Math in… Leaving a Tip

Before everyone had a calculator on their phones and receipts featured suggested tips, calculating the tip at the end of the meal used to require a little mental math.

If you ask for separate checks at a restaurant, the suggested tips can be based on the full check, so it’s still wise to at least estimate the amount.

When the standard in tipping was 15%, a nice trick was to take the amount owed, move the decimal left one position, and then tip that amount plus another half of that amount.

For example, to tip 15% on a $27 check:

1. Move the decimal left in $27.00 to get $2.70

2. Divide $2.70 by 2 to get $1.35

3. Add $2.70 and $1.35 to get $4.05

The tip would then be $4.05.

This trick is based on some algebra. Let’s walk through what it’s doing. First, note that moving the decimal left one space is equivalent to dividing by 10. If x is the amount owed, the steps are:

1. Find x / 10, or 0.10x

2. Find 0.10x / 2, or 0.05x

3. Find 0.10x + 0.05x, or 0.15x

The big tool here is the distributive property, since the calculation hinges on (a + b)x = ax + bx in the form (0.10+0.05)x = 0.10x + 0.05x.

In terms of percentages, these steps are:

1. Find 10% of x

2. Find 5% of x

3. Add 10% of x to 5% of x to get 15% of x

For a 20% tip, things are simpler! After we find 10%, we could double that amount (add it to itself) to get 20%. On a $33 check, this looks like

1. Move the decimal left in $33.00 to get $3.30

2. Double $3.30 to get $6.60

The tip is then $6.60.

Percentages that aren’t multiples of 5 can be a little tricky to calculate in your head. However, if the number you’re trying to take the percentage of is nice, here’s a cool rule that can make the calculation easier:

x percent of y = y percent of x

Suppose you want to find 23% of 15, for example. Not a particularly nice calculation! However, the rule above says that we could instead find 15% of 23. Using our tip calculating trick, that’s 2.30 + 1.15 = 3.45.

Why does this work? Again, it’s a nice little algebra trick! This time it’s based on the commutative and associative properties of multiplication.

If we want to find x percent of y, this is the same as multiplying y by x/100. Similarly, finding y percent of x is equivalent to multiplying x by y/100. Since we can reorder factors in products,

(⅟₁₀₀x) ∙ y = (⅟₁₀₀y) ∙ x