Problem 62 - Elementary Algebra

Question

\(12 \div(6.2)\)

Step-by-Step Solution

Answer

The quotient is \( \frac{60}{31} \)

Step 1: Understand the Division Problem

The problem is to divide 12 by 6.2. Write down the division expression: \( \frac{12}{6.2} \)

Step 2: Convert Decimal to Fraction

To make the calculation easier, convert the decimal 6.2 to a fraction. \( 6.2 = \frac{62}{10} \)

Step 3: Rewrite Division as Multiplication

Rewrite the division of fractions as a multiplication by the reciprocal. \( \frac{12}{\frac{62}{10}} = 12 \times \frac{10}{62} \)

Step 4: Simplify the Multiplication

Multiply the numbers: \( 12 \times \frac{10}{62} = \frac{120}{62} \)

Step 5: Simplify the Fraction

Simplify \( \frac{120}{62} \) by finding the greatest common divisor (GCD) of 120 and 62, which is 2. \( \frac{120 \div 2}{62 \div 2} = \frac{60}{31} \)

Key Concepts

Division of Decimals

When you need to divide a number by a decimal, it can become tricky. But don't worry! There's a simple way to handle it. First, we turn the decimal into a fraction. For example, in the problem above, we have 6.2. This can be written as a fraction: \( \frac{62}{10} \). By doing this, we transform the division problem into something more familiar—you guessed it, fraction division! Turning decimals into fractions is very handy. Just remember, the decimal point indicates how many zeros we need in the denominator. Once the decimal is converted to a fraction, we can proceed to the next steps easily.

Multiplication by Reciprocal

Dividing fractions might seem hard, but here’s the trick: you don’t actually divide. You multiply! More precisely, you multiply by the reciprocal. What's a reciprocal, you ask? It's just flipping the fraction. So, for \( \frac{12}{\frac{62}{10}} \), you keep the 12 and switch \( \frac{62}{10} \) to \( \frac{10}{62} \). Now, the problem turns into this multiplication: \( 12 \times \frac{10}{62} \). Doesn’t that look simpler already? This technique is always your go-to method for fraction division.

Simplifying Fractions

After multiplying, you often end up with a fraction that can be simplified. It’s crucial to simplify to get the cleanest answer. For instance, \( 12 \times \frac{10}{62} = \frac{120}{62} \). The numbers 120 and 62 both have a common factor, which makes the fraction reducible. The greatest common divisor (GCD) of 120 and 62 is 2. So, dividing both the numerator and denominator by their GCD simplifies the fraction: \( \frac{120 \ \div \ 2}{62 \ \div \ 2} = \frac{60}{31} \). Simplifying fractions ensures your answer is in its simplest form, making your result precise and clear.