Problem 14 - Calculate With Confidence

Question

½ is what percentage of 60? ___________________________

Step-by-Step Solution

Answer

0.8%

Step 1: Understand the Problem

We want to find what percentage \( \frac{1}{2} \) is of 60. This involves expressing \( \frac{1}{2} \) as a fraction of 60, then converting that fraction into a percentage.

Step 2: Express the Fraction

Express \( \frac{1}{2} \) as a fraction of 60 by dividing 1 by 2 to get 0.5.

Step 3: Calculate the Ratio

To find the effective percentage, calculate the ratio of 0.5 to 60. Divide 0.5 by 60: \( \frac{0.5}{60} = \frac{1}{120} \).

Step 4: Convert to Percentage

To convert the result from Step 3 into a percentage, multiply by 100. \( \frac{1}{120} \times 100 = \frac{100}{120} = 0.8333\ldots \).

Step 5: Final Result

Round the percentage to the nearest tenth. Hence, \( \frac{1}{120} \times 100 \approx 0.8\% \).

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. When you see a fraction like \( \frac{1}{2} \), it means "one out of two equal parts". Imagine cutting a pizza into two equally sized slices and taking one slice; that one slice is \( \frac{1}{2} \) of the pizza. Fractions are expressed with a numerator (top number) and a denominator (bottom number):

• Numerator: Represents how many parts you are considering.
• Denominator: Represents how many equal parts the whole is divided into.

Fractions are a way to show division, hence \( \frac{1}{2} \) can also be written as 1 divided by 2, which equals 0.5. This understanding is fundamental when working with percentages because percentages are essentially fractions with a denominator of 100.

Grasping the Concept of Ratio

A ratio is a way to compare quantities. When we talk about the ratio of \( \frac{1}{2} \) to 60, we are comparing one part to the whole (60 in this case). Ratios tell us how much of one thing there is compared to another.You can think of ratios similar to a fraction but used in different contexts. For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2:3. In problems involving "percent of", you typically need to transform a ratio into a percentage.In this exercise, we found the ratio by dividing the fraction by 60:\[ \text{Ratio} = \frac{0.5}{60} = \frac{1}{120} \]This ratio \( \frac{1}{120} \) suggests that for every 120 units, there is 1 unit of the fraction part. This step helps bridge the gap towards finding a percentage, where we scale our ratio appropriately.

Conversion to Percentage

Converting a number into a percentage is like converting it to an equivalent fraction out of 100. This is because percentages are essentially "per hundred". To convert a fraction or ratio into a percentage, you multiply by 100.For instance, using our calculated ratio of \( \frac{1}{120} \), you convert it to a percentage:\[ \frac{1}{120} \times 100 \approx 0.8333 \ldots \]This result can further be simplified by rounding. In practice, it is common to round percentages to the nearest tenth or whole number for simplicity. Thus, \( 0.8333 \ldots\% \) becomes \( 0.8\% \). Rounding ensures the resulting percentage is easier to read and use in everyday life.Understanding these conversions is crucial, as percentages are a universal way to convey proportions across different contexts.