Problem 107 - Algebra 1

Question

You leave a 1.50 dollars tip for a lunch of \(\$ 6.00 .\) What percent tip did you leave? (Review 3.8).

Step-by-Step Solution

Answer

The tip left was 25 percent of the cost of the lunch.

Step 1: Understanding the Concept of Percent

Firstly, understanding percent is important. Percent means per hundred. It is a ratio that compares a number to 100. Therefore, finding the percent of an amount is equivalent to finding what fraction of 100 the amount is. In this case, the question is: \$1.50 is what percentage of \$6.00? This involves calculating a ratio, where \$1.50 is the part and \$6.00 is the whole.

Step 2: Setting Up the Ratio

Next, set up the ratio. According to how percent is defined, the percent tip, which will be represented as 'p', can be calculated with the formula: p = (1.50/6.00) x 100.

Step 3: Calculating the Percent

Finally, substitute the given values into the formula and calculate: p = (1.50/6.00) x 100 = 25.

Key Concepts

Ratio

Ratios are a way of comparing two quantities, usually characterized by a division relationship between them.
For example, when you want to find the tip you left for lunch, you can view it as comparing the amount of the tip to the total cost of the lunch. In our example, the tip is \(1.50 and the lunch cost is \)6.00. So, the ratio of tip to cost is written as \( \frac{1.50}{6.00} \). This ratio tells us how much the tip is compared to the entire cost, showing the relationship between a part and the whole.
Understanding ratios is important as they are the building blocks for calculating percentages in real-world problems.

Percent

A percent is a special kind of ratio; it means 'per hundred.'
This means when you have a percentage, you are talking about how many out of every hundred. When calculating percentages, you're essentially asking, "If the whole were 100, what part would this be?".
In our tip example, the calculation finds out what portion \(\frac{1.50}{6.00}\) is of a hundred.
The formula to find the percentage from a ratio is:

• Multiply the ratio by 100.
• So, \( \frac{1.50}{6.00} \times 100\) gives us 25.

Therefore, the percentage of the tip you left is 25%. If you imagine the total bill of $6.00 as comprising 100 equal parts, you left a tip that covers 25 of those parts. Understanding percent helps in making comparisons and quick calculations in everyday life, from discounts to statistical data.

Fraction

Fractions represent a part of a whole and are expressed as one number over another, like \(\frac{1.50}{6.00}\).
In this example, the fraction signifies the same relationship as the ratio: the part (tip) over the whole (cost of lunch).
Understanding fractions is crucial because they form the groundwork for understanding more complex mathematical concepts like ratios and percentages.
When converting fractions to percentages, it’s about finding out what the fraction would be if the whole is considered to be 100 parts.
For instance, the fraction \(\frac{1.50}{6.00}\) indicates the division of \(1.50 by \)6.00 to obtain a smaller scale comparison. Multiplying by 100 converts this fraction into a percent, illustrating it in a way that’s easier to compare with other percents or to visualize quickly.
Recognizing how fractions play out in real-life situations like tipping can make fractions feel like an intuitive part of everyday computation.