急! F4 Quadratic Equations

13.
Let n be the number of apples he bought.

Amount he earned:
(10n/3) - 900 = 900 * (300/n)
[(10n/3) - 900] * 3n = [900 * (300/n)] * 3n
10n² - 2700n = 810000
n² - 270 - 81000 = 0
(n + 180)(n - 450) = 0
n = -180 (rejected) or n = 450
Hence, he bought 450 apples.


====
14.
(a)
Let n be the number of glasses he bought.

Total profit:
(n - 2)[(48/n) + 3] - 48 = 22
{(n - 2)[(48/n) + 3] - 48} * n = 22 * n
(n - 2)(48 + 3n) - 48n = 22n
3n² + 42n - 96 - 48n = 22n
3n² - 28n - 96 = 0
(n - 12)(3n + 8) = 0
n = 12 or n = -8/3 (rejected)
Hence, she bought 12 glasses.

(b)
Price per glass that she sold them
= $[(48/12) + 3]
= $7


=====
15.
Let t h be the time of the longer route (24 km).

Difference in speed:
(24/t) - [20/(t + 0.5)] = 4
{(24/t) - [20/(t + 0.5)]} * t(t + 0.5) = 4 *t(t + 0.5)
24(t + 0.5) - 20t = 4t(t + 0.5)
24t + 12 - 20t = 4t² + 2t
4t² - 2t - 12 = 0
2t² - t - 6 = 0
(2t + 3)(t - 2) = 0
t = -3/2 (rejected) or t = 2
Hence, the longer route takes him 2 hours.


=====
16.
Let t hours be the time that Peter requires to load the truck alone.
Time that Tom requires to load the truck alone = (t - 3) hours

Fraction of the truck that can load in 1 hour when they work together:
(1 / t) + [1 / (t - 3)] = 1 / 2
[(t - 3) / t(t - 3)] + [t / t(t - 3)] = 1 / 2
[(t - 3) + t] / t(t - 3) = 1 / 2
(2t - 3) / (t² - 3t) = 1 / 2
t² - 3t = 4t - 6
t² - 7t + 6 = 0
(t - 6)(t - 1) = 0
t = 6 or t = 1 (rejected for t - 3 < 0)
Hence, Peter alone requires 6 hours to load the truck.


=====
17.
Let t minutes be the time taken for Tap A alone to fill up the tank.

To filled up the tank:
{(1/t) + [1/(t - 25)]} * (20 + 10) = 1
{(1/t) + [1/(t - 25)]} * 30 * t(t - 25) = 1 * t(t - 25)
30(t - 25) + 30t = t² - 25t
30t - 750 + 30t = t² - 25t
t² - 85t + 750 = 0
(t - 75)(t - 10) = 0
t = 75 or t = 10 (rejected for t - 25 < 0)
Hence, Tap A alone takes 75 minutes to fill up the tank.