(a) Simplify : (frac{1}{3^{5n}} times 9^{n – 1} times 27^{n + 1})
(b) The sum of the ages of a woman and her daughter is 46 years. In 4 years’ time, the ratio of their ages will be 7 : 2. Find their present ages.
Explanation
(a) (frac{1}{3^{5n}} times 9^{n – 1} times 27^{n + 1})
(3^{-5n} times (3^{2})^{n – 1} times (3^{3})^{n + 1})
(3^{-5n} times 3^{2n – 2} times 3^{3n + 3})
(3^{-5n + 2n – 2 + 3n + 3})
= (3^{1} = 3)
(b) Let the daughter’s age be c and the woman’s age be d.
(c + d = 46 …. (1))
In 4 years time, the daughter’s age = c + 4
The woman’s age = d + 4
(frac{d + 4}{c + 4} = frac{7}{2})
(2(d + 4) = 7(c + 4) implies 2d + 8 = 7c + 28)
(2d – 7c = 28 – 8 implies 2d – 7c = 20 … (2))
(c + d = 46 implies d = 46 – c)
(therefore 2(46 – c) – 7c = 20)
(92 – 2c – 7c = 20 implies 92 – 20 = 9c)
(72 = 9c implies c = 8)
(d = 46 – c implies d = 46 – 8 = 38)
Therefore, the daughter is 8 years old and the woman is 38 years.