7. A cylindrical tank whose diameter is 1.4 metres and height 80 cm is initially empty. Water whose volume is

Question

492.8 litres is poured into the tank. Determine the fraction of the tank filled with water. (4 marks)
8. An article which is marked for sh. 450 is sold to a customer for sh. 393.75. What percentage discount is the
customer allowed? (3 marks)
9. Syengo spends one-third of his salary on food, one-quarter on rent, three-fifths of the remainder on transport
and saves the rest. If he spends sh 1 800 on transport, find how much money he saves

Answer 1

7. Fraction of the Tank Filled with Water

First, we need to calculate the volume of the cylindrical tank.

Given:

• Diameter = 1.4 m
• Radius (r) = Diameter / 2 = 1.4 m / 2 = 0.7 m
• Height (h) = 80 cm = 0.8 m

Volume of the Cylinder (V) is given by the formula: \[ V = \pi r^2 h \] Substituting the values: \[ V = \pi (0.7)^2 (0.8) \] Calculating: \[ V = \pi (0.49) (0.8) \approx \pi (0.392) \approx 1.23 \text{ m}^3 \] (Using \(\pi \approx 3.14\))

Now, convert this volume to litres (since 1 m³ = 1000 litres): \[ V \approx 1.23 \times 1000 = 1230 \text{ litres} \]

Now, the volume of water poured into the tank is 492.8 litres.

The fraction of the tank filled with water is: \[ \text{Fraction} = \frac{\text{Volume of water}}{\text{Volume of tank}} = \frac{492.8}{1230} \] Calculating the fraction: \[ \text{Fraction} \approx \frac{492.8}{1230} \approx 0.4 \]

So, the tank is approximately 0.4 filled with water.

8. Percentage Discount

To find the percentage discount, we first need to calculate the discount amount.

Given:

• Marked Price = sh. 450
• Selling Price = sh. 393.75

Discount Amount: \[ \text{Discount} = \text{Marked Price} - \text{Selling Price} = 450 - 393.75 = sh. 56.25 \]

Percentage Discount: \[ \text{Percentage Discount} = \left(\frac{\text{Discount}}{\text{Marked Price}}\right) \times 100 = \left(\frac{56.25}{450}\right) \times 100 \] Calculating the percentage: \[ \text{Percentage Discount} \approx 0.125 \times 100 = 12.5% \]

So, the percentage discount is 12.5%.

9. Syengo's Savings

Let \( S \) be the total salary of Syengo.

Distribution:

1. Spends on food: \( \frac{1}{3} S \)
2. Spends on rent: \( \frac{1}{4} S \)

Remainder after food and rent: \[ \text{Remainder} = S - \left(\frac{1}{3} S + \frac{1}{4} S\right) \] Finding a common denominator (12): \[ \frac{1}{3} S = \frac{4}{12} S, \quad \frac{1}{4} S = \frac{3}{12} S \] Thus: \[ \text{Remainder} = S - \left(\frac{4}{12} S + \frac{3}{12} S\right) = S - \frac{7}{12} S = \frac{5}{12} S \]

Spending on transport: He spends \( \frac{3}{5} \) of the remainder: \[ \text{Spent on transport} = \frac{3}{5} \times \frac{5}{12} S = \frac{3}{12} S = \frac{1}{4} S \] Given: \[ \text{Spent on transport} = 1800 \text{ sh} \] So, \[ \frac{1}{4} S = 1800 \] Thus, \[ S = 1800 \times 4 = 7200 \text{ sh} \]

Calculating savings: \[ \text{Savings} = \text{Remainder} - \text{Spent on transport} \] Finding the remaining amount: \[ \text{Remainder} = \frac{5}{12} \times 7200 = 3000 \text{ sh} \] So, \[ \text{Savings} = 3000 - 1800 = 1200 \text{ sh} \]

Therefore, Syengo saves sh. 1200.