Problem 36 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{42}{6} $$

Step-by-Step Solution

Answer

Answer: The value of the expression \(\frac{42}{6}\) is 7.

Step 1: Identify the numbers in the fraction

In the given expression, \(\frac{42}{6}\), the numerator is 42, and the denominator is 6.

Step 2: Perform the division

Now we need to divide the numerator (42) by the denominator (6). This can be done using simple division or multiplication tables. $$ \frac{42}{6} = 42 \div 6 $$ By dividing, we find that 42 divided by 6 is equal to 7.

Step 3: Write the final answer

The value of the given expression \(\frac{42}{6}\) is 7.

Key Concepts

Fraction Division

When working with fraction division, the process is all about dividing the numerator by the denominator. The numerator is the top number, representing how many parts we have, while the denominator is the bottom number, indicating into how many parts the whole is divided.

Let's take \( \frac{42}{6} \) as an example. This fraction means we have 42 parts, and we want to know how many groups of 6 are in those 42 parts. Dividing the two numbers simplifies the fraction, effectively reducing it to its simplest form.

Here's a tip: Always check if you can simplify the fraction before you divide. Simplification might involve finding a common factor of the numerator and the denominator. However, in this case, 42 and 6 are already in their simplest form concerning each other, as 6 is a divisor of 42.

Therefore, simplifying \( \frac{42}{6} \) is achieved by performing the division, resulting in a whole number, or an integer, which makes the concept of 'fraction' vanish since the division yields no remainder.

Numerators and Denominators

Understanding numerators and denominators is critical when working with fractions. The numerator, as mentioned before, tells us how many parts we are dealing with, while the denominator tells us the size of the groups we're dividing those parts into.

In the context of our example, the numerator 42 indicates that we have 42 individual units. The denominator 6 denotes that these units are being grouped into parts of 6. Imagine you have 42 apples, and you want to pack them equally in bags, with each bag holding 6 apples. The number of bags you would end up with is the same as the value of the fraction \( \frac{42}{6} \).

This relationship is handy in many real-world situations, from dividing up slices of pizza to distributing materials evenly in a classroom. Being able to identify and differentiate the numerator from the denominator is a foundational skill in understanding how to work with any type of fractions.

Simple Division

Simple division is one of the basic arithmetic operations and is crucial when dealing with fraction division. It involves dividing one number (the dividend) by another (the divisor) to find out how many times the divisor fits into the dividend.

In our example \( \frac{42}{6} \), simple division is used to divide 42 (the dividend) by 6 (the divisor). To perform this division, you can use long division, a calculator, or mental math if the numbers are familiar to you, as they should be in this case. You simply ask yourself 'How many times does 6 go into 42?' The answer, 7, signifies that 6 can fit into 42 exactly 7 times without any remainder.

Simple division is a key ingredient in simplifying fractions because it helps to break down larger numbers into more manageable quantities. It's a useful skill that applies not just in mathematics but also in everyday life, such as when sharing equally among friends or calculating rates and proportions.