We are given the ratios of how Amanda, Brandon, and Charlene shared the cost of a present, and we know the total contribution of Amanda and Charlene. Let's denote their contributions as follows:

• Let Amanda's share be $A$,
• Brandon's share be $B$,
• Charlene's share be $C$.

We are given two ratios:

1. $\frac{A}{B} = \frac{4}{5}$, which means Amanda's share is 4 parts for every 5 parts of Brandon's share.
2. $\frac{B}{C} = \frac{2}{3}$, which means Brandon's share is 2 parts for every 3 parts of Charlene's share.

Additionally, we know that Amanda and Charlene together contributed R46: $A + C = 46$

Step 1: Express shares in terms of a common variable

We first express all shares in terms of a common variable. From the ratio $\frac{A}{B} = \frac{4}{5}$, we can express $A$ and $B$ in terms of a variable $x$: $A = 4x, \quad B = 5x$

From the ratio $\frac{B}{C} = \frac{2}{3}$, we can express $B$ and $C$ in terms of another variable $y$: $B = 2y, \quad C = 3y$

Step 2: Relate the two expressions for $B$

We now have two expressions for $B$: one from the first ratio and another from the second ratio: $5x = 2y$ Solve for $y$ in terms of $x$: $y = \frac{5x}{2}$

Step 3: Substitute $y$ into the equation for $C$

Now substitute $y = \frac{5x}{2}$ into the expression for $C$: $C = 3y = 3 \times \frac{5x}{2} = \frac{15x}{2}$

Step 4: Use the total contribution of Amanda and Charlene

We know that Amanda and Charlene's combined contribution is R46: $A + C = 46$ Substitute $A = 4x$ and $C = \frac{15x}{2}$ into this equation: $4x + \frac{15x}{2} = 46$ Multiply through by 2 to eliminate the fraction: $8x + 15x = 92$ Combine like terms: $23x = 92$ Solve for $x$: $x = \frac{92}{23} = 4$

Step 5: Find the total cost of the present

Now that we have $x = 4$, we can find the shares of Amanda, Brandon, and Charlene:

• $A = 4x = 4 \times 4 = 16$,
• $B = 5x = 5 \times 4 = 20$,
• $C = \frac{15x}{2} = \frac{15 \times 4}{2} = 30$.

The total cost of the present is the sum of their contributions: $A + B + C = 16 + 20 + 30 = 66$

Thus, the total cost of the present is R66.

Would you like a more detailed explanation of any step, or do you have any questions?

Here are five related questions to consider:

1. If the total contribution was R92 instead of R46, what would be the new total cost of the present?
2. How would the problem change if Amanda and Charlene gave a total of R60 for the present?
3. If the ratio between Amanda's and Brandon's share was 5:6, how would the calculations change?
4. What if Charlene's contribution was doubled—how would that affect the total cost?
5. How would the solution differ if all three contributed equally?

Tip: Always check if the sum of individual shares adds up correctly to the total cost when dealing with ratios.