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Find x:y:z given that x:y=12:15 and y:z=21:25.

Answer
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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 15×7=105,21×5=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,
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.
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