United States
Hong KongFTCE Order of Operations
Skill 3 of Competency 1 is "Evaluate expressions involving order of operations", one of only three skills in that competency.
- Multiplication and division rank equally and run left to right, as do addition and subtraction.
- That equal ranking is where most errors come from.
On this page7 sections
It is a skill of its own in a competency of three. The rule is telling you something with that.
The order, with the part people misread
- Parentheses and other grouping symbols, working from the inside out
- Exponents and roots
- Multiplication and division, ranked equally, left to right
- Addition and subtraction, ranked equally, left to right
Steps three and four are two steps, not four. The mnemonic most people learned suggests multiplication comes before division and addition before subtraction, and neither is true.
So 20 divided by 5 times 2 is 8, not 2. Left to right: divide first because it appears first. This one construction produces more wrong answers than everything else in the competency put together.
Evaluate: 2 + 3 x 4 - 6 / 2
- 7
- 11
- 14
- 16
The calculator will not save you
An on-screen calculator is supplied for this subtest, per the Test Structure document. A calculator that follows the standard order will get the right answer from a correctly typed expression, and it cannot tell you whether the expression you typed matches the one on the screen.
The failure is parentheses. A fraction bar groups everything above it and everything below it, and typing that as a flat line drops the grouping. If the question shows a numerator of a plus b over c, you have to type the brackets yourself.
Watch how the calculator treats a negative base. Squaring negative 3 gives 9 if the negative is inside a bracket and negative 9 if it is not, because the exponent binds tighter than the sign. Both are legitimate readings of different expressions, so the brackets you type are the whole question.
Number properties
Competency 1 does not name these. Competency 3 does: skill 1 there asks you to "determine whether two algebraic expressions are equivalent by applying properties of operations or equality". So the properties are tested, in the algebra competency, as a tool for judging equivalence.
| Property | What it says | Where it applies |
|---|---|---|
| Commutative | Order does not matter | Addition and multiplication only |
| Associative | Grouping does not matter | Addition and multiplication only |
| Distributive | Multiplication spreads over addition | Expanding and factoring |
| Identity | Adding zero or multiplying by one changes nothing | Simplifying |
| Inverse | A number plus its opposite is zero | Solving equations |
The first two rows carry a restriction worth remembering: subtraction and division are neither commutative nor associative. Five minus 3 is not 3 minus 5, and that is exactly the kind of thing an equivalence item is built on.
If you failed the Mathematics subtest
Check your working on the questions you thought were easy. Order-of-operations slips do not feel like errors while you are making them, and on a subtest where about 22 of 35 correct passes, three careless arithmetic questions are a quarter of your margin.
The published data is encouraging. Mathematics passed 71% of first-attempt candidates in 2025 and 83% at best attempt, a 12-point gain. A retake is a fresh single-subtest registration at USD 32.50, and Rule 6A-4.0021(3)(d) requires at least thirty days since your previous administration of that subtest.
The honest caution: if you failed by a wide margin, order of operations is not the problem and algebra probably is. It is more than twice the size of this competency on our estimate of how the 35 questions divide, and that estimate is ours.
Follow the standard order of operations
Evaluate expressions in this order: grouping symbols, exponents, multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not automatically come before division, nor does addition automatically come before subtraction; operations at the same level are evaluated left to right. For 18 ÷ 3 × 2, compute 6 × 2 = 12, not 18 ÷ 6.
Grouping symbols include parentheses, brackets, and braces. Evaluate nested groups from the innermost outward. In 5 + 2(8 − 3)², first subtract to get 5, then square to get 25, multiply by 2 to get 50, and add 5 to get 55. Writing intermediate steps reduces errors when several operations appear.
Use properties to simplify accurately
The commutative property allows order to change for addition and multiplication: a + b = b + a and ab = ba. The associative property changes grouping: (a + b) + c = a + (b + c). Neither property applies to subtraction or division in the same way. For example, 10 − 3 is not equal to 3 − 10, and 24 ÷ 6 is not generally equal to 6 ÷ 24.
The distributive property is a(b + c) = ab + ac. For 4(3 + 5), either add inside first to get 4 × 8 = 32 or distribute to get 12 + 20 = 32. When a negative is distributed, it changes each term’s sign: −2(x − 4) = −2x + 8. A common error is multiplying only the first term inside the parentheses.
Respect signs, fractions, and exponents
A negative sign and an exponent require careful grouping. −3² means −(3²) = −9, while (−3)² = 9. Parentheses show whether the negative number itself is squared. For fractions, a bar acts as a grouping symbol: (6 + 2)/4 = 2, while 6 + 2/4 = 6.5. Use parentheses to make the intended numerator and denominator clear.
Number properties can provide a check. Multiplying by zero gives zero; multiplying by one preserves a value; adding zero preserves it; and a number plus its additive inverse gives zero. Estimate the size of a result before calculating exactly. If multiplying two positive numbers less than 1 gives a value larger than both, check the arithmetic.
- Evaluate grouping and exponents before lower-level operations.
- Handle multiplication/division and addition/subtraction left to right.
- Use commutative and associative properties only for addition or multiplication.
- Distribute to every term and track negative signs.
- Use parentheses to distinguish a negative base from a negative result.
Common questions
Does multiplication come before division on the FTCE?
No. They rank equally and you work left to right, and the same is true of addition and subtraction. That is why 20 divided by 5 times 2 is 8 rather than 2, and it produces more wrong answers than anything else in this competency.
Will the calculator handle order of operations?
It follows the standard order on a correctly typed expression, which is not the same as reading the question for you. Fraction bars group their numerator and denominator, and a flat line loses that, so the brackets you type are the real test.
Are number properties tested on the FTCE?
Yes, but in the algebra competency. Skill 1 of Competency 3 asks you to determine whether two algebraic expressions are equivalent by applying properties of operations or equality, so the properties appear as a tool rather than as a topic.
Is subtraction commutative?
No, and neither is division. Order matters for both, and grouping matters too, so neither is associative. Equivalence items are frequently built on exactly that restriction, offering a rearrangement that would be legal for addition but is not for subtraction.