Question

What is \(30 \%\) of \(160 ?\) 105 is what percent of \(240 ?\)

Short answer

30% of 160 is 48. 105 is 43.75% of 240.

Step-by-step solution

Find 30% of 160

To find 30% of 160, multiply 160 by 0.30 (since percent means per hundred and 30% equals 0.30 in decimal form).\n Thus, \(160 \times 0.30 = 48\)

Express 105 as a percent of 240

To express 105 as a percentage of 240, divide 105 by 240 and multiply by 100 (since percent means per hundred). Thus, \(\frac{105}{240} \times 100 = 43.75\%.\)

Key concepts

Percentage Calculation

Understanding percentage calculation is essential in various aspects of everyday life, from calculating discounts while shopping to assessing the growth or decrease in quantities. The term 'percentage' literally means 'per hundred', and it provides a way to express a number as a fraction of 100.

To calculate a percentage of a number, like in the exercise where we find 30% of 160, you take the percentage and convert it into decimal form—30% becomes 0.30. Then, you multiply this decimal by the number you're interested in. Here's the formula in a general form:
\[ \text{Percentage of a number} = (\text{Percent converted to decimal}) \times (\text{Number}) \]
That's how we obtained 48 in our solution when multiplying 0.30 by 160. This simple method allows you to calculate percentages for any number, making it a versatile tool for financial calculations, statistics, and more.

Converting Percents to Decimals

When working with percent problems, it often becomes necessary to convert percents to decimals. This skill simplifies calculations and can be applied across a wide range of problems. The conversion process is straightforward: simply divide the percentage by 100 or shift the decimal point two places to the left.

Here's the method:

• For percentages greater than 1%, such as 30%, you remove the percent sign and divide by 100, which gives you 0.30.
• For percentages less than 1%, like 0.25%, you would shift the decimal point two places to the left, giving you 0.0025.

Once in decimal form, percentages become much easier to work with, especially when performing multiplications or divisions in different numerical problems.

Expressing Numbers as a Percentage

Expressing one number as a percentage of another is a common requirement in mathematical problems, and it is also widely used in comparative analysis across different domains such as finance and economics. To convert a number into a percentage, you need to understand the relationship between the part and the whole.

You divide the 'part' (the number you want to convert) by the 'whole' (the reference number), and then multiply by 100. This yields the percentage that the 'part' is of the 'whole'.
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]
For instance, in the exercise, to find what percent 105 is of 240, divide 105 by 240 to get approximately 0.4375. Multiplying this by 100 gives you the percentage: 43.75%. This technique can be applied universally, whether it's about the percentage of a budget spent or the percentage of a test correctly answered.