"To divide by a fraction, invert and multiply."
Suppose we have a calculation like . But we know that is a fraction, and that . In order to understand our calculation, we can replace the in the calculation with the thing it equals, thus: . It is evident now that in our calculation, we are dividing by a fraction.
To divide by a fraction, use the rule they teach in elementary school: invert and multiply. Inverting the fraction gives us . Now, multiply by the inverted fraction: .
In this form our calculation is simpler because there is only one division. It is okay to remove the parentheses: , and to rearrange terms: . We can add parentheses to show we do the division first: . All of this is valid arithmetic.
Our new formula will produce the same result as , our original calculation.
is the quantity of money.
is real output.
is output in actual prices.
is the price level (the "deflator").
In its final version, our calculation divides the quantity of money by output in actual prices, and multiplies the result by the price level.
The calculation we started with was used by Milton Friedman in Money Mischief to produce all those graphs that show how "money relative to output" follows the same trend-line as prices.
His numbers follow the price trend because he multiplies by the price level.
You can mimic the price trend in many ways. You don't have to use the quantity of money and output. You can use any number you want, pretty much. Just multiply by the price level, and your answer takes on the shape of the price trend.