Problem 36 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{23}{5}$$

Step-by-Step Solution

Answer

The mixed number is \( 4\frac{3}{5} \).

Step 1: Divide the Numerator by the Denominator

To convert the improper fraction \( \frac{23}{5} \) into a mixed number, first divide 23 by 5. The quotient will be the whole number part of the mixed number. When 23 is divided by 5, the quotient is 4, because 5 goes into 23 four times.

Step 2: Calculate the Remainder

After finding that the quotient is 4, calculate the remainder of this division. Multiply 4 by 5 (the divisor) to get 20. Subtract this product from the original numerator (23) to find the remainder: \( 23 - 20 = 3 \).

Step 3: Write the Mixed Number

The mixed number is composed of the quotient and the remainder. The quotient (4) is the whole number, and the remainder (3) becomes the numerator of the fraction part, while the denominator remains the same (5). Thus, the mixed number is \( 4\frac{3}{5} \).

Key Concepts

Mixed Numbers

Mixed numbers are expressions that consist of a whole number and a fraction. They are useful for representing quantities greater than one that are not whole numbers. For example, if you have an improper fraction like \( \frac{23}{5} \), converting it to a mixed number gives you a more intuitive idea of the amount, which is \( 4\frac{3}{5} \) in this case. Mixed numbers make understanding portions easier and are commonly used in everyday life, such as in cooking or measuring distances.

Division of Fractions

The division of fractions is essential in many mathematical conversions, including changing improper fractions to mixed numbers. When dividing fractions, you divide the numerator by the denominator. For \( \frac{23}{5} \), you divide 23 by 5, simplifying how this fraction can be expressed. It's like splitting a total into equal parts. Division helps identify the whole number part and calculate any remainder, key steps when you're turning an improper fraction into a mixed number.

Remainders in Division

Remainders play a crucial role when converting improper fractions to mixed numbers. When you divide and can't divide exactly, what’s left over is the remainder. In our example with \( \frac{23}{5} \), dividing 23 by 5 gives a quotient of 4, with a remainder of 3. This remainder becomes the numerator in the fractional part of the mixed number. Understanding remainders makes the conversion process clearer, showing how much is left after forming the whole number part.

Numerator and Denominator

These two components are fundamental in fractions. The numerator is the top number that represents how many parts you have. The denominator, the bottom number, indicates how many parts make a whole. For \( \frac{23}{5} \), 23 is the numerator, and 5 is the denominator. When converting to a mixed number, the denominator remains the same, while the numerator is adjusted to reflect any leftover portions after full divisions. Understanding these terms is key to working with fractions efficiently.