Problem 86 - Elementary Algebra

Question

The division problem \(35 \div 7\) can be interpreted as "how many \(7 \mathrm{~s}\) are there in \(35 \mathrm{~m}^{\text {" Likewise, a division }}\) problem such as \(3 \div \frac{1}{2}\) can be interpreted as, "how tion to do the following division problems. (a) \(4 \div \frac{1}{2}\) (b) \(3 \div \frac{1}{4}\) (c) \(5 \div \frac{1}{8}\) (d) \(6 \div \frac{1}{7}\) (e) \(\frac{5}{6} \div \frac{1}{6}\) (f) \(\frac{7}{8} \div \frac{1}{8}\)

Step-by-Step Solution

Answer

(a) 8, (b) 12, (c) 40, (d) 42, (e) 5, (f) 7.

Step 1: Understand Division by a Fraction

When you divide by a fraction, you can instead multiply by its reciprocal. The reciprocal of a fraction is obtained by switching the numerator and the denominator.

Step 2: Solve Part (a)

For the problem \(4 \div \frac{1}{2}\), replace division by multiplication with the reciprocal of \(\frac{1}{2}\). So it becomes \(4 \times 2\). The result is \(8\).

Step 3: Solve Part (b)

For \(3 \div \frac{1}{4}\), we rewrite this as \(3 \times 4\). This equates to \(12\).

Step 4: Solve Part (c)

The division \(5 \div \frac{1}{8}\) is equivalent to \(5 \times 8\). This results in \(40\).

Step 5: Solve Part (d)

Convert \(6 \div \frac{1}{7}\) to a multiplication: \(6 \times 7\). This gives us \(42\).

Step 6: Solve Part (e)

For \(\frac{5}{6} \div \frac{1}{6}\), multiply \(\frac{5}{6}\) by the reciprocal of \(\frac{1}{6}\), which is \(6\). This simplifies to \(5\).

Step 7: Solve Part (f)

\(\frac{7}{8} \div \frac{1}{8}\) becomes \(\frac{7}{8} \times 8\). Simplifying, the \(8\) cancels with the denominator, resulting in \(7\).

Key Concepts

Division by Fractions

Division by fractions might seem tricky at first, but there’s a simple trick to make it easier: turn it into multiplication! Whenever you see a problem asking you to divide by a fraction, you can actually solve it by multiplying. This conversion shifts the focus from division to a more familiar operation—you just have to know how to work with reciprocals.
This process eliminates confusion and makes solving these types of problems a breeze.

Multiplication by Reciprocals

When dealing with fractions, you often hear about multiplying by the reciprocal. But what exactly is a reciprocal? Simply put, the reciprocal of a fraction is what you get when you flip the numerator and the denominator. For example, the reciprocal of \( \frac{1}{2} \) is \( 2 \). So, if you have a division problem like \( 4 \div \frac{1}{2} \), you can simplify it to \( 4 \times 2 \) by using the reciprocal.
This approach emphasizes that multiplication and division are closely linked. By swapping the sign, you convert a division task into one of multiplication. Solving becomes straightforward, as handling multiplication is often more intuitive than division.

Understanding Reciprocals

Reciprocals are fundamental when working with fractions, especially in division. Understanding reciprocals boils down to grasping the concept of 'flipping' a fraction. Here's a vital thing to remember:

• To find the reciprocal of a number, swap the top (numerator) and the bottom (denominator).
  
• If you start with a whole number like 3, its reciprocal is \( \frac{1}{3} \).
• For any non-zero number, multiplying it by its reciprocal gives you 1. This is because they essentially "cancel" each other out.

Reciprocals are handy not only in division but also simplify many complex operations involving fractions.

Fraction Operations

Operating with fractions often involves addition, subtraction, multiplication, and division. Each has its own set of rules, and understanding these is crucial for successfully managing fractions. Here's a brief overview:

• **Addition & Subtraction:** Work only when denominators are the same. To add or subtract fractions with different denominators, first find a common denominator and adjust the fractions accordingly.
• **Multiplication:** Much simpler! Just multiply the numerators together and the denominators together.
• **Division:** Instead of directly dividing, multiply by the reciprocal of the divisor.

Mastering these operations is essential for handling fractions efficiently. Each rule provides a clear path for solving math problems that involve fractions.