# Understanding this ratio trick

Two numbers are in the ratio $$3:5.$$ If $$9$$ is subtracted from each, they become in the ratio $$12:23.$$ Find the smaller number.

Solution

Let's denote the two numbers as $$3x$$ and $$5x$$.

According to the problem: $$\dfrac{3x - 9}{5x - 9} = \dfrac{12}{23}$$

Cross-multiplying: $$23(3x - 9) = 12(5x - 9)\\69x - 207 = 60x - 108\\x = 11$$

Therefore, the smaller number $$3x$$ is $$33$$.

There is a trick to solving this problem. The difference between $$12$$ and $$23$$ is $$11.$$ If you multiply $$11$$ to $$3$$ and $$5$$ you get $$33$$ and $$55.$$ So, the smaller number is $$33.$$

I understand the above solution. How does the trick work, though?

• Think about it for a moment. How can this "trick" be correct advice when it does not consider the fact that the number we subtracted was $9$ and not some completely different number? Commented May 26 at 15:01

The trick is fake: it claims that $$\dfrac{3x - 18}{5x - 18} = \dfrac{12}{23}$$ has solution $$23-12=11$$ while its actual solution is $$22.$$
hint: try solving for $$\frac{ax-c}{bx-c}$$=$$\frac{d}{e}$$
when you will try to solve it you'll get $$x=\frac{c(e-d)}{ae-bd}$$ and when you'll do that you'll find that the trick is just a coincidence because the numbers in the original ratio will be $$ax:bx$$ and in your case $$x=|d-e|$$ which is not true for all cases
• as a final nod try this trick for other numbers,this trick will be false where $\frac{c}{ae-bd}$ =1 Commented May 26 at 6:07
• Do you mean the trick will be false unless $\frac{c}{ae-bd}=1$? That equation happens to be true in the original question. Commented May 26 at 14:59