Solving Ratio and Sum Problems with Three Numbers

This problem involves finding the value of the second number when the sum of three numbers is given, along with the ratios between the first and second, and the second and third numbers.

We are given the following information:

• Sum of three numbers = 280
• Ratio between the first and second numbers = 2 : 3
• Ratio between the second and third numbers = 4 : 5

To find the individual numbers, we first need to find a combined ratio for all three numbers (First : Second : Third). We can do this by making the ratio part corresponding to the second number the same in both given ratios.

The two ratios are:

1. First : Second = 2 : 3
2. Second : Third = 4 : 5

The second number is common to both ratios. The ratio parts for the second number are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

We will now adjust both ratios so that the second number's ratio part becomes 12:

• For the ratio First : Second = 2 : 3, multiply both parts by 4 to make the second number's part 12: First : Second = $(2 \times 4) : (3 \times 4) = 8 : 12$
• For the ratio Second : Third = 4 : 5, multiply both parts by 3 to make the second number's part 12: Second : Third = $(4 \times 3) : (5 \times 3) = 12 : 15$

Now we have the ratios relative to a common value for the second number:

• First : Second = 8 : 12
• Second : Third = 12 : 15

Therefore, the combined ratio for the three numbers is First : Second : Third = 8 : 12 : 15.

Let the numbers be $8x$, $12x$, and $15x$, where $x$ is a common multiple. The sum of these three numbers is given as 280.

So, we can write the equation:

Sum of numbers = First Number + Second Number + Third Number

$280 = 8x + 12x + 15x$

Combine the terms with $x$:

$280 = (8 + 12 + 15)x$

$280 = 35x$

Now, solve for $x$ by dividing both sides by 35:

$x = \frac{280}{35}$

To simplify $\frac{280}{35}$, we can divide both numerator and denominator by 5:

$\frac{280 \div 5}{35 \div 5} = \frac{56}{7}$

Now, divide 56 by 7:

$x = 8$

We need to find the second number. From the combined ratio, the second number is $12x$.

Substitute the value of $x$ into the expression for the second number:

Second Number = $12x = 12 \times 8$

Second Number = $96$

Thus, the second number is 96.

We can verify this by finding the other two numbers:

• First Number = $8x = 8 \times 8 = 64$
• Third Number = $15x = 15 \times 8 = 120$

Let's check the sum: $64 + 96 + 120 = 280$. The sum matches the given information.

Let's check the ratios:

• First : Second = 64 : 96. Dividing both by 32, we get $64 \div 32 : 96 \div 32 = 2 : 3$. This matches the given ratio.
• Second : Third = 96 : 120. Dividing both by 24, we get $96 \div 24 : 120 \div 24 = 4 : 5$. This also matches the given ratio.

All conditions are met, confirming the calculated numbers are correct.

Revision Table: Key Concepts

Understanding ratios and how to combine them is crucial for solving this type of problem. Here are some key concepts:

• Ratio: A comparison of two quantities. It can be written as a:b or a/b.
• Combined Ratio: When you have ratios involving common elements (like the second number here), you can combine them to find the ratio of all elements together. This requires making the ratio part of the common element equal in all ratios.
• LCM (Least Common Multiple): Used to find the smallest common value for the ratio parts when combining ratios.
• Using a Common Multiple ($x$): Representing the numbers as $ax, bx, cx$ where a:b:c is the combined ratio and $x$ is a common multiple allows you to use the sum to find the value of $x$.

Additional Information: Applications of Ratios

Ratio problems are common in various fields and everyday life:

• Mixing Ingredients: Recipes often use ratios, e.g., a ratio of flour to sugar.
• Scaling Maps and Models: Maps use a scale ratio to represent real-world distances. Models are built to scale using ratios.
• Financial Analysis: Financial ratios are used to evaluate a company's performance.
• Science: Ratios are used in chemistry (e.g., mole ratios in reactions), physics (e.g., gear ratios), and other scientific disciplines.

Mastering ratio problems like this one builds a strong foundation for tackling more complex applications.