Ratios & Proportions: Combining Ratios with Different Multipliers - Equate the Common Member
At a certain stage of a soccer tournament, the score ratio of teams A, B, and C was $$3{:}4{:}5$$. Eventually, the score ratio of A to B has doubled while the score ratio of A to C has halved. If the final score of team C was 40, what was the final score of team B?
Incorrect.
[[snippet]]Incorrect.
[[snippet]]Incorrect.
[[snippet]]Incorrect.
[[snippet]]| Ratio | Multiplier | Real | |
| A | $$3$$ |
||
| B | $$2$$ | $$\color{red}{\times4}$$ |
$$\color{red}{2\times4=8}$$ |
| C |
$$10$$ |
$$\color{red}{\times4}$$ |
$$40$$ |
| Total |
The real number of team C is 40, so the multiplier is $$\color{red}{\times4}$$. Therefore the score of team B is $$4\times\color{red}{2}=\color{purple}{8}$$.
If we were to multiply the ratio of B as well, then the ratio of A:B would not be doubled; it would be expanded from 3:4 to 6:8, but would remain essentially the same ratio with no change.
Remember that a ratio is like a fraction. When we say that A:B has a ratio of 3:4, we are saying that A is $$\frac{3}{4}$$ of B. When we now want to change the ratio to a new one by doubling it, we need to reach double the original fraction: from $$\frac{3}{4}$$ we move to $$\frac{6}{4}$$, or $$\frac{3}{2}$$. Thus, when a question asks for ratio changes (and not merely ratio expansion, which is the same ratio with a different multiplier) through multiplication/division (for example, the ratio is doubled, or tripled, or halved) do the operation only on the left side, and keep the other side as is.
Let's do so.
Correct.
[[snippet]]A : B : C
Initial ratio $$3$$ : $$4$$ : $$5$$
Double the ratio A:B, and halve the ratio A:C. Halving a ratio is equivalent to multiplying the second part by 2. Notice that there is no way to know how the scores change in order to make these ratio changes, so the ratios must be treated separately.
A : B : C
Ratio I $$3\times2= \color{orange}{6}$$ : $$4$$
Ratio II $$3$$ : : $$5\times 2= \color{orange}{10}$$
Equate the common quantity, A, to find the ratio of B:C. Do this by reducing Ratio I by 2 to get A:B is equal to 3:2.
A : B : C
Ratio I $$3$$ : $$2$$
Ratio II $$3$$ : : $$10$$
So the final ratio of A:B:C is 3:2:10. To be sure, check that the changes have indeed been made correctly—the ratio of A:B is 3:2, which is indeed twice the original ratio of 3:4.
