Solving the Sum of Three Numbers with Ratios

This problem involves finding a specific number when the sum of three numbers and the ratios between pairs of these numbers are given. We are told the sum of three numbers is 126, and we have two ratios: the ratio of the first to the third number, and the ratio of the second to the third number.

Understanding the Given Information

• The sum of the three numbers \(N_1, N_2, N_3\) is 126. So, \(N_1 + N_2 + N_3 = 126\).
• The ratio of the first number (\(N_1\)) to the third number (\(N_3\)) is 3 ∶ 7. This can be written as \(N_1 : N_3 = 3 : 7\).
• The ratio of the second number (\(N_2\)) to the third number (\(N_3\)) is 4 ∶ 7. This can be written as \(N_2 : N_3 = 4 : 7\).

Representing the Numbers Using Ratios

Since both ratios involve the third number (\(N_3\)) and the ratio part for \(N_3\) is the same (7) in both cases, we can represent the three numbers using a common variable, let's say \(x\).

• From \(N_1 : N_3 = 3 : 7\), we can say \(N_1 = 3x\) and \(N_3 = 7x\) for some constant \(x\).
• From \(N_2 : N_3 = 4 : 7\), we can say \(N_2 = 4x\) and \(N_3 = 7x\) for the same constant \(x\), as the ratio part for \(N_3\) is 7.

So, the three numbers can be represented as \(3x\), \(4x\), and \(7x\).

Setting up the Equation

The sum of the three numbers is 126. Using our representations, we can write the equation:

\(N_1 + N_2 + N_3 = 126\)

\(3x + 4x + 7x = 126\)

Solving for the Variable

Combine the terms on the left side of the equation:

\((3 + 4 + 7)x = 126\)

\(14x = 126\)

Now, solve for \(x\):

\(x = \frac{126}{14}\)

\(x = 9\)

Finding the Second Number

The second number is represented by \(4x\). Substitute the value of \(x\) we found:

\(N_2 = 4x = 4 \times 9\)

\(N_2 = 36\)

Verification

Let's find all three numbers and check if their sum is 126.

• First number \(N_1 = 3x = 3 \times 9 = 27\)
• Second number \(N_2 = 4x = 4 \times 9 = 36\)
• Third number \(N_3 = 7x = 7 \times 9 = 63\)

Sum = \(N_1 + N_2 + N_3 = 27 + 36 + 63\)

Sum = \(63 + 63 = 126\)

The sum is indeed 126, which matches the information given in the problem. The second number is 36.

Revision Table: Key Concepts

Additional Information: Ratio and Proportion

Ratio and proportion is a fundamental concept in mathematics used to compare quantities. A ratio can be written as a fraction or using the colon symbol. For example, a ratio of 3 to 7 can be written as \(3/7\) or \(3:7\).

When multiple quantities are related through ratios involving a common quantity, like in this problem where both \(N_1\) and \(N_2\) are compared to \(N_3\), we can often express all quantities in terms of a single variable multiplied by their respective ratio parts. This simplifies the problem, allowing us to use a single equation based on the total sum or difference of the quantities.

If the common quantity had different ratio parts (e.g., \(N_1:N_3 = 3:7\) and \(N_2:N_3 = 5:14\)), you would first need to find a common value for the ratio part of \(N_3\) (the least common multiple of 7 and 14, which is 14) and adjust the other ratio parts accordingly before introducing the variable \(x\).