Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{100}{13}=\) ______

Short answer

The mixed number is \(7 \frac{9}{13}\).

Step-by-step solution

Divide the Numerator by the Denominator

To convert an improper fraction to a mixed number, start by dividing the numerator (100) by the denominator (13). We perform the division: \[ 100 \div 13 \approx 7.692 \]The whole number part of this division is 7, which will be the whole number in our mixed number.

Calculate the Remainder

After determining the whole number quotient, calculate the remainder of the division. Subtract \(13 \times 7 = 91\) from 100 to find the remainder: \[ 100 - 91 = 9 \]This remainder becomes the new numerator of the fractional part of the mixed number.

Write the Mixed Number

Combine the whole number part with the new fraction. The fraction consists of the remainder as the numerator and the original denominator 13 as the denominator: \[ 7 \frac{9}{13} \] Thus, the mixed number is \(7 \frac{9}{13}\).

Reduce the Fraction to Lowest Terms (If Possible)

Check if the fraction \(\frac{9}{13}\) can be reduced further. The greatest common divisor (GCD) of 9 and 13 is 1, so the fraction is already in its simplest form. Thus, the final answer remains \(7 \frac{9}{13}\).

Key concepts

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. They are often used to express improper fractions, making them easier to understand and work with. For instance, if you have an improper fraction like \(\frac{100}{13}\), it can be converted into a mixed number.

This conversion involves two parts:

• The whole number: obtained from the division of the numerator by the denominator.
• The fractional part: derived from the remainder of this division over the original denominator.

When we convert \(\frac{100}{13}\) to a mixed number, it becomes \(7 \frac{9}{13}\), where 7 is the whole number and \(\frac{9}{13}\) is the fraction.

Reducing Fractions

Reducing a fraction means to simplify it to its lowest terms. This is accomplished by dividing both the numerator and the denominator by their greatest common divisor (GCD).

This simplification helps in making calculations easier and interpretations clearer. For the fraction \(\frac{9}{13}\), the numbers 9 and 13 have a GCD of 1, which means the fraction is already as simple as it can get.

Whenever you're reducing fractions, ensure to:

• Determine the GCD of the numerator and denominator.
• Divide the numerator and denominator by their GCD.

In our case, since the GCD was 1, \(\frac{9}{13}\) remains unchanged.

Numerator and Denominator Explained

In fractions, the numerator and denominator have specific roles. The numerator is the top number, representing how many parts of a whole you have. The denominator, on the other hand, is the bottom number indicating into how many parts the whole is divided.

In the improper fraction \(\frac{100}{13}\):

• 100 is the numerator.
• 13 is the denominator.

Understanding these elements is crucial for converting fractions and dividing numbers correctly. In the mixed number \(7 \frac{9}{13}\), 9 becomes the numerator of the fractional part, and 13 remains as the denominator.

Mastering Division and Remainder

Division and remainder are fundamental concepts used in converting improper fractions to mixed numbers. When dividing, the number you're dividing is called the dividend, while the number you divide by is the divisor.

For \(\frac{100}{13}\):

• 100 is the dividend.
• 13 is the divisor.

The division doesn't result in a whole number, but in this case, it yields a quotient of 7, with a remainder of 9. This remainder is crucial as it forms the numerator of the fractional part in a mixed number.

Always remember:

• The whole number result of the division becomes the whole number part of the mixed number.
• The remainder becomes the numerator of the fractional part.

Understanding division and remainders can unlock the logic behind mixed numbers.