Problem 69 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ 7 \frac{2}{5} \div \frac{1}{5} $$

Step-by-Step Solution

Answer

The result is 37.

Step 1: Convert the Mixed Number to an Improper Fraction

To divide by a fraction, first convert the mixed number to an improper fraction. In this case, 7 2/5 is the mixed number. Multiply the whole number (7) by the denominator (5) and add the numerator (2). This gives you: \[ 7 \times 5 + 2 = 35 + 2 = 37 \]So, the improper fraction is \( \frac{37}{5} \).

Step 2: Perform the Division

Dividing by a fraction is the same as multiplying by its reciprocal. In this case, divide \( \frac{37}{5} \) by \( \frac{1}{5} \), which is the same as multiplying \( \frac{37}{5} \) by \( 5 \) (the reciprocal of \( \frac{1}{5} \)).\[ \frac{37}{5} \div \frac{1}{5} = \frac{37}{5} \times 5 = \frac{37 \times 5}{5} = 37 \]The result is 37.

Key Concepts

Mixed Numbers

In mathematics, mixed numbers combine a whole number with a fraction, offering a clear representation of numbers that are not entirely whole, nor solely fractional. Mixed numbers, like 7 \(\frac{2}{5}\), are expressed as a combination of a whole number (7 in this case) and a proper fraction (\(\frac{2}{5}\)). To work with mixed numbers, it can be useful to convert them into improper fractions.

• Identify the whole number and the fraction parts of the mixed number.
• Multiply the whole number by the denominator of the fraction.
• Add the numerator of the fraction to this product.
• Express the result as an improper fraction with the original denominator.

For example, in 7 \(\frac{2}{5}\), multiplying 7 by the denominator 5 gives 35. Adding 2 (the numerator) results in 37, thus forming the improper fraction \(\frac{37}{5}\). This step simplifies arithmetic operations like division.

Reciprocal

The concept of reciprocals plays an essential role in mathematics, especially in division of fractions. The reciprocal of a number flips the numerator and the denominator of a fraction. Essentially, when you multiply a number by its reciprocal, the product is 1.

• To find the reciprocal of a fraction, swap its numerator and denominator.
• For example, the reciprocal of \(\frac{1}{5}\) is \(5\) (or \(\frac{5}{1}\)).
• Applying reciprocals simplifies division problems, transforming them into multiplication.

In the exercise, dividing by \(\frac{1}{5}\) turns into multiplying by its reciprocal, 5. Thus, the original problem \(\frac{37}{5} \div \frac{1}{5}\) becomes \(\frac{37}{5} \times 5\). Using reciprocals makes computations more straightforward, helping to avoid division which can be more complex.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator, signifying a value equal to or greater than one whole unit. These fractions are a simpler form for completing operations like addition, subtraction, multiplication, and particularly division.

• An improper fraction consolidates mixed numbers into a single fraction.
• This can simplify arithmetic, as it allows operations without needing to manage separate whole and fractional parts.
• For instance, convert 7 \(\frac{2}{5}\) into \(\frac{37}{5}\), which is easier to use in subsequent division tasks.

In the example exercise, changing the mixed number to an improper fraction such as \(\frac{37}{5}\) efficiently sets the stage for division. It allows for seamless application of reciprocals and straightforward multiplication, ensuring clarity through fewer steps and lesser complexity.