Unit 2 · Topic 05 · Operations
Ms. Rao wrote 2 + 3 × 4 on the board and asked for the answer.
Half the class said 20. The other half said 14.
Both halves had done their arithmetic perfectly.
Ms. Rao wrote a single line on the board: 2 + 3 × 4.
Anaya worked left to right, the way you read. Two plus three is five, five times four is twenty. She said twenty.
Kabir multiplied first. Three fours are twelve, plus two is fourteen. He said fourteen.
"Nobody has made an arithmetic mistake," said Ms. Rao. "Anaya's addition is right and Kabir's multiplication is right. So why are you two apart?"
"We did them in a different order," said Kabir.
"And if the world let everyone choose their own order, this line would mean two different things depending on who read it. A recipe would produce two different cakes. So mathematicians agreed on one order, and everybody follows it."
She wrote BODMAS down the side of the board. Brackets. Of. Division and Multiplication. Addition and Subtraction.
"Fourteen is correct," she said. "Multiplication is settled before addition."
Anaya was not satisfied. "But what if you WANT the addition first?"
"Then you say so — with brackets. That is exactly what brackets are for." She wrote (2 + 3) × 4 = 20. "Your answer was not wrong. It was the answer to a different question, and now the line asks that question properly."
Then she wrote 8 − 4 + 2 and asked for the answer. Kabir, now confident that BODMAS meant addition came before subtraction, said 8 − 6 = 2.
"Two is wrong," said Ms. Rao. "Addition and subtraction sit at the SAME level. So do division and multiplication. When two operations are on the same level, you go left to right — and no further rule applies."
8 − 4 = 4, then 4 + 2 = 6.
"So BODMAS is four levels, not six," said Anaya slowly. "DM is one level and AS is one level."
"That is the part almost everybody gets wrong," said Ms. Rao. "They read six letters and assume six steps."
When a line mixes operations, the order they are carried out in changes the answer. BODMAS is the agreed order so that one line means one thing to everybody.
B is brackets, worked from the innermost outwards. O is 'of', as in half OF ten, and powers. Then DM, then AS.
The crucial point is that BODMAS has FOUR levels, not six. Division and Multiplication share a level, and Addition and Subtraction share a level. Operations on the same level are done LEFT TO RIGHT, in the order they appear.
So 8 − 4 + 2 is 6, not 2, because the subtraction comes first simply by being further left. And 48 ÷ 8 ÷ 2 is 3, not 12, for the same reason. Brackets exist so you can override all of this and say exactly what you mean.
Write 6 2 3 in a row and put + and × between them in both possible ways. Work out both answers.
Then add brackets to one of them so it gives the other answer. You have just proved why the rule has to exist.
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