Problem 71 - Elementary Algebra

Question

Divide and reduce to lowest terms. $$ 212 \div 53 $$

Step-by-Step Solution

Answer

212 divided by 53 equals to 4.

Step 1: Understand the Problem

We are asked to divide 212 by 53 and simplify the result to its lowest terms, if possible. This involves performing the division and then checking if the result needs further simplification.

Step 2: Perform the Division

Calculate the division of 212 by 53. Divide 212 by 53 to get:\[ 212 \div 53 = 4 \] with a remainder. Therefore, 212 divided by 53 gives a quotient of 4 with an additional remainder.

Step 3: Determine the Remainder

Find the remainder of the division by multiplying the quotient by 53 and subtracting from 212: \[ 53 \times 4 = 212 \] Thus, there is no remainder, making 4 an exact answer without any fractional part.

Step 4: Simplify the Result

Since the division did not include any remainder, the result of dividing 212 by 53 is an exact integer and is already in its simplest terms.

Key Concepts

Simplification

Simplification in mathematics means breaking down a complex mathematical expression or problem into the simplest form possible. This is done to make problems easier to understand and solve.
In the context of division, simplification may mean reducing the quotient to the lowest and most understandable form. When you are dividing and the result is an integer with no remainder, as in the case of 212 \( \div \) 53, the quotient (which is 4) is already in its simplest form.

• Identify the problem to simplify.
• Perform the mathematical operations needed (e.g., division).
• Check if the result can be reduced or expressed in simpler terms.

Simplifying expressions and results makes them easier to manage and work with in further calculations. It means there'll be no additional fractional parts or decimals involved, simplifying further mathematical operations.

Integer

An integer is a whole number that can be either positive, negative, or zero. In the division process, when both the dividend and the divisor are integers, the quotient can be:

• An integer with no remainder.
• An integer with a remainder expressed as a fraction if not fully divisible.

In this example, 212 \( \div \) 53 results in 4, which is a positive integer. This outcome indicates that the division is complete and exact with no less-than-one fractional part left over.
Whole numbers like this are often easier to deal with in math, as they eliminate the complications associated with fractions or decimal numbers. This makes integer calculations straightforward and reliable when performing arithmetic operations like addition, subtraction, multiplication, or further division.

Remainders

Remainders in division refer to what is left over after dividing one number by another. When a number does not divide evenly into another, the amount that is left is called the remainder. In our example calculation where 212 is divided by 53, we find there is no remainder, which signifies that 53 divides 212 perfectly without leaving anything aside. If there were a remainder, the solution would have to include this as part of the quotient.

• The presence of a remainder indicates that the division is not exact.
• An absence of a remainder shows that the division has been completed precisely.

When encountering divisions that result in remainders, it's often useful to know the remainder to better understand the scaling and approximation possible in more complex problems. Remainders can be expressed as fractions or decimals, depending on the context of the problem.