Finding a unified ratio from two separate ratios

Question

I'm self-studying with Stroud & Booth's amazing Engineering Mathemathics, 7th edition. I'm stuck at a problem set that gives me two ratios of variables A and B, and B and C respectively, an expects me to "unify" them, so to speak, in order to produce an A:B:C ratio. To be more precise:

A, B and C are mixed according to the ratios $A:B = 2:5$ and $B:C = 10:11$. Find the common ratio $A:B:C$.

Just please keep in mind that I'm still on the first chapter, Arithmetic.

Answer 1

$A:B = 2:5$ and $B:C = 10:11$

$A:B = 4:10$ and $B:C = 10:11$

$A:B:C = 4:10:11$

Answer 2

I work this kind of problem out by imagining totals that turn the ratios into amounts. I choose totals to make the arithmetic easy. Since I see $2$ and $5$ and $10$ and $11$ I'll suppose I have $2 \times 10 \times 11 = 220$ things. Then using $x$ to represent the number of things of type $X$ I know $$ a + b + c = 220. $$ I also know that since $a/b = 2/5$, $$ 2b = 5a $$ and, similarly, $$ 11b = 10c. $$ Now solve those three linear equations ih three unknowns.

If you don't like the $220$ total I started with you can use $$ a + b + c = 1 $$ instead. Then the answers will be the appropriate fractions of the whole.

Answer 3

You have 2 of A per every 5 of B and you have 10 of B per every 11 of C. It's not hard to see that you need twice the amount of A per B to cover the amount of B per C:

$$ 2\cdot 2:2 \cdot 5:11\implies\\ 4:10:11 $$