Question

At a given time, what was the ratio of the number of sailboats to the number of motorboats on Lake \(X ?\) (1) If the number of motorboats on Lake \(X\) had been \(25 \%\) greater, the number of sailboats on Lake \(X\) would have been \(110 \%\) of the number of motorboats on Lake \(X\) (2) The positive difference between the number of motorboats on Lake \(X\) and the number of sailboats on Lake \(X\) was 30 .

Short answer

The ratio is \(\frac{11}{8}\).

Step-by-step solution

Define Variables

Let the number of sailboats be denoted as \(S\) and the number of motorboats be denoted as \(M\).

Translate Statement 1 into an Equation

Statement 1 mentions that if the number of motorboats was 25\% greater, then the number of sailboats would be 110\% of the number of motorboats. This translates to an equation: \(S = 1.1 \times 1.25M\). Simplifying, we get: \(S = 1.375M\).

Translate Statement 2 into an Equation

Statement 2 mentions that the positive difference between the number of motorboats and sailboats is 30. This translates to the equation: \(S - M = 30\).

Substitute and Solve for M

Using the equation from Step 2 (\(S = 1.375M\)) and substituting into the equation from Step 3: \(1.375M - M = 30\). Simplifying this, we get: \(0.375M = 30\). Solving for \(M\), we find: \(M = \frac{30}{0.375} = 80\).

Solve for S

Now that \(M = 80\), use the equation \(S = 1.375M\) to find \(S\): \(S = 1.375 \times 80 = 110\).

Find the Ratio

The ratio of the number of sailboats to the number of motorboats is given by \(\frac{S}{M}\). Substitute the values \(S = 110\) and \(M = 80\): \(\frac{110}{80} = \frac{11}{8}\).

Key concepts

ratio problems

Ratio problems are common in GMAT quantitative reasoning. Ratios compare two quantities, showing their relative sizes per unit. For example, in the given exercise, the ratio of sailboats to motorboats helps to understand how many sailboats exist per motorboat. To solve ratio problems:

• Identify the quantities you are comparing.
• Express the comparison as a fraction.
• Simplify the fraction if possible.

In our exercise, the ratio of sailboats to motorboats was calculated as \(\frac{110}{80} = \frac{11}{8}\). Here, \(\frac{11}{8}\) tells us there are 11 sailboats for every 8 motorboats. Making sense of ratios can be challenging, but focusing on how the quantities relate to each other simplifies the process.

algebraic equations

Algebraic equations are crucial for solving problems requiring relationships between variables. In the exercise, we translated verbal statements into algebraic expressions. Step 1 involved defining variables. Let:

• \(S\) be the number of sailboats
• \(M\) be the number of motorboats.

Statement 1 translated to \(S = 1.375M\). Statement 2 translated to \(S - M = 30\). By substituting and solving these equations, we determined the individual values of \(S\) and \(M\). Algebra helps to:

• Formulate equations based on given conditions.
• Simplify and solve for unknowns.

This approach makes solving quantitative problems on the GMAT efficient and systematic.

percentages in GMAT

Understanding percentages is crucial for solving many GMAT questions. A percentage represents a part of a whole, expressed per 100. In our exercise, the given statements involved percentages:

• 25% greater number of motorboats.
• 110% of the number of motorboats becoming sailboats.

To convert these percentages into equations:

• 25% greater means \(1 + \frac{25}{100} = 1.25M\).
• 110% of \(M\) is \(1.1M\).

By substituting these values, we derived equations to find the actual numbers. Practicing percentage problems will help you quickly convert verbal statements into mathematical expressions, facilitating easy problem-solving.

GMAT problem-solving

Effective GMAT problem-solving involves several steps:

• Understanding the problem statement.
• Identifying known and unknown variables.
• Translating verbal statements into mathematical expressions.
• Formulating and solving equations.
• Verifying the solution.

In our exercise, understanding the ratios, percentages, and how they translate mathematically was key. Define your variables first, then methodically work through the related equations. Practice these strategies to enhance clarity and efficiency in solving GMAT quantitative reasoning problems.