A man works for *T* hours daily and earns Rs *P* per hour. For every hour worked beyond *T* hours, he is paid twice his normal hourly rate. One day, he works for *N* hours and earns Rs 1200 more than his usual daily wage. Given that *P* can be either Rs 100 or Rs 150, which of the following pairs (*N*, *T*) satisfy this condition?

(i) (12, 8)

(ii) (10, 4)

(iii) (9, 5)

(iv) (13, 7)

(v) (17, 13)

Which of the following options lists the correct pairs?

Question

A man works for *T* hours daily and earns Rs *P* per hour. For every hour worked beyond *T* hours, he is paid twice his normal hourly rate. One day, he works for *N* hours and earns Rs 1200 more than his usual daily wage. Given that *P* can be either Rs 100 or Rs 150, which of the following pairs (*N*, *T*) satisfy this condition?

(i) (12, 8)

(ii) (10, 4)

(iii) (9, 5)

(iv) (13, 7)

(v) (17, 13)

Which of the following options lists the correct pairs?

Accepted Answer

**Solution:**

1. Let the man work *T* hours daily at Rs *P* per hour. So, his usual daily earning is:
   
   $$	ext{Daily earning} = P 	imes T$$

2. On a day he works *N* hours, where *N > T*. For the extra hours *(N - T)*, he earns double the rate, i.e., Rs $2P$ per hour.

3. Total earning on that day:
   
   $$	ext{Total earning} = P 	imes T + 2P 	imes (N - T) = P 	imes T + 2P(N - T)$$

4. The extra amount earned compared to usual daily earning is Rs 1200:
   
   $$	ext{Extra} = 	ext{Total earning} - 	ext{Daily earning} = 1200$$

Substitute the values:

$$P 	imes T + 2P (N - T) - P 	imes T = 1200$$

Simplifies to:

$$2P (N - T) = 1200$$

$$N - T = \frac{1200}{2P} = \frac{600}{P}$$

5. Since *P* can be Rs 100 or Rs 150, calculate *N - T* for both:

- For $P = 100$:
     $$N - T = \frac{600}{100} = 6$$

- For $P = 150$:
     $$N - T = \frac{600}{150} = 4$$

6. Now check each pair (*N*, *T*) to see if $N - T$ equals 6 or 4:

- (i) 12 - 8 = 4 ✔
   - (ii) 10 - 4 = 6 ✔
   - (iii) 9 - 5 = 4 ✔
   - (iv) 13 - 7 = 6 ✔
   - (v) 17 - 13 = 4 ✔

All pairs satisfy the condition for either $P=100$ or $P=150$.

**Answer:** (e) All of the above

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