Answer:
The time it would take the clerk to count a sum of Rs. 4,500 is 34 minutes
Step-by-step explanation:
The given parameters are;
The rate at which the clerk in the bank counted one rupee currency notes for the first 10 minutes = Rs. 150/minute
After the first 10 minutes, the rate at which the clerk counted = Rs. 2 less than the previous minute
The time it will take the clerk to count Rs. 4,500 is given as follows;
The number of one rupee currency notes the clerk counts in the first 10 minutes = 10 × Rs. 150 = Rs. 1,500
The amount in one rupee currency notes the clerk counted at a decreasing rate of Rs. 2 less every minute = Rs. 4,500 - Rs. 1,500 = Rs. 3,000
From the 11th minute the number of rupees the clerk counts = Rs. 150 - Rs. 2 = Rs. 148
We have an arithmetic progression with the first term, a = 148, and the common difference, d = -2
The sum of n terms of an AP, [tex]S_n = \dfrac{n}{2} \times \left [ 2 \cdot a + (n - 1)\cdot d\right ][/tex]
Which gives;
[tex]3000 = \dfrac{t}{2} \times \left [ 2 \times 148 + (t - 1)\times (-2)\right ] = 148 \cdot t - t^2 + t[/tex]
3000 = 149·t - t²
Which gives;
t² - 149·t + 3,000 = 0
(t - 24) × (t - 125) = 0, therefore, t = 24 minutes or t = 125 minutes
The time it would take the clerk to count a sum of Rs. 4,500 = 10 + 24 = 34 minutes.
