Why do we multiply before adding when doing PEMDAS?

It makes a lot of sense to multiply before adding when we follow the order of operations or PEMDAS.

Say for example, you want to calculate 6 × 2 + 5

The order of operations tells us to  multiply 6 and 2 first. However, the following question remains:

Why can't we add 2 and 5 first and then multiply by 6?

In order to see why it makes sense to multiply 6 and 2 first and then add 5, we need to turn the expression 6 × 2 + 5 into a word problem.

Maria went to the supermarket and bought 6 bottles of water and 1 gallon of milk. If a bottle of water cost 2 dollars and a gallon of milk costs 5 dollars, how much money does she owe to the cashier.

The numerical expression for this word problem is 6 × 2 + 5

If you add 2 to 5 first and then multiply the result by 6, you will get 42 dollars. This does not make sense since that is way too much money for 1 gallon of milk and 6 bottles of water. What went wrong here?

After adding 2 and 5, we get 7. However, 7 includes the price of 1 gallon of water or 5 dollars. Therefore, if you multiply 7 by 6, you are in fact multiplying the price of 1 gallon of milk or 5 by 6. In other words, this means that you purchased 6 gallons of water or
6 × 5.

If indeed you did purchase 6 bottles of water and 6 gallons of milk, then the expression will look like this: 6 × (2 + 5)

Putting parentheses ensure that both 2 and 5 will be multiplied by 6. Without the parentheses
or if the problem is 6 × 2 + 5, first you get the cost of buying 6 bottles of water and then add that to the cost of buying 1 gallon of milk.

Why it makes sense to multiply before adding



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