Can a/b and (a+b)/b ever form a proportion? Why or why not?
To determine whether a/b and (a+b)/b can form a proportion, let's first understand what it means for two ratios to be in proportion. In a proportion, the fractions on each side of the equation must have equal ratios. In other words, if we have two ratios a/b and c/d, they form a proportion when a/b = c/d.
Now, let's apply this to the given expressions: a/b and (a+b)/b.
To check if these two ratios can form a proportion, we need to equate them and see if they are equal:
a/b = (a+b)/b
To simplify this equation, we can cross multiply:
a * b = (a+b) * 1 (since b/b equals 1)
Expanding the equation:
ab = a + b
Now, let's rearrange the equation and try to solve for either 'a' or 'b':
ab - a = b
Factoring 'a' from the left-hand side:
a(b - 1) = b
To solve for 'a', divide both sides by (b - 1):
a = b / (b - 1)
From this equation, we can conclude that 'a' and 'b' can be in proportion when the value of 'a' is equal to b / (b - 1).
So, in summary, a/b and (a+b)/b can form a proportion if 'a' is equal to b / (b - 1). Otherwise, they will not be in proportion.
well, let's try it ...
a/b = (a+b)/b
times b, if b ≠0
a = a+b
then b = 0
but now we have a contraction, so .....
or
cross - multiply
ab = ab + b^2
b^2 = 0
b = 0 , but that would make a/b undefined,
so what do you think?