Final answer:
To find the ratio t:m:w in its simplest form when t:m is 3:8 and m:w is 4:7, we normalize the 'm' in both ratios to the same value by multiplying the ratios to get a common 'm' and then combine them into a single ratio, which is then simplified to get the final answer of 3:8:14.
Step-by-step explanation:
To find the ratio t:m:w in its simplest form, given that the ratio t:m is 3:8 and the ratio m:w is 4:7, we need to make the 'm' in both ratios correspond to the same value. We can accomplish this by finding a common multiple of the two 'm' values in the given ratios, which are 8 and 4.
To find the common multiple, we can multiply the first ratio t:m by 4 and the second ratio m:w by 8. This allows the 'm' from both ratios to have the same value, thus being able to combine them into a single triplet ratio:
t:m = 3:8 becomes 3\(\times\)4 : 8\(\times\)4 => 12:32,
m:w = 4:7 becomes 4\(\times\)8 : 7\(\times\)8 => 32:56.
Now that the 'm' is the same in both ratios, we can write the combined ratio:
t:m:w = 12:32:56.
Lastly, to find the ratio in its simplest form, we divide each part by the greatest common divisor of the three numbers, which is 4:
t:m:w becomes 12/4:32/4:56/4 => 3:8:14 which is the ratio in its simplest form.