(a) The first term of an Arithmetic Progression (AP) is 8, the ratio of the 7th term to the 9th term is 5 : 8, find the common difference of the AP.
(b) A trader bought 30 baskets of pawpaw and 100 baskets of mangoes for N2,450.00. She sold the pawpaw at a profit of 40% and the mangoes at a profit of 30%. If her profit on the entire transaction was N855.00, find the (i) cost price of a basket of pawpaw ; (ii) selling price of the 100 baskets of mangoes.
(a) For an AP, the \(n\)th term is \(a + (n-1)d\), with \(a = 8\).
7th term \(= 8 + 6d\); 9th term \(= 8 + 8d\). Given ratio \(5:8\):
\(\dfrac{8 + 6d}{8 + 8d} = \dfrac{5}{8}\)
\(8(8 + 6d) = 5(8 + 8d)\)
\(64 + 48d = 40 + 40d\)
\(8d = -24 \ \Rightarrow\ \mathbf{d = -3}\)
(b) Let cost price be \(N\,p\) per basket of pawpaw and \(N\,m\) per basket of mango.
\(30p + 100m = 2450 \quad(1)\)
Profit: \(0.40(30p) + 0.30(100m) = 855 \ \Rightarrow\ 12p + 30m = 855 \quad(2)\)
Simplify: \((1)\div 10: 3p + 10m = 245\); \((2)\div 6: 2p + 5m = 142.5\).
Multiply the last by 2: \(4p + 10m = 285\). Subtract \(3p + 10m = 245\): \(p = 40\).
Then \(2(40) + 5m = 142.5 \ \Rightarrow\ 5m = 62.5 \ \Rightarrow\ m = 12.5\).
(i) Cost price of a basket of pawpaw \(= \mathbf{N40.00}\).
(ii) Selling price of 100 baskets of mango \(= 100m \times 1.30 = 1250 \times 1.30 = \mathbf{N1{,}625.00}\).