Rahul can finish one-fifth of his homework in one hour. Neha can finish three-sevenths of her homework in one hour thirty minutes and Riya can finish three fourth of her homework in three hours thirty minutes. If all of them start their homework at 12.00 p.m. And can go to play as soon as they all finish their homework, when can they start to play if they take a break at 3.30 p.m. For thirty minutes?

Rahul can finish one-fifth of his homework in 1 hour.

So, the time needed to complete his full homework is

$$\frac{1}{\left (\frac{1}{5} \right)}$$

= 5 hours

Neha can finish three-sevenths of her homework in 1 hour 30 minutes (1.5 hours)

So, the time needed to complete her full homework

$$\frac{1.5}{\left ( \frac{3}{7} \right )}$$

⇒ $$1.5\times \frac{7}{3}$$

= 3.5 hours or 3 hours 30 minutes

Riya can finish three fourth of her homework in 3 hours 30 minutes (3.5 hours)

So, the time needed to complete her full homework

$$\frac{3.5}{\left ( \frac{3}{4} \right )}$$

⇒ $$3.5\times \frac{4}{3}$$

⇒ $$4\frac{2}{3}$$

= 4 hours 40 mins

All of them began their homework at 12.00 p.m. and they take a break at 3.30 p.m. for 30 minutes.

As Neha is taking 3 hours 30 minutes to finish the homework, that implies she can start to play at

12 p.m. + 3 hours 30 minutes = 3:30 p.m.

Rahul can start to play at

12 p.m. + 5 hours +30 minutes break = 5: 30 p.m.

Riya can start to play at

12 p.m + 4 hours 40 minutes +30 minutes break = 5: 10 p.m.