Question

The difference of a number times `8` and `9` is `4`. What is the number?

Answer 1

The number is `13/8`

Explanation:

'Difference' means you have to subtract two numbers.

Look for the word 'and' to find out which two numbers are being subtracted.

We have: The difference of (a number times 8) AND (9)

Let the number be `x`

The number times `8` is `8x`

So now:

The difference of (a number times 8) AND (9) `rarr 8x-9`

Now you can form an equation because we know that the difference is equal to `4`

`8x-9=4" "larr` solve for `x`

`8x = 4+9" "` ( add 9 to both sides)

`8x = 13" "`(divide both sides by `8`

`x = 13/8`

Check:

`13/cancel8 xx cancel8 -9`

`=13-9`

`=4`

Answer 2

Perhaps there is some ambiguity here...

Explanation:

This question seems to admit several interpretations.

Here are a few possibilities based on common English usage rather than pure mathematical considerations...

If `n` stands for the number, then we could interpret as follows:

First possible interpretation

"The difference between ( `8` times a number ) and `9` is `4`."

In symbols: `8n-9 = 4`

Then adding `9` to both sides of the equation, we get:

`8n = 13`

Then dividing both sides by `8` we get:

`n = 13/8`

This is a good "mathematical" interpretation.

Second possible interpretation

"The difference between ( `8` times a number ) and ( `9` times the same number ) is `4`."

In symbols: `8n - 9n = 4`

The left hand side can be simplified:

`8n - 9n = (8-9)n = -1 * n = -n`

So we have:

`-n = 4`

and hence:

`n = -4`

This interpretation assumes that "difference between `a` and `b`" means the result of subtracting `b` from `a`.

Third possible interpretation

"The size of the difference between ( `8` times a number ) and ( `9` times the same number ) is `4`."

This interpretation is based on a common English usage that differences are measured as positive quantities - the smaller quantity is always subtracted from the larger.

In symbols: `abs(8n - 9n) = 4`

In this case we find `n = 4` or `n = -4`

In practice, there would be a slight preference towards the positive solution, i.e. `4`.