Problem 68 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. An unknown quantity is decreased by six. This result is then divided by twenty. Ten is subtracted from this result and negative two is obtained.

Step-by-Step Solution

Answer

Answer: The value of the unknown quantity x is 166.

Step 1: Write down the unknown quantity as a variable

Let's use x as the variable to represent the unknown quantity.

Step 2: Translate the first operation

"An unknown quantity is decreased by six." This can be represented as x - 6.

Step 3: Translate the second operation

"This result is then divided by twenty." Take the result of x - 6 and divide it by 20. This can be represented as (x - 6)/20.

Step 4: Translate the third operation

"Ten is subtracted from this result." This means we need to take the result of step 3, (x - 6)/20, and subtract 10 from it. So it can be represented as ((x - 6)/20) - 10.

Step 5: Translate the final operation

"Negative two is obtained." The result of step 4 is equal to -2. This can be represented as ((x - 6)/20) - 10 = -2.

Step 6: Solve the equation

Now that we have the equation, we can solve it for x. ((x - 6)/20) - 10 = -2 First, let's get rid of the fraction by multiplying both sides of the equation by 20: 20 * (((x - 6)/20) - 10) = 20 * (-2) (x - 6) - 200 = -40 Now, let's isolate x by adding 6 and 200 to both sides of the equation: x - 200 + 6 = -40 + 200 + 6 x = 166 So, the value of the unknown quantity x is 166.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols, but it does not have an equal sign like an equation would. When we translate words into algebraic expressions, it's important to recognize the key terms that indicate specific mathematical operations. For example, decreased by suggests subtraction, while divided by indicates division.

Let's look at our original example: 'An unknown quantity is decreased by six. This result is then divided by twenty.' As an algebraic expression, this is not yet an equation because we don't know what the unknown quantity is equal to. We represent the unknown with a variable, often 'x' or 'y', and in this case, the phrase translates to \(\frac{x - 6}{20}\), which succinctly captures the operations described.

Equation Solving

The process of equation solving involves finding the value of the variable that makes the equation true. It is essentially a puzzle where we apply various mathematical operations to isolate the variable. In our example, we have the equation \(\frac{x - 6}{20} - 10 = -2\). To solve for 'x', we follow a set of steps that reverse the operations applied to 'x'.

We start by eliminating the fraction, followed by adding or subtracting constants to both sides of the equation. The goal is always to get 'x' by itself. Remember, whatever you do to one side of the equation, you must also do to the other to maintain balance. Solving equations requires a systematic approach, and with practice, it becomes an intuitive process.

Variable Representation

In algebra, variable representation is the use of letters to stand in for unknown quantities, which allows us to work with quantities that might not be specified. Variables can represent numbers that are unknown, can change, or can be applied to different situations.

In the example provided, 'x' is used to represent 'an unknown quantity.' Variables give us flexibility and the ability to generalize problems, making them powerful tools in algebra. When dealing with word problems, identifying what the variable represents is crucial to translating the words into a mathematical model, which is the first step in problem solving.

Mathematical Operations

Understanding mathematical operations is fundamental in algebra. The basic operations include addition, subtraction, multiplication, and division. However, mathematical expressions often involve combining these operations and following the order of operations, known as PEMDAS or BIDMAS, to calculate the correct value.

For instance, the word 'divided' in a word problem indicates division, while 'subtracted from' indicates subtraction. Recognizing these terms allows students to construct algebraic expressions accurately, leading to the successful solving of equations. Breaking down complex operations into simpler steps is key in learning to manipulate and solve algebraic expressions and equations.