Counting hops along a segment

A ribbon is cut so one piece is twice the other. In the lab, drag B. Slide P along the equal hops to the ratio asked.

The rule

Let P cut the segment AB so that AP : PB = m : n. Here A is the first point and B is the last. The ratio splits AB into m + n equal hops. A hop is one equal step along AB. The cut point is m hops from the first point.

Hop rule: make m + n equal hops. Walk m of them from the first point.

Take A(1, 1) and B(13, 10). The whole jump is 12 across and 9 up. Here 13 − 1 = 12 and 10 − 1 = 9. The ratio 2 : 1 makes 3 hops. One hop is (12 ÷ 3, 9 ÷ 3) = (4, 3). Two hops from A reach (1 + 8, 1 + 6) = (9, 7).

Divide the whole jump by m + n. Do not use m or n alone. Together, m + n hops fill the whole jump.

The ratio 4 : 2 is the same cut as 2 : 1. Reduce a ratio by dividing both parts by the same number. Reduce first, so there are fewer, bigger hops.