2/8 + 1/2 with EXPLANATION
Answers
Answer:Answer To 8÷2(2 + 2) = ?
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
The correct answer is 16 according to the modern interpretation of the order of operations.
The order of operations
The expression can be simplified by the order of operations, often remembered by the acronyms PEMDAS/BODMAS.
First evaluate Parentheses/Brackets, then evaluate Exponents/Orders, then evaluate Multiplication-Division, and finally evaluate Addition-Subtraction.
Everyone is in agreement about the first step: simplify the addition inside the parentheses.
8÷2(2 + 2)
= 8÷2(4)
This is where the debate starts.
The answer is 16
If you type 8÷2(4) into a calculator, the input has to be parsed and then computed. Most calculators will convert the parentheses into an implied multiplication, so we get
8÷2(4)
= 8÷2×4
According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 8 and divide it by 2, and then multiply by 2.
8÷2×4
= 4×4
= 16
This gets to the correct answer of 16.
This is without argument the correct answer of how to evaluate this expression according to current usage.
Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago. Some people may have learned this other interpretation more recently too, but this is not the way calculators would evaluate the expression today.
The other result of 1
Suppose it was 1917 and you saw 8÷2(4) in a textbook. What would you think the author was trying to write?
Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below).
Under that interpretation:
8÷2(4)
= 8÷(2(4))
(Important: this is outdated usage!)
From this stage, the rest of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 2 by 4 to get 8. And then we divide 8 by 8.
8÷(2(4))
= 8÷8
= 1
This gives the result of 1. This is not the correct answer that calculators will evaluate; rather it is what someone might have interpreted the expression according to older usage.
Binary expression trees
Since some people think the answer is 16, and others think it is 1, many people argue this problem is ambiguous: it is a poorly written expression with no single correct answer.
But here’s my counter-point: a calculator is not going to say “it’s an ambiguous expression.” Just as courts rule about ambiguous legal sentences, calculators evaluate seemingly ambiguous numerical expressions. So if we take the expression as written, what would a calculator evaluate it as?
There are two possible binary expression trees.
Answer:
2/8+ 1/2 = 2/8 + 1/2 × 4 = 4/8 because same denominators then 2/8 + 4/8 = 6/8