Question 1:
Table below shows the price indices and percentage of usage of four items, P , Q , R and S , which are the main ingredients in the production of a type of toy.


(a)  Calculate
(i) the price of S in the year 2004 if its price in the year 2007 is RM43.20 ,
(ii) the price index of P in the year 2007 based on the year 2002 if its price index in the year 2004 based on the year 2002 is 110.

(b) The composite index of the cost of toy production for the year 2007 based on the year 2004 is 125.
Calculate
(i) the value of  x ,
(ii) the price of the toy in the year 2004 if the corresponding price in the year 2007 is RM75.
(a)(i)
$\begin{array}{l}135=\frac{43.2}{P_{2004}}\times 100\\ P_{2004}=\frac{43.2\times 100}{135}=32\\ \therefore \text{The price of }S\text{ in the year 2004 is RM32}\end{array}$

(a)(ii)
$\begin{array}{l}\text{Given }\frac{P_{2004}}{P_{2002}}\times 100=110\to \frac{P_{2004}}{P_{2002}}=\frac{110}{100}\\ \text{and }\frac{P_{2007}}{P_{2004}}\times 100=125\to \frac{P_{2007}}{P_{2004}}=\frac{125}{100}\\ \\ \text{Price index of }P\text{ in the year 2007 based on the year 2002,}\\ I=\frac{P_{2007}}{P_{2002}}\times 100\\ I=\frac{P_{2007}}{P_{2004}}\times \frac{P_{2004}}{P_{2002}}\times 100\\ I=\frac{125}{100}\times \frac{110}{100}\times 100\\ I=137.50\end{array}$

(b)(i)
$\begin{array}{l}\text{Given composite index }\overline{I}=125\\ \overline{I}=\frac{\sum IW}{\sum W}\\ 125=\frac{\left(125\right)\left(30\right)+\left(x\right)\left(20\right)+\left(115\right)\left(10\right)+\left(135\right)\left(40\right)}{30+20+10+40}\\ 125=\frac{10300+20x}{100}\\ 12500=10300+20x\\ x=110\end{array}$

(b)(ii)
$\begin{array}{l}\text{Lets price of the toy in the year 2004 is }{P}_{2004}\\ P_{2004}\times \frac{125}{100}=75\\ P_{2004}=RM60\end{array}$

11.2 Composite Index

Example 1

Noodle	Price index	Weightage
Fried noodle	112	4
Tomyam noodle	104	3
Seafood noodle	109	2
Wantan noodle	111	1

Solution:
Composite index for the year 2000 based on the year 1996
$\begin{array}{l}\overline{I}=\frac{\sum IW}{\sum W}\\ \overline{I}=\frac{\left(112\right)\left(4\right)+\left(104\right)\left(3\right)+\left(109\right)\left(2\right)+\left(111\right)\left(1\right)}{4+3+2+1}\\ \overline{I}=\frac{448+312+218+111}{10}\\ \overline{I}=\frac{1089}{10}=108.9\end{array}$

Example 2

Item	Price index	Weightage
P	110	4
Q	x	2
R	120	1
S	115	3

Solution:
$\begin{array}{l}\overline{I}=\frac{\sum IW}{\sum W}\\ 116.5=\frac{\left(110\right)\left(4\right)+\left(x\right)\left(2\right)+\left(120\right)\left(1\right)+\left(115\right)\left(3\right)}{4+2+1+3}\\ 116.5=\frac{440+2x+120+345}{10}\\ 116.5=\frac{905+2x}{10}\\ 1165=905+2x\\ 2x=260\\ x=130\end{array}$

11.1 Index Number

Example 1
The price of the house Sulaiman bought in the year 1998 was RM 110000. In the year 2000, the price of the house went up to RM 130000. Calculate the price index of the house for the year 2000 based on the year 1998.

Solution:
$\begin{array}{l}{P}_0=RM110000\\ P_1=RM130000\\ I=\frac{P_1}{P_0}\times 100\\ I=\frac{130000}{110000}\times 100\\ I=118.18\end{array}$

Example 2
The price of a bottle of cooking oil in the year 1999 was RM15.00. Given that the price index for the year 2000 based on the year 1999 is 105, calculate the price of the cooking oil in the year 2000.

Solution:
$\begin{array}{l}{P}_0=RM15\\ P_1=x{\text{ (Price}}_{2000})\\ I=\frac{P_1}{P_0}\times 100\\ 105=\frac{x}{15}\times 100\\ 1575=100x\\ x=RM15.75\end{array}$

Example 3
Based on the year 1995, the index number of an oven for the years 1998 and 2001 are 110 and 130 respectively. Find the index number for the year 2001 based on the year 1998.

Solution:
$\begin{array}{l}\text{Given, }\frac{P_{1998}}{P_{1995}}\times 100=110\text{ and }\frac{P_{2001}}{P_{1995}}\times 100=130\\ \therefore \frac{P_{2001}}{P_{1998}}\times 100=\frac{P_{2001}}{P_{1995}}\times \frac{P_{1995}}{P_{1998}}\times 100\\ =\frac{130}{100}\times \frac{100}{110}\times 100\\ \text{ =118}\text{.18}\end{array}$