# Ask Uncle Colin: A Rational Mess

Dear Uncle Colin, I’ve got a ratios question, and I don’t understand the solution.

The question is: Three numbers, $x$, $y$, and $z$, have a sum of 871. The ratio $x:y$ is 4:5 and the ratio $y:z$ is 3:8. What is the value of $y$?

Their solution is to say the ratio $x:y = 4:5 = \left(4\times\frac{3}{5}\right):\left(5\times\frac{3}{5}\right) = \frac{12}{5}:3$ (*). The ratio $y:z = 3:8$, therefore, the ratio $x:y:z = (12/5):3:8 = 12:15:40$. The value of $y$ is $\frac{15}{67}$ of the sum of 871, which is 195.

I don’t understand why they multiplied the ratio $x:y$ by (3/5) in (*). I assume it has something to do with the ratio of $y:z$ and how they both have a 3.

-- Ratio Understanding Not Generally Excellent

Hi, RUNGE, and thanks for your message! You’re quite right about it being something to do with the 3; they’ve done that to get the same value for the $y$ part in each expression.

I’d have done it slightly differently, and aimed to make the $y$ part 15 (3 × 5) in both parts of the ratio. $x:y = 12:15$ and $y:z = 15:40$ directly, without all that messing about with fractions.

I would also note that 871 is 67 \times 13 (because I know 67 × 12 = 804), so $y$ = 15 × 13 = 195.

Hope that helps!

-- Uncle Colin