Question

The ratio of \(a\) to \(b\) is \(\frac{2}{3}\). The sum of \(a\) and \(b\) is \(10 .\) What is the ratio of \(a+b\) to \(b-a ?\)

Short answer

The ratio is 5.

Step-by-step solution

Identify Given Information

The ratio of \(a\) to \(b\) is \(\frac{2}{3}\), and the sum of \(a\) and \(b\) is 10. This translates to two equations: \( \frac{a}{b} = \frac{2}{3} \) and \( a + b = 10 \).

Express One Variable in Terms of Another

From the ratio \( \frac{a}{b} = \frac{2}{3} \), we can express \(a\) in terms of \(b\): \(a = \frac{2}{3}b\).

Substitute into the Sum Equation

Substitute \(a = \frac{2}{3}b\) into \(a + b = 10\): \( \frac{2}{3}b + b = 10 \).

Solve for \(b\)

Combine like terms: \( \frac{2}{3}b + \frac{3}{3}b = 10 \), which simplifies to \( \frac{5}{3}b = 10 \). Multiply both sides by 3/5: \( b = 6 \).

Find \(a\)

Use the relationship \(a = \frac{2}{3}b\) to find \(a\): \( a = \frac{2}{3} \times 6 = 4 \).

Compute \(a + b\) and \(b - a\)

Calculate \(a + b\): \(4 + 6 = 10\), and \(b - a\): \(6 - 4 = 2\).

Find the Ratio

The ratio of \(a + b\) to \(b - a\) is \( \frac{10}{2} = 5 \).

Key concepts

Ratios

Ratios are a way to compare two quantities by expressing one quantity as a fraction of the other. For example, the ratio \[ \frac{a}{b} = \frac{2}{3} \] means that for every 2 parts of \(a\), there are 3 parts of \(b\). Ratios help us understand the relative size of two quantities. To solve problems involving ratios, convert the ratio into an equation if it isn't already.

Variable Substitution

Variable substitution involves replacing one variable with an expression involving another variable. This is especially useful in solving systems of equations. In our example, we knew that \[ \frac{a}{b} = \frac{2}{3} \] and used this to express \(a\) in terms of \(b\): \[ a = \frac{2}{3}b \]. This substitution allowed us to solve the problem more easily by reducing the number of unknowns.

Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. They generally take the form \(ax + by = c\). In our problem, we used the sum equation \[ a + b = 10 \] and the ratio equation \[ a = \frac{2}{3}b \] to form a system of linear equations. Solving these equations involves combining, substituting, and simplifying terms to find the values of the unknown variables.

Step-by-Step Solution

Breaking down the problem into smaller, manageable steps helps ensure a correct solution. In our problem, we:

• Identified Given Information: The ratio and the sum.
• Expressed one variable in terms of another using the ratio.
• Substituted this expression into the sum equation to solve for \(b\).
• Used the value of \(b\) to find \(a\).
• Computed \(a + b\) and \(b - a\) to find the ratio.

This methodical approach makes it easier to follow and verify each step of the solution.