Understanding the Ratio Problem

The problem describes a scenario involving two numbers that are initially in a specific ratio. A constant number is added to each of these original numbers. This changes both the individual numbers and their ratio, as well as their sum. We need to find the value of the number that was added.

Setting up the Equations

Let the original two numbers be in the ratio 4 ∶ 5. We can represent these numbers as \(4x\) and \(5x\), where \(x\) is a common factor.

Let the number added to each of these numbers be \(k\). After adding \(k\), the new numbers become \(4x + k\) and \(5x + k\).

We are given two pieces of information about these resulting numbers:

1. The sum of the resulting numbers is 39.
2. Their ratio (in the same order) is 6 ∶ 7.

From the first piece of information, we can write the sum equation:

\((4x + k) + (5x + k) = 39\)

Simplifying this, we get:

\(9x + 2k = 39\)    (Equation 1)

From the second piece of information, we can write the ratio equation:

\(\frac{4x + k}{5x + k} = \frac{6}{7}\)

Solving for the Unknowns

Now we have a system of two linear equations with two variables, \(x\) and \(k\). We can solve this system to find the values of \(x\) and \(k\).

Let's solve the ratio equation first. Cross-multiplying gives us:

\(7(4x + k) = 6(5x + k)\)

Distributing the numbers on both sides:

\(28x + 7k = 30x + 6k\)

Now, let's rearrange the terms to find a relationship between \(x\) and \(k\). Subtract \(28x\) from both sides and subtract \(6k\) from both sides:

\(7k - 6k = 30x - 28x\)

\(k = 2x\)

This equation tells us that the number added (\(k\)) is twice the common factor (\(x\)) of the original ratio.

Now, substitute this relationship (\(k = 2x\)) into Equation 1:

\(9x + 2(2x) = 39\)

Simplify and solve for \(x\):

\(9x + 4x = 39\)

\(13x = 39\)

\(x = \frac{39}{13}\)

\(x = 3\)

We have found the value of \(x\). Now we can find the value of \(k\) using the relationship \(k = 2x\):

\(k = 2 \times 3\)

\(k = 6\)

The number added is 6.

Verifying the Solution

Let's check if adding 6 to the original numbers (based on \(x=3\)) satisfies the given conditions.

• Original numbers: \(4x = 4(3) = 12\) and \(5x = 5(3) = 15\). Their ratio is 12:15, which simplifies to 4:5. Correct.
• Number added: \(k = 6\).
• Resulting numbers: \(12 + 6 = 18\) and \(15 + 6 = 21\).
• Sum of resulting numbers: \(18 + 21 = 39\). Correct.
• Ratio of resulting numbers: 18:21. Dividing both by 3, we get 6:7. Correct.

All conditions are met, confirming that the number added is 6.

Revision Table

Additional Information: Ratio and Proportion Concepts

A ratio is a way to compare two quantities. If the ratio of two numbers is \(a:b\), it means the first number is \(\frac{a}{b}\) times the second number. We can represent the numbers as \(ak\) and \(bk\) for some constant \(k\). This allows us to work with the absolute values of the numbers.

A proportion is an equation that states that two ratios are equal, for example, \(\frac{a}{b} = \frac{c}{d}\). In this problem, the ratio of the new numbers formed a proportion with 6:7.

When a number is added to (or subtracted from) each term in a ratio, the ratio changes unless the added/subtracted number is zero. This problem demonstrates how such an addition affects both the ratio and the sum, requiring algebraic methods to find the unknown value.