Question

Translate to an equation and solve. \(10 \frac{1}{4} \%\) of 68 is what number?

Short answer

Answer: Approximately 6.97.

Step-by-step solution

Convert percentage to decimal

To convert a percentage into a decimal, divide the percentage by 100. In this case, we have a mixed number (10 and 1/4) as a percentage. First, let's convert the mixed number to an improper fraction: \(\frac{41}{4}\). Now, divide by 100: \(\frac{41}{4}\div100=\frac{41}{400}\).

Multiply the decimal by 68

Now, we need to find \(\frac{41}{400}\) of 68. To do this, we multiply the decimal (expressed as a fraction) by 68: $$ \frac{41}{400} \times 68 = \frac{41 \times 68}{400}. $$ After calculating the numerator (41 × 68) and simplifying the fraction, we get: $$ \frac{2788}{400} = 6.97. $$ So, \(10 \frac{1}{4}\%\) of 68 is approximately \(6.97\).

Key concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is essential for solving many algebraic problems, particularly percentage word problems. When you see a percentage, it represents a number out of 100. To convert it into a decimal, you simply divide the percentage by 100.

For instance, if you have a simple percentage like 50%, you would divide 50 by 100 to get 0.5 as a decimal. However, when dealing with more complex percentages, such as mixed numbers like in our exercise (\(10 \frac{1}{4} \text{%}\)), you first convert the mixed number to an improper fraction. You could express that mixed number as \(\frac{41}{4} \) before dividing by 100, which yields \(\frac{41}{400} \) as a decimal in fractional form.

This step is critical because it provides a precise value that you then use in subsequent calculations. It's like translating words so that an equation can understand them!

Solving Equations

When approaching word problems that require you to solve equations, it's vital to identify and translate the words into mathematical expressions. In our example, we're tasked with finding 'what number' represents \(10 \frac{1}{4} \text{%}\) of 68. This 'of' translates to multiplication in mathematics.

After converting the percentage to a decimal, you've prepared the equation for solving. The next step is the multiplication operation. By multiplying the decimal equivalent by the whole number (in this case, 68), you're effectively finding the part of the whole number that the percentage represents. Solving this gives you the final answer, which answers the question posed by the word problem.

Basic Arithmetic Operations

The heart of most mathematical problems, including percentage word problems, lies within the basic arithmetic operations: addition, subtraction, multiplication, and division. These fundamental operations allow us to simplify complex problems into manageable steps.

In the context of our exercise, after converting percentages to decimals, we use multiplication to find the amount that the percentage represents. The problem \(\frac{41}{400} \times 68\) may look daunting at first, but it's just an application of basic multiplication of fractions. Once you multiply the numerators (41 and 68), you divide by the denominator (400) to find the decimal answer. The ability to perform these operations accurately and with understanding is crucial for solving a wide array of mathematical challenges.