a) 1\;:\;4\;:\;5

b) 2\;:\;3\;:\;5

c) 1\;:\;3\;:\;7

d) 1\;:\;3\;:\;5

correct answer is: d) 1\;:\;3\;:\;5

Explanation

According to the question,
The ratio of ∠BAC, ∠ABC and ∠ACB is 3\;:\;5\;:\;10\;.
Let,
∠BAC = 3x .
∠ABC = 5x .
∠ACB = 10x .
So,
3x+5x+10x=180°
\rightarrow 18x=180°
\rightarrow x=\large\frac{180^\circ}{18}
\rightarrow x=10°
So,
∠BAC = 3x or 3×10° or 30°.
∠ABC = 5x or 5×10° or 50°.
∠ACB = 10x or 10×10° or 100°.

Now, ∠BAC is decreased 10° = \left(30^\circ-10^\circ\right)=20°.
∠ACB is increased 10° = \left(50^\circ+10^\circ\right)=60°.
So that, the new ratio of the three angles :
∠BAC : ∠ABC : ∠ACB = 20° : 60° : 100°
\rightarrow 1\;:\;3\;:\;5\;.
Ans: The new ratio of the three angles = 1\;:\;3\;:\;5\;.

An alternative way to phrase this particular math query is possible:

1. In triangle ABC, the sum of the three angles is 180°. The ratio of ∠BAC, ∠ABC, and ∠ACB is 3 : 5 : 10. If the measure of ∠BAC is reduced by 10° and the measure of ∠ABC is increased by 10°, determine the new ratio of the three angles.
2. The total of the three angles in triangle ABC is 180°, with the ratio of ∠BAC, ∠ABC, and ∠ACB being 3 : 5 : 10. If the measure of ∠BAC is decreased by 10° and the measure of ∠ABC is increased by 10°, find the new ratio of the three angles.
3. In triangle ABC, the sum of the angles is 180°, and the ratio of ∠BAC, ∠ABC, and ∠ACB is 3 : 5 : 10. Calculate the new ratio of the angles if the measure of ∠BAC is reduced by 10° and the measure of ∠ABC is increased by 10°.