Master properties of ratio and proportion including componendo-dividendo, and apply them to solve algebraic problems.
If a:b=c:d, then properties like alternendo (a:c=b:d) and invertendo (b:a=d:c) let us rearrange the proportion for easier problem solving.
If a:b=3:4, find b:a.
By invertendo, b:a=4:3
If a:b=c:d, show a:c=b:d (alternendo) using a=3,b=6,c=4,d=8.
3:6=4:8=1:2, both sides simplify to 1:2, confirming alternendo
If a/b=c/d, then (a+b)/(a−b)=(c+d)/(c−d) — this identity, called componendo-dividendo, simplifies many proportion problems.
If (x+3)/(x−3)=5/2, find x using componendo-dividendo principles in reverse (cross multiply).
2(x+3)=5(x−3); 2x+6=5x−15; 21=3x; x=7
If a/b=3/5, find (a+b)/(a−b) using componendo-dividendo.
(3+5)/(3−5)=8/−2=−4
1. If a:b=5:7, find b:a.
2. If x/y=2/3, find (x+y)/(x−y) using componendo-dividendo.
3. If (x+5)/(x−5)=3/2, find x.
4. If a:b=4:9, find a²:b².
5. If a/b=c/d=2/3, and a+c=10, find b+d.