Question

John spent 40 percent of his earnings last month on rent and 30 percent less than what he spent on rent to purchase a new dishwasher. What percent of last month's earnings did John have left over? A. $$30 \%$$ B. $$32 \%$$ C. $$45 \%$$ D. $$68 \%$$ E. $$70 \%$$

Short answer

48%

Step-by-step solution

- Determine Rent Expense

John spent 40 percent of his earnings on rent. This means the amount spent on rent is 40% of his total earnings, which can be expressed as 0.40 E, where E represents his total earnings.

- Determine Amount Spent on Dishwasher

John spent 30 percent less on the dishwasher than what he spent on rent. This can be calculated as 0.30 × (0.40 E). Since 0.30 × 0.40 = 0.12, the amount spent on the dishwasher is 0.12 E.

- Calculate Total Amount Spent

Add the amounts spent on rent and the dishwasher to find the total amount spent: 0.40 E + 0.12 E = 0.52 E.

- Determine Leftover Amount

Subtract the total amount spent from the total earnings to find the leftover: 1 E - 0.52 E = 0.48 E.

- Convert Leftover Amount to Percentage

Convert the leftover amount to a percentage: 0.48 E represents 48% of the total earnings.

Key concepts

percentages

Percentages are a way to express parts of a whole. They are especially useful in problems involving division of quantities. In this exercise, we see two percentages: 40% of earnings spent on rent and 30% less than that on a dishwasher. A percentage represents a number out of 100. For example, 40% is written mathematically as 0.40. This helps in easily calculating parts of a whole. Here, use the formula: \(\text{percentage} \times \text{total amount}\). Practice makes this basic concept clear and easy to apply.

GMAT arithmetic

The GMAT tests many arithmetic concepts. One common type of problem is percentage-based, like the exercise we have. To solve the problem: get the percentage amount, account for percentage differences, and do arithmetic operations like addition or subtraction. In this example, John’s expenses show how easy-to-follow steps lead to the solution. Start by converting percentages to decimal form. Next, subtract and add these amounts as required, which leads to finding what's left. Knowing these basic operations ensures you can tackle similar problems with confidence. Breaking down steps, like converting percentages and simple multiplication, helps avoid confusion.

algebra

Algebra often simplifies complex problems. By using variables like E for earnings, you turn a real-world problem into a solvable equation. This exercise uses algebraic expressions to find how much John spent and what's left. For example: let John’s total earnings be E. He spends 40% on rent: \(0.40E\). He then spends less on a dishwasher: \(0.12E\). Adding these gives the total expense: \(0.52E\). Subtract from total earnings: \(1E - 0.52E = 0.48E\). Converting \(0.48E\) to a percentage, you get 48%. Using algebra simplifies the calculation process and ensures clarity. Algebra is powerful in showing relationships and solving problems step-by-step.