Question

If a bicyclist in motion increases his speed by 30 percent and then increases this speed by 10 percent, what percent of the original speed is the total increase in speed? A \(10 \%\) B \(40 \%\) C \(43 \%\) D \(64 \%\) E \(140 \%\)

Short answer

C 43%

Step-by-step solution

Define initial speed

Let the original speed of the bicyclist be denoted as \(v\).

Calculate first speed increase

When the speed is increased by 30%, the new speed is \(v + 0.3v = 1.3v\).

Calculate second speed increase

Next, increase the new speed \(1.3v\) by 10%. The increased speed becomes \(1.3v + 0.1(1.3v) = 1.3v + 0.13v = 1.43v\).

Determine the total percentage increase

The total increased speed \(1.43v\) compared to the original speed \(v\) is \[\frac{1.43v - v}{v} = 0.43 or 43\%\].

Key concepts

percentage increase

A percentage increase is a useful mathematical tool to determine by how much a value has grown relative to its original amount. To calculate a percentage increase, follow these steps:

1. Determine the original amount. Let's call it 'A'.
2. Identify the new amount. We'll call it 'B'.
3. Use the formula \((\frac{B - A}{A}) \times 100\) to find the percentage increase.

This formula shows you how much the original value has increased in terms of percentages. For instance, in the bicyclist problem, the original speed increased from 'v' to '1.43v'. By applying the formula, you can measure the increase precisely.

compound growth

Compound growth refers to the growth of an initial amount by applying successive increases. This often involves calculating percentages more than once over time, which accumulates to a larger overall increase.

In the bicycle speed problem:

• First, the speed increased by 30%, resulting in \((1 + 0.3) \times v = 1.3v\).
• Second, this new speed was again increased by 10%, giving \((1 + 0.1) \times 1.3v = 1.43v\).

By sequentially applying these increases, we get a total compound growth that is more than just adding the two percentages (30% + 10% ≠ 40%). Instead, it is 43%, as demonstrated by our step-by-step solution.

arithmetic operations

Arithmetic operations like addition, subtraction, multiplication, and division form the basis of all percentage and growth calculations.

In our bicyclist example:

• First, we multiply the original speed 'v' by the percentage increases. This is an application of multiplication (e.g., 0.3v and 0.1(1.3v)).
• Second, we add the increase to the current speed. This is an application of addition (e.g., 1.3v + 0.13v = 1.43v).
• Finally, we subtract the original speed from the compounded speed to find the total increase, followed by converting that figure to a percentage using division and multiplication (e.g., \(\frac{1.43v - v}{v} = 0.43\) or 43%).

Understanding these core arithmetic processes is essential to solving GMAT problems accurately and efficiently.