Question

Answer the questions below. Explain at each step and show your solutions by writing step by step.
QUESTION FIVE
a) The prices and quantities of three (3) products supplied to a chain store in two years are shown in the table below;

2017 2018
Product Price ( PO) Quantity (QO) Price (p1) Quantity (Q1)
A 30 4 45 3.5
B 25 6 10 5.5
C 40 7 50 8.0

Using 2017 as base year, calculate the following index numbers;
(i) Simple price index of product B using 2017 as base year.

(ii) Simple quantity index of product C using 2017 as base year.

(iii) Paasche price index using 2017 as base year.

b) In 2016 the price of a 5 litre container of cooking oil was 45% more than it was in 2013. If the price in 2016 is k120, find the price in 2013.

c) Evaluate ; Log327 + Log39 – Log33

Answer 1

Let's tackle each part of QUESTION FIVE step by step.

### Part a

We will calculate the required index numbers using the formulas for simple price index, simple quantity index, and Paasche price index.

#### (i) Simple Price Index of Product B

The formula for the Simple Price Index is:

\[
\text{Simple Price Index} = \left( \frac{P_1}{P_0} \right) \times 100
\]

Where:
- \( P_1 \) = Price in 2018
- \( P_0 \) = Price in 2017

For Product B:
- Price in 2017 (\( P_0 \)) = 25
- Price in 2018 (\( P_1 \)) = 10

Now, plug in the values:

\[
\text{Simple Price Index for Product B} = \left( \frac{10}{25} \right) \times 100 = \left( 0.4 \right) \times 100 = 40
\]

So, the Simple Price Index of Product B using 2017 as the base year is **40**.

#### (ii) Simple Quantity Index of Product C

The formula for the Simple Quantity Index is:

\[
\text{Simple Quantity Index} = \left( \frac{Q_1}{Q_0} \right) \times 100
\]

Where:
- \( Q_1 \) = Quantity in 2018
- \( Q_0 \) = Quantity in 2017

For Product C:
- Quantity in 2017 (\( Q_0 \)) = 7
- Quantity in 2018 (\( Q_1 \)) = 8

Now, plug in the values:

\[
\text{Simple Quantity Index for Product C} = \left( \frac{8}{7} \right) \times 100 \approx 114.29
\]

So, the Simple Quantity Index of Product C using 2017 as the base year is approximately **114.29**.

#### (iii) Paasche Price Index using 2017 as base year

The formula for the Paasche Price Index is:

\[
P = \frac{\sum (P_1 \times Q_1)}{\sum (P_0 \times Q_1)} \times 100
\]

Where:
- \( P_1 \) = Price in the current year (2018)
- \( Q_1 \) = Quantity in the current year (2018)
- \( P_0 \) = Price in the base year (2017)

Calculate \( \sum (P_1 \times Q_1) \) and \( \sum (P_0 \times Q_1) \).

1. **Calculating \( \sum (P_1 \times Q_1) \)**:
- For Product A: \( 45 \times 3.5 = 157.5 \)
- For Product B: \( 10 \times 5.5 = 55 \)
- For Product C: \( 50 \times 8.0 = 400 \)

Total = \( 157.5 + 55 + 400 = 612.5 \)

2. **Calculating \( \sum (P_0 \times Q_1) \)**:
- For Product A: \( 30 \times 3.5 = 105 \)
- For Product B: \( 25 \times 5.5 = 137.5 \)
- For Product C: \( 40 \times 8.0 = 320 \)

Total = \( 105 + 137.5 + 320 = 562.5 \)

Now, substitute into the Paasche formula:

\[
P = \frac{612.5}{562.5} \times 100 \approx 108.89
\]

So, the Paasche Price Index using 2017 as the base year is approximately **108.89**.

### Part b

Let the price in 2013 be \( P \).

According to the problem, the price in 2016 is 45% more than it was in 2013.

\[
\text{Price in 2016} = P + 0.45P = 1.45P
\]

Given that the price in 2016 is k120:

\[
1.45P = 120
\]

To find \( P \):

\[
P = \frac{120}{1.45} \approx 82.76
\]

So, the price in 2013 is approximately **k82.76**.

### Part c

Evaluate \( \log_3 27 + \log_3 9 - \log_3 3^3 \).

First, simplify each term:

1. \( \log_3 27 = \log_3 (3^3) = 3 \)
2. \( \log_3 9 = \log_3 (3^2) = 2 \)
3. \( \log_3 3^3 = 3 \)

Now, substitute back into the expression:

\[
3 + 2 - 3 = 2
\]

Therefore, the value is **2**.

### Final Answers
a) (i) 40, (ii) 114.29, (iii) 108.89
b) k82.76
c) 2