AI Tutor
How could you calculate 7×487 \times 48 in your head in under five seconds without paper or a phone?
Instead of grueling long multiplication, you can think of 4848 as (50−2)(50 - 2), compute 7×50=3507 \times 50 = 350 and 7×2=147 \times 2 = 14, and subtract them to get 336336 instantly.
Why are we mathematically allowed to split numbers apart like this?

The Area Model and Euclid's Insight

Around 300 BCE, the Greek mathematician Euclid demonstrated in Book II of his Elements that the total area of a large rectangle equals the sum of its smaller side-by-side pieces.
This geometric truth is the distributive property, which states that multiplying a number by a sum equals the sum of each individual product: a(b+c)=ab+aca(b + c) = ab + ac.
Does this same splitting shortcut hold true when we deal with subtraction instead of addition?

Distribution Over Subtraction

The rule extends directly to subtraction: multiplying by a difference gives the difference of the individual products, written as a(b−c)=ab−aca(b - c) = ab - ac.
For instance, to evaluate 6(30−4)6(30 - 4), we distribute the multiplier 66 across both inner terms to get (6×30)−(6×4)=180−24=156(6 \times 30) - (6 \times 4) = 180 - 24 = 156.
A frequent error is multiplying the outside number by only the first term inside the parentheses while leaving the second term untouched.