How do parentheses alter the outcome for expressions like (143 + 305) * 70 versus 143 + 305 * 70?
Executive summary
Parentheses change the result by forcing the enclosed operations to be performed first; in (143 + 305) 70 you add 143 and 305 first, then multiply the sum by 70, while without parentheses — 143 + 305 70 — multiplication binds before addition and you multiply 305 by 70 first and then add 143 (order-of-operations convention: parentheses, exponents, multiply/divide left-to-right, add/subtract left-to-right) [1] [2].
1. Why parentheses exist: a packaging rule that controls order
Parentheses are a deliberate notation that tell the reader “treat what’s inside as one packaged quantity and evaluate it before anything outside”; textbooks and teaching resources consistently list parentheses as the first step in PEMDAS/BODMAS and instruct students to simplify inside them first, even when nested parentheses appear [3] [4].
2. Two different computations, two different outcomes — a concrete comparison
If you compute (143 + 305) 70 you first do the addition 143 + 305 = 448 and then multiply 448 70. If you compute 143 + 305 70 you follow the rule that multiplication comes before addition, so you do 305 70 = 21,350 and then add 143 to get 21,493. The different placement of parentheses changes the grouping and therefore the numeric outcome [1] [2].
3. The convention behind the rule: widely taught, but ultimately a human agreement
The “order of operations” convention—parentheses, exponents, multiplication/division, addition/subtraction—is not a law of nature but a widely adopted agreement to avoid ambiguity in expressions; educational resources and encyclopedic entries explain and apply that convention to guide consistent interpretation [5] [6].
4. Ambiguities that look similar but are tricky in practice
Some expressions remain ambiguous unless parentheses are added; classic controversies (like the 48/2(9+3) debate) show that when notation is compact or implies grouping, different readers can parse the same string differently. Mathematicians and educators therefore advise inserting parentheses to make intent explicit — e.g., write (48/2)(9+3) or 48/(2(9+3)) to show which meaning you intend [7].
5. How calculators and programming languages treat parentheses and precedence
Most calculators and programming languages implement the same basic precedence rules—evaluate parenthesized subexpressions first, then apply exponentiation, then multiplication/division (left-to-right), then addition/subtraction (left-to-right). Because some notation (implied multiplication, juxtaposition next to fractions) can be interpreted differently across systems, explicit parentheses remove any implementation-specific guessing [5] [8].
6. Teaching and learning: why instructors emphasize parentheses early
Elementary and middle‑school materials present parentheses as the first rule in mnemonic devices like PEMDAS or BODMAS and use metaphors such as “packages” to help learners understand that everything inside must be simplified before proceeding; worksheets and guides repeatedly stress “parentheses first” to prevent mistakes when operations mix [9] [10].
7. Practical advice: write to avoid misreading and show intent
Because conventions can be misremembered or applied inconsistently in edge cases (noted in sources discussing ambiguous forms), the safest practice is to use parentheses whenever the intended order matters. When in doubt — or when sharing calculations with others or machines — add parentheses to make the grouping explicit [7] [1].
Limitations and source notes: I draw solely on classroom guides, educational websites and a university math discussion that explain the order-of-operations convention and known ambiguities; these sources teach that parentheses force evaluation first and recommend explicit grouping for clarity [3] [7]. Available sources do not mention the exact arithmetic results for the 143/305 example in any single source, but they state the rules that determine those outcomes [2] [1].