Problem 74 - Prealgebra

Question

Write as a percent. Write the remainder in fractional form. $$\frac{17}{50}$$

Step-by-Step Solution

Answer

\( \frac{17}{50} = 34\% \).

Step 1: Understanding the Fraction

We start with the fraction \( \frac{17}{50} \). This fraction represents 17 parts out of 50.

Step 2: Converting Fraction to Percent

To convert a fraction into a percentage, we multiply it by 100. So, we calculate \( \frac{17}{50} \times 100 = 34 \). This calculation shows that \( \frac{17}{50} \) is equivalent to 34%.

Step 3: Analyze the Remainder

Since \( \frac{17}{50} \) is a simplified fraction that converts directly to a percentage without any remainder, the remainder is 0 in this case. Hence, a fractional remainder isn't applicable here as there is no leftover part.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They are used extensively in mathematics to express numbers that are not whole numbers. A fraction consists of two elements: the numerator and the denominator. The numerator, written on top, indicates how many parts are being considered, while the denominator, on the bottom, tells us how many equal parts the whole is divided into. For example, in the fraction \( \frac{17}{50} \), 17 is the numerator, and 50 is the denominator. This means we have 17 parts out of a total 50 parts.
Understanding fractions is fundamental in many math applications, including conversion to percentages. When we talk about percentages, we are essentially talking about fractions with a denominator of 100. This is why understanding how to manipulate fractions, such as converting them into different forms, is crucial.

Simplified Fractions

Simplifying fractions means reducing them to their simplest form. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. A simplified fraction is the one which cannot be reduced any further.
In our example of \( \frac{17}{50} \), this fraction is already in its simplest form. Since 17 is a prime number (only divisible by 1 and 17) and does not have any common factors with 50 other than 1, the fraction cannot be reduced further. Keeping fractions in their simplest form is important because it makes them easier to interpret and use in further calculations.
Simplified fractions often make calculations like converting to percentages or comparing fractions much more straightforward.

Mathematical Remainder Analysis

Mathematical Remainder Analysis involves looking at what remains after a division process where one number is divided by another. In terms of fractions, the remainder can be seen as how much of the numerator is left after dividing it by the denominator. However, this concept applies more commonly in integer division, where division may not always yield a perfect integer result.
For the fraction \( \frac{17}{50} \), when converting to percent, multiplying by 100 gives us an exact whole number (34), indicating no remainder in fractional form. When the conversion results in a non-integer percentage, the leftover would typically be expressed as a fractional remainder.
In scenarios where a percentage conversion leaves a remainder, understanding this concept can help us express the exact extent of what is leftover, should any exist. Thus, even though a remainder may not always be present, knowing how to analyze it is beneficial for more complicated calculations.