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#### 8.6.1 Percentages

The `M-%` (`calc-percent`) command takes a percentage value, say 5.4, and converts it to an equivalent actual number. For example, `5.4 M-%` enters 0.054 on the stack. (That's the <META> or <ESC> key combined with `%`.)

Actually, `M-%` creates a formula of the form ‘`5.4%`’. You can enter ‘`5.4%`’ yourself during algebraic entry. The ‘`%`’ operator simply means, “the preceding value divided by 100.” The ‘`%`’ operator has very high precedence, so that ‘`1+8%`’ is interpreted as ‘`1+(8%)`’, not as ‘`(1+8)%`’. (The ‘`%`’ operator is just a postfix notation for the `percent` function, just like ‘`20!`’ is the notation for ‘`fact(20)`’, or twenty-factorial.)

The formula ‘`5.4%`’ would normally evaluate immediately to 0.054, but the `M-%` command suppresses evaluation as it puts the formula onto the stack. However, the next Calc command that uses the formula ‘`5.4%`’ will evaluate it as its first step. The net effect is that you get to look at ‘`5.4%`’ on the stack, but Calc commands see it as ‘`0.054`’, which is what they expect.

In particular, ‘`5.4%`’ and ‘`0.054`’ are suitable values for the rate arguments of the various financial functions, but the number ‘`5.4`’ is probably not suitable—it represents a rate of 540 percent!

The key sequence `M-% *` effectively means “percent-of.” For example, `68 <RET> 25 M-% *` computes 17, which is 25% of 68 (and also 68% of 25, which comes out to the same thing).

The `c %` (`calc-convert-percent`) command converts the value on the top of the stack from numeric to percentage form. For example, if 0.08 is on the stack, `c %` converts it to ‘`8%`’. The quantity is the same, it's just represented differently. (Contrast this with `M-%`, which would convert this number to ‘`0.08%`’.) The `=` key is a convenient way to convert a formula like ‘`8%`’ back to numeric form, 0.08.

To compute what percentage one quantity is of another quantity, use `/ c %`. For example, `17 <RET> 68 / c %` displays ‘`25%`’.

The `b %` (`calc-percent-change`) [`relch`] command calculates the percentage change from one number to another. For example, `40 <RET> 50 b %` produces the answer ‘`25%`’, since 50 is 25% larger than 40. A negative result represents a decrease: `50 <RET> 40 b %` produces ‘`-20%`’, since 40 is 20% smaller than 50. (The answers are different in magnitude because, in the first case, we're increasing by 25% of 40, but in the second case, we're decreasing by 20% of 50.) The effect of `40 <RET> 50 b %` is to compute ‘`(50-40)/40`’, converting the answer to percentage form as if by `c %`.