- Thread starter porknbeans
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To do the multiplication first, you have to switch them around, so you do 10 x 5 then /2 to get 25.$10 \div 2 \times 5$

Do the division first and you get 25. Do the multiplication first and you get 1.

Another way to do it is convert the divide into a multiply:

$10 \div 2 \times 5$

is the same as

$10 \times 0.5 \times 5$

An unbroken string of multiplies and divides all act on the leading number

for example:

$10 \div 2 \times 5 \div 7 \div 4 \times 49$

I can do these in any order I like, so I'm gonna start with:

$10 \times 49$ to give me 490. Then I'm going to:

$490 \div 7$ to give me 70. Next:

$70 \div 2$ to give me 35. Next:

$35 \times 5$ to give me 175. Next:

$175 \div 4$ to give me 175/4 or 43.75 if you prefer.

The same is true of addition/subtraction

for example, this is an unbroken string of additions and subtractions:

$1 + 1 - 1 - 1$

They all act on the leading number, so I can do these in reverse if I wish and still get 0 as the answer.

If we replace the the leading 10 in my example, with 1 x 10, we can then start with the $\div 7$

Let's start with that then do the rest of them in reverse order, the new order is:

$1 \div 7 \times 49 \div 4 \times 5 \div 2 \times 10$

The result is still 43.75

The problem is that the multiplication isn't really $2 \times 5$, the symbol to the left matters! There are 2 ways to proceed if you really do want to evaluate those two numbers first.$10 \div 2 \times 5$

Do the division first and you get 25. Do the multiplication first and you get 1.

Method 1:

we don't care about the 10 at the start of the problem, let's replace that with N

we now have:

$N \div 2 \times 5$

N is the same a N x 1, so we can say:

$N \times 1\div 2 \times 5$

So the bit we're interested in is

$1\div 2 \times 5$

This is why taking the reciprocal and changing the divide to a multiply works....

Method 2:

Using the same starting point:

$N \div 2 \times 5$

We can change the order to be:

$N \times 5 \div 2$

So the bit we're interested in is

$5 \div 2$

Note that these methods do actually result in doing a divide first.....

Having gone through this, I can see why people are taught to do the 'D' before the 'M', if you can't rearrange correctly you'll get it wrong.

That's just a convention. Possible interpretations of 10 ÷ 2 × 5 are (10 ÷ 2) × 5 and 10 ÷ (2 × 5). There is a convention that the first of those possibilities is used, but that convention is separate from BODMAS.The problem is that the multiplication isn't really $2 \times 5$, the symbol to the left matters!

I was thinking more along the lines of:

$-x^2$

But I think this works too... (because it's really $-1 \times x^2$)

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