Understanding the Ratio Problem

This problem involves working with ratios of two numbers. We are given an initial ratio and a new ratio after a specific value is added to both numbers. Our goal is to find the difference between the original two numbers.

Let the two numbers be represented by A and B.

The initial ratio of A to B is given as 5:8.

This means that for some common factor, let's call it \(x\), we can write the numbers as:

• A = \(5x\)
• B = \(8x\)

We are told that if 5 is added to each number, A and B, the new ratio becomes 2:3.

So, after adding 5 to both A and B, the new numbers are \(A+5\) and \(B+5\). The new ratio is \((A+5):(B+5) = 2:3\).

We can write this as an equation:

\[ \frac{A+5}{B+5} = \frac{2}{3} \]

Setting up the Equation with the Common Factor

Now, we substitute the expressions for A and B in terms of \(x\) into this new ratio equation:

\[ \frac{5x+5}{8x+5} = \frac{2}{3} \]

To solve for \(x\), we can cross-multiply:

\[ 3 \times (5x+5) = 2 \times (8x+5) \]

Solving for the Common Factor \(x\)

Let's distribute the numbers on both sides of the equation:

\[ 15x + 15 = 16x + 10 \]

Now, we need to isolate the term with \(x\). We can subtract \(15x\) from both sides:

\[ 15 = 16x - 15x + 10 \]

\[ 15 = x + 10 \]

Finally, subtract 10 from both sides to find the value of \(x\):

\[ 15 - 10 = x \]

\[ 5 = x \]

So, the common factor \(x\) is 5.

Finding the Original Numbers A and B

Now that we have the value of \(x\), we can find the original numbers A and B using our initial definitions:

• A = \(5x = 5 \times 5 = 25\)
• B = \(8x = 8 \times 5 = 40\)

The two original numbers are 25 and 40.

Calculating the Difference Between A and B

The question asks for the difference between A and B. Since B is larger than A, the difference is \(B - A\).

Difference = \(40 - 25\)

Difference = \(15\)

Let's quickly check if adding 5 to 25 and 40 gives the new ratio 2:3.

• A + 5 = 25 + 5 = 30
• B + 5 = 40 + 5 = 45

The new ratio is 30:45. Dividing both numbers by their greatest common divisor, which is 15:

\[ \frac{30}{15} = 2 \]

\[ \frac{45}{15} = 3 \]

The new ratio is 2:3, which matches the information given in the problem. This confirms our values for A and B are correct.

The difference between the two numbers A and B is 15.

Revision Table: Ratio Problem Solving

Additional Information on Ratios and Proportions

A ratio is a comparison of two quantities. It can be written as \(a:b\) or \(\frac{a}{b}\), where \(b \neq 0\). Ratios can be simplified by dividing both parts by their greatest common divisor.

A proportion is an equation stating that two ratios are equal. For example, \(\frac{a}{b} = \frac{c}{d}\) is a proportion. Proportion problems can often be solved using cross-multiplication (\(ad = bc\)).

When dealing with ratio problems where a value is added or subtracted from the original quantities, representing the original quantities using a common factor (\(ax, bx\)) is a standard algebraic approach. This allows you to set up an equation based on the new ratio and solve for the common factor.