Question

Find \[x:y:z\] given that $x:y = 12:15$ and $y:z = 21:25$.

Answer 1

Hint: Here $x:y$ and $y:z$ are given. We can find the combined ratio using the common variable $y$. For that we have to make the number corresponding to $y$ the same in both the ratios. Multiplying both numbers in a ratio by an integer will not change the ratio.

Useful formula:
A ratio does not change if we multiply both the numbers by the same integer.
That is, $m:n = am:an$, for any $m,n,a$.

Complete step-by-step answer:
We are given that $x:y = 12:15$ and $y:z = 21:25$.
We have to find \[x:y:z\].
Here we can see $y$ is common in both the ratios.
So we can find the ratio using it.
We know that a ratio does not change if we multiply both the numbers by the same integer.
So we can multiply the given ratios so that the number corresponding to $y$ will be the same in both the ratios.
Now the numbers corresponding to $y$ are $15$ and $21$.
We can see their least common multiple is 105, where $15 \times 7 = 105,21 \times 5 = 105$
Consider $12:15$.
Multiplying both numbers by $7$ we get, $84:105$.
This gives, $x:y = 12:15 = 84:105$
Consider $21:25$.
Multiplying both numbers by $5$ we get, $105:125$.
This gives, $y:z = 21:25 = 105:125$
Now we have, $x:y = 84:105$ and $y:z = 105:125$.
Combining these two we get,
$\therefore x:y:z = 84:105:125$

Note: Here, to make the numbers corresponding to $y$ the same, we multiply both the ratios by $7$ and $5$ respectively. Likewise we can divide the terms in a ratio without changing the ratio. That is, $12:15 = 4:5$, by cancelling the common factor $3$. But in this case, by dividing we cannot make the terms in the ratio the same.