Equations and Inequalities (Grade 11 NSC Matric Mathematics): Revision Notes

Worked Example 1: Gym Membership Pricing

Problem: A gym charges R1000 annually for single membership and R1500 for family membership. The gym wants to increase both membership types by the same amount. After the increase, a single membership should cost $\frac{5}{7}$ of a family membership. Find the proposed increase amount.

Solution:

Step 1: Let the increase amount be $x$ Rand.

Step 2: Create a table to organise the information:

Step 3: Set up the equation based on the given relationship:

• $1000 + x = \frac{5}{7}(1500 + x)$

Step 4: Solve for $x$:

• $7000 + 7x = 7500 + 5x$
• $2x = 500$
• $x = 250$

Step 5: The proposed increase is R250.

This problem demonstrates how to use fractions in word problems and how creating a table can help organise information clearly.

Worked Example 2: Coffee Shop Pricing

Problem: Erica pays R54.00 for four cappuccinos and three filter coffees. A cappuccino costs R3.00 more than a filter coffee. Calculate the cost of each type of coffee.

Solution:

Method 1: Two variables

Step 1: Let cappuccino cost = $x$ and filter coffee cost = $y$

Step 2: Set up the system of equations:

• $4x + 3y = 54 \text{ ... (1)}$
• $x = y + 3 \text{ ... (2)}$

Step 3: Substitute equation (2) into equation (1):

• $4(y + 3) + 3y = 54$
• $4y + 12 + 3y = 54$
• $7y = 42$
• $y = 6$

Step 4: Find $x$: $x = 6 + 3 = 9$

Method 2: One variable

Step 1: Let filter coffee cost = $x$ and cappuccino cost = $x + 3$

Step 2: Set up one equation:

• $4(x + 3) + 3x = 54$
• $7x = 42$
• $x = 6$

Final answer: A cappuccino costs R9 and filter coffee costs R6.

This example shows two different approaches to the same problem, demonstrating flexibility in problem-solving methods.

Worked Example 3: Container Filling Rates

Problem: Two taps fill a container at different rates. The less powerful tap takes 2 hours longer than the more powerful tap. Together, they fill the container in 1 hour, 52 minutes, and 30 seconds. Find how long each tap takes individually.

Solution:

Step 1: Let the more powerful tap take $x - 2$ hours and the less powerful tap take $x$ hours.

Step 2: Convert the combined time to hours:

• $1 + \frac{52}{60} + \frac{30}{3600} = 1.875 \text{ hours}$

Step 3: Set up the rate equation:

• $\frac{1}{x} + \frac{1}{x-2} = \frac{1}{1.875}$

Step 4: Multiply through by the common denominator and simplify:

• $1.875(x-2) + 1.875x = x(x-2)$
• $0 = x^2 - 5.75x + 3.75$

Step 5: Multiply by 4 to eliminate decimals:

• $0 = 4x^2 - 23x + 15$
• $0 = (4x - 3)(x - 5)$

Step 6: Solve: $x = \frac{3}{4}$ or $x = 5$

Since both taps working together take 1.875 hours, the solution $x = \frac{3}{4}$ doesn't make physical sense.

Final answer: The less powerful tap takes 5 hours and the more powerful tap takes 3 hours.

This problem illustrates how to work with rates and why checking solutions for physical reasonableness is crucial.