=0.125 ; 1/9≈0.111 ; 1/11≈0.0909
am × an ⇒ am+n ; am ÷ an ⇒ am−n
(am)n ⇒ amn ; a−n ⇒ 1/an ; a0 ⇒ 1
√ in denominator × → kill it via conjugate
a) √x → × √x/√x
b) a + √b → × (a − √b)/(a − √b)
(a±b)² ⇒ a² ± 2ab + b²
a² − b² ⇒ (a+b)(a−b)
(a+b)³ ⇒ a³ + b³ + 3ab(a+b)
a³ ± b³ ⇒ (a±b)(a² &minus/+ ab + b²) ; upper/lower signs together
Rounding ⇒ 49.99→50 ; 12.01→12 ; 3.94×15.02 → 60
% shortcuts :- 10% → decimal left 1 ; 5% → half of 10% ; 1% → decimal left 2
Continued fractions → solve bottom-up
Digit sum (mod 9) ⇒ verify when options far apart
Squares till 30 → root estimates (eg √50 ≈ √49 → 7.1)
Logs :- log10100 = 2 ; log1045 → between 1 and 2
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