Sharing the Spoils: A Ratio, Investment, and Compound Interest Adventure!

Let’s tackle a problem that combines ratios, algebra, and compound interest, all essential skills for GCSE Maths. This problem involves sharing money, figuring out amounts, and then calculating investment growth.

The Problem:

Fred, John, and Kathy share some money in the ratio 3:2:7. Kathy has £600 more than John. Fred invests his share in a bank account paying 5% compound interest a year. How much will Fred’s investment be worth after 8 years? Give your answer to the nearest penny.

Unlocking the Ratios:

The first part of the problem involves understanding how ratios represent proportions of a whole.

Solving the Problem:

1. Determine the Value of One Ratio Part:

Let ‘x’ be the value of one part of the ratio. Then:

• Fred’s share = 3x
• John’s share = 2x
• Kathy’s share = 7x

We know that Kathy has £600 more than John, so:

7x = 2x + £600

Subtracting 2x from both sides:

5x = £600

Dividing by 5:

x = £120

Therefore, one part of the ratio is worth £120.

2. Calculate Fred’s Share:

Fred’s share is 3x, so:

Fred’s share = 3 * £120 = £360

3. Calculate Fred’s Investment After 8 Years:

Fred invests his £360 in a bank account paying 5% compound interest per year. The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (as a decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

In this case:

• P = £360
• r = 5% = 0.05
• n = 1 (compounded annually)
• t = 8 years

So, the formula becomes:

A = 360 (1 + 0.05)^(1*8)

A = 360 (1.05)^8

A ≈ 360 * 1.477455

A ≈ £531.8838

4. Round to the Nearest Penny:

Rounding to the nearest penny, Fred’s investment will be worth £531.88.

The Answer:

Fred’s investment will be worth £531.88 after 8 years.

Key Concepts:

• Ratios and Proportions: Using ratios to represent and calculate proportions of a whole.
• Algebra: Setting up and solving algebraic equations.
• Compound Interest: Applying the compound interest formula to calculate investment growth.

This problem demonstrates how different mathematical concepts can be combined to solve a complex real-world scenario. By mastering these skills, you’ll be well-prepared for tackling more challenging questions.