The table below shows the rand equivalent of one British pound and one US dollar.
| COUNTRY | CURRENCY | RATE OF EXCHANGE OF THE RAND |
| --------------------------- | ----------- | ------------------------------ |
| Britain (United Kingdom) | Pound (£) | 21.41 |
| United States of America | Dollar ($) | 12.45 |
A South African nurse works in the United States of America.
4.2.1 The nurse saves the equivalent of R4 800 per month.
Calculate the amount, in US (American) dollars, that she saves per month.
4.2.2 She ordered a book from the United Kingdom (Britain) and paid 385 for it. Calculate the price of the book in pounds (£).
A sum of money doubles in 5 years when the interest is compounded. Calculate the rate of interest.
Answer (2)
Calculate the nurse's savings in US dollars: 12.45 4800 = 385.54 .
Calculate the price of the book in British Pounds: 21.41 385 = 17.98 .
Calculate the annual interest rate: r = 5 2 − 1 = 0.1487 .
State the final answers: The nurse saves 385.54 USD, the book costs 17.98 GBP, and the interest rate is 14.87% .
Explanation
Understanding the Problem We are given the exchange rates between the South African Rand (R) and the British Pound (£), and between the Rand and the US Dollar ( $). We need to calculate the nurse's savings in US dollars, the price of the book in British Pounds, and the rate of interest for a sum of money that doubles in 5 years with compound interest.
Calculating Savings in US Dollars First, let's calculate the nurse's savings in US dollars. The nurse saves R4 800 per month, and 1 US Dollar is equivalent to R12.45. To find the equivalent amount in US dollars, we divide the savings in Rand by the Rand-to-Dollar exchange rate: Savings in US dollars = Rand equivalent of one US dollar Savings in Rand = 12.45 4800 = 385.54 So, the nurse saves approximately $385.54 per month.
Calculating Price in British Pounds Next, let's calculate the price of the book in British Pounds. The book costs R385, and 1 British Pound is equivalent to R21.41. To find the equivalent price in British Pounds, we divide the price in Rand by the Rand-to-Pound exchange rate: Price in British Pounds = Rand equivalent of one British Pound Price in Rand = 21.41 385 = 17.98 So, the price of the book is approximately £17.98.
Calculating the Interest Rate Finally, let's calculate the rate of interest. We know that a sum of money doubles in 5 years with compound interest. The formula for compound interest is: A = P ( 1 + r ) n where:
A is the amount after n years
P is the principal amount
r is the annual interest rate
n is the number of years In this case, A = 2 P (since the sum doubles) and n = 5 . Substituting these values into the formula, we get: 2 P = P ( 1 + r ) 5 Divide both sides by P :
2 = ( 1 + r ) 5 Take the fifth root of both sides: 5 2 = 1 + r Solve for r :
r = 5 2 − 1 r ≈ 1.1487 − 1 = 0.1487 So, the annual interest rate is approximately 14.87%.
Final Answer The nurse saves $385.54 per month, the book costs £17.98, and the annual interest rate is 14.87%.
Examples
Understanding currency exchange rates is crucial for international travelers and businesses. For example, if you're planning a trip from South Africa to the United States, knowing the Rand to Dollar exchange rate helps you budget effectively. Similarly, businesses importing goods from the UK need to understand the Rand to Pound exchange rate to determine the cost of goods. Compound interest calculations are essential for financial planning, helping individuals understand how their investments grow over time.
The nurse saves about $385.54 per month, the book costs approximately £17.98, and the annual interest rate is about 14.87%.
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