What is the value of (143 + 305) * 70 compared to 143 + (305 * 70)?
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Executive summary
The two expressions are not equal because parentheses change the order in which operations are performed; (143 + 305) 70 evaluates to 44 8 times 70 = 31,360, while 143 + (305 70) evaluates to 143 plus 21,350 = 21,493, so the first is larger by 9,867. The difference follows directly from the standard order of operations: parentheses first, then multiplication before the remaining addition (PEMDAS/BODMAS) [1] [2].
1. Why the question matters: parentheses as mathematical switches
Parentheses in arithmetic act as explicit instructions to perform the enclosed operations first, changing how the rest of an expression is evaluated, which is why (143 + 305) 70 is fundamentally different from 143 + (305 70); textbooks and pedagogy stress this because failing to respect the order of operations yields incorrect results [1] [3].
2. Apply the rule: step-by-step evaluation and authoritative sources
Following the standard rule—parentheses first, then multiplication and division, then addition and subtraction—one computes (143 + 305) 70 by first adding 143 and 305 to get 448, then multiplying by 70 to get 31,360; conversely, 143 + (305 70) does the multiplication inside its parentheses first (305 * 70 = 21,350) and then adds 143, yielding 21,493, which demonstrates a clear numeric divergence driven solely by grouping [1] [4].
3. The arithmetic and the gap: how big and why
The numerical gap between the two results is 31,360 − 21,493 = 9,867, entirely attributable to whether the 70 multiplies the sum (143+305) or only 305; when the multiplier applies to the whole sum, it scales both terms, producing a larger product than when it applies only to the single term 305—this is the distributive effect of multiplication over addition made concrete (the calculation itself is standard arithmetic and not directly quoted from the provided sources) [2].
4. Common confusion and real-world analogies
Learners often misread expressions like these because written layout can obscure which operations the multiplier applies to; pedagogical resources use concrete analogies—if 70 is a price per unit and the parenthesis counts units from two groups combined, multiplying the combined total yields a different bill than multiplying only one group's units and then adding the other group's fixed amount—educational sites and worksheets frame these examples to prevent such mistakes [5] [6].
5. Verification and tools: how to check independently
Order-of-operations calculators and solvers (such as the ones that show PEMDAS steps) will reproduce these steps and results step-by-step, reinforcing that parentheses alter evaluation and that the two expressions produce distinct numeric outcomes; online solvers and PEMDAS explainers provide both the procedural rule and tools to confirm the arithmetic [7] [8] [4].
6. Alternative viewpoints and limitations in sourcing
There is no meaningful alternative mathematical interpretation under conventional arithmetic rules—any disagreement would imply a redefinition of notation rather than a disagreement about calculation—however, the provided sources are explanatory and tool-based materials about order of operations rather than repositories of these exact numeric examples, so while the rules and verification tools are cited, the stepwise numeric arithmetic above is shown as direct calculation rather than quoted from a specific example in the supplied snippets [1] [3].